Daily Papers

The present work investigates two properties of level crossings of a stationary Gaussian process X(t) with autocorrelation function R_X(tau). We show firstly that if R_X(tau) admits finite second and fourth derivatives at the origin, the length of up-excursions above a large negative level -gamma is asymptotically exponential as -gamma to -infty. Secondly, assuming that R_X(tau) admits a finite second derivative at the origin and some defined properties, we derive the mean number of crossings as well as the length of successive excursions above two subsequent large levels. The asymptotic results are shown to be effective even for moderate values of crossing level. An application of the developed results is proposed to derive the probability of successive excursions above adjacent levels during a time window.

Group Relative Policy Optimization (GRPO) was introduced and used recently for promoting reasoning in LLMs under verifiable (binary) rewards. We show that the mean + variance calibration of these rewards induces a weighted contrastive loss in which the contrastive samples are synthetic data drawn from the previous policy. While GRPO was originally paired with clipping to keep updates near the old policy, we analyze variants that differ in reward normalization (mean-only vs mean + variance) and in how they regularize updates using KL divergence: either penalizing divergence from the previous model (mirror), penalizing divergence from a fixed reference model π_{ref}, or combining both forms of regularization. For each, the optimal policy π_n admits an explicit form in terms of the binary reward and the first and second order statistics of the reward under π_{n-1}, as well as the policies π_{n-1} and π_{ref}. Iterating results in a sequence {π_n} whose probability of success (PoS) obeys a simple recurrence that converges to a fixed point determined by the reference PoS and the regularization strength. We further show that this fixed point exceeds the reference, demonstrating that GRPO amplifies the policy's probability of success.

Climbing grades are used to classify a climbing route based on its perceived difficulty, and have come to play a central role in the sport of rock climbing. Recently, the first statistically rigorous method for estimating climbing grades from whole-history ascent data was described, based on the dynamic Bradley-Terry model for games between players of time-varying ability. In this paper, we implement inference under the whole-history rating model using Markov chain Monte Carlo and apply the method to a curated data set made up of climbers who climb regularly. We use these data to get an estimate of the model's fundamental scale parameter m, which defines the proportional increase in difficulty associated with an increment of grade. We show that the data conform to assumptions that the climbing grade scale is a logarithmic scale of difficulty, like decibels or stellar magnitude. We estimate that an increment in Ewbank, French and UIAA climbing grade systems corresponds to 2.1, 2.09 and 2.13 times increase in difficulty respectively, assuming a logistic model of probability of success as a function of grade. Whereas we find that the Vermin scale for bouldering (V-grade scale) corresponds to a 3.17 increase in difficulty per grade increment. In addition, we highlight potential connections between the logarithmic properties of climbing grade scales and the psychophysical laws of Weber and Fechner.

In this paper, we propose feedback designs for manipulating a quantum state to a target state by performing sequential measurements. In light of Belavkin's quantum feedback control theory, for a given set of (projective or non-projective) measurements and a given time horizon, we show that finding the measurement selection policy that maximizes the probability of successful state manipulation is an optimal control problem for a controlled Markovian process. The optimal policy is Markovian and can be solved by dynamical programming. Numerical examples indicate that making use of feedback information significantly improves the success probability compared to classical scheme without taking feedback. We also consider other objective functionals including maximizing the expected fidelity to the target state as well as minimizing the expected arrival time. The connections and differences among these objectives are also discussed.

Varying dynamics parameters in simulation is a popular Domain Randomization (DR) approach for overcoming the reality gap in Reinforcement Learning (RL). Nevertheless, DR heavily hinges on the choice of the sampling distribution of the dynamics parameters, since high variability is crucial to regularize the agent's behavior but notoriously leads to overly conservative policies when randomizing excessively. In this paper, we propose a novel approach to address sim-to-real transfer, which automatically shapes dynamics distributions during training in simulation without requiring real-world data. We introduce DOmain RAndomization via Entropy MaximizatiON (DORAEMON), a constrained optimization problem that directly maximizes the entropy of the training distribution while retaining generalization capabilities. In achieving this, DORAEMON gradually increases the diversity of sampled dynamics parameters as long as the probability of success of the current policy is sufficiently high. We empirically validate the consistent benefits of DORAEMON in obtaining highly adaptive and generalizable policies, i.e. solving the task at hand across the widest range of dynamics parameters, as opposed to representative baselines from the DR literature. Notably, we also demonstrate the Sim2Real applicability of DORAEMON through its successful zero-shot transfer in a robotic manipulation setup under unknown real-world parameters.

Image steganography is an information-hiding technique that involves the surreptitious concealment of covert informational content within digital images. In this paper, we introduce {rm SCR{small EED}S{small OLO}}, a novel framework for concealing arbitrary binary data within images. Our approach synergistically leverages Random Shuffling, Fernet Symmetric Encryption, and Reed-Solomon Error Correction Codes to encode the secret payload, which is then discretely embedded into the carrier image using LSB (Least Significant Bit) Steganography. The combination of these methods addresses the vulnerability vectors of both security and resilience against bit-level corruption in the resultant stego-images. We show that our framework achieves a data payload of 3 bits per pixel for an RGB image, and mathematically assess the probability of successful transmission for the amalgamated n message bits and k error correction bits. Additionally, we find that {rm SCR{small EED}S{small OLO}} yields good results upon being evaluated with multiple performance metrics, successfully eludes detection by various passive steganalysis tools, and is immune to simple active steganalysis attacks. Our code and data are available at https://github.com/Starscream-11813/SCReedSolo-Steganography.

On-policy distillation offers dense, per-token supervision for training reasoning models; however, it remains unclear under which conditions this signal is beneficial and under which it is detrimental. Which teacher model should be used, and in the case of self-distillation, which specific context should serve as the supervisory signal? Does the optimal choice vary from one token to the next? At present, addressing these questions typically requires costly training runs whose aggregate performance metrics obscure the dynamics at the level of individual tokens. We introduce a training-free diagnostic framework that operates at the highest resolution: per token, per question, and per teacher. We derive an ideal per-node gradient defined as the parameter update that maximally increases the student's probability of success. We then develop a scalable targeted-rollout algorithm to estimate this gradient efficiently, even for long chains of intermediate thoughts. The gradient alignment score, defined as the cosine similarity between this ideal gradient and any given distillation gradient, quantifies the extent to which a particular configuration approximates the ideal signal. Across a range of self-distillation settings and external teacher models, we observe that distillation guidance exhibits substantially higher alignment with the ideal on incorrect rollouts than on correct ones, where the student already performs well and the teacher's signal tends to become noisy. Furthermore, we find that the optimal distillation context depends jointly on the student model's capacity and the target task, and that no single universally effective configuration emerges. These findings motivate the use of per-task, per-token diagnostic analyses for distillation.

Shot charts in basketball analytics provide an indispensable tool for evaluating players' shooting performance by visually representing the distribution of field goal attempts across different court locations. However, conventional methods often overlook the bounded nature of the basketball court, leading to inaccurate representations, particularly along the boundaries and corners. In this paper, we propose a novel model-based approach to shot chart estimation and visualization that explicitly considers the physical boundaries of the basketball court. By employing Gaussian mixtures for bounded data, our methodology allows to obtain more accurate estimation of shot density distributions for both made and missed shots. Bayes' rule is then applied to derive estimates for the probability of successful shooting from any given locations, and to identify the regions with the highest expected scores. To illustrate the efficacy of our proposal, we apply it to data from the 2022-23 NBA regular season, showing its usefulness through detailed analyses of shot patterns for two prominent players.

Reasoning prefixes shape the future trajectory of LLM problem solving, yet existing process reward models usually evaluate them through local step correctness. We argue that correctness is a useful but indirect proxy for the effect we ultimately care about: whether a prefix increases the probability of successful completion. We define this effect as prefix gain, the solve-rate improvement induced by conditioning lightweight student model group on a prefix, and use it to train a Prefix Utility Model (PUM) with a simple pairwise ranking objective. PUM learns outcome-grounded prefix utility and can score both complete trajectories and partial reasoning prefixes. Across Best-of-N selection, beam search, and reinforcement learning on mathematical reasoning, PUM provides a strong prefix-level supervision signal, especially when candidate pools are large, search budgets increase, or rule-based rewards are sparse. We release all data, models, and code at https://zhiqix.github.io/pum-project-page.

Quantum architecture Search (QAS) is a promising direction for optimization and automated design of quantum circuits towards quantum advantage. Recent techniques in QAS emphasize Multi-Layer Perceptron (MLP)-based deep Q-networks. However, their interpretability remains challenging due to the large number of learnable parameters and the complexities involved in selecting appropriate activation functions. In this work, to overcome these challenges, we utilize the Kolmogorov-Arnold Network (KAN) in the QAS algorithm, analyzing their efficiency in the task of quantum state preparation and quantum chemistry. In quantum state preparation, our results show that in a noiseless scenario, the probability of success is 2 to 5 times higher than MLPs. In noisy environments, KAN outperforms MLPs in fidelity when approximating these states, showcasing its robustness against noise. In tackling quantum chemistry problems, we enhance the recently proposed QAS algorithm by integrating curriculum reinforcement learning with a KAN structure. This facilitates a more efficient design of parameterized quantum circuits by reducing the number of required 2-qubit gates and circuit depth. Further investigation reveals that KAN requires a significantly smaller number of learnable parameters compared to MLPs; however, the average time of executing each episode for KAN is higher.

We study the problem of learning to perform multi-stage robotic manipulation tasks, with applications to cable routing, where the robot must route a cable through a series of clips. This setting presents challenges representative of complex multi-stage robotic manipulation scenarios: handling deformable objects, closing the loop on visual perception, and handling extended behaviors consisting of multiple steps that must be executed successfully to complete the entire task. In such settings, learning individual primitives for each stage that succeed with a high enough rate to perform a complete temporally extended task is impractical: if each stage must be completed successfully and has a non-negligible probability of failure, the likelihood of successful completion of the entire task becomes negligible. Therefore, successful controllers for such multi-stage tasks must be able to recover from failure and compensate for imperfections in low-level controllers by smartly choosing which controllers to trigger at any given time, retrying, or taking corrective action as needed. To this end, we describe an imitation learning system that uses vision-based policies trained from demonstrations at both the lower (motor control) and the upper (sequencing) level, present a system for instantiating this method to learn the cable routing task, and perform evaluations showing great performance in generalizing to very challenging clip placement variations. Supplementary videos, datasets, and code can be found at https://sites.google.com/view/cablerouting.

We introduce a unified mathematical and probabilistic framework for understanding and comparing diverse AI agent strategies. We bridge the gap between high-level agent design concepts, such as ReAct, multi-agent systems, and control flows, and a rigorous mathematical formulation. Our approach frames agentic processes as a chain of probabilities, enabling a detailed analysis of how different strategies manipulate these probabilities to achieve desired outcomes. Our framework provides a common language for discussing the trade-offs inherent in various agent architectures. One of our many key contributions is the introduction of the "Degrees of Freedom" concept, which intuitively differentiates the optimizable levers available for each approach, thereby guiding the selection of appropriate strategies for specific tasks. This work aims to enhance the clarity and precision in designing and evaluating AI agents, offering insights into maximizing the probability of successful actions within complex agentic systems.

Existing Large Language Model (LLM) routing methods score LLMs independently to select top-k models. However, this ignores model correlations and enforces a rigid computational budget. Consequently, routers often select redundant models that share failure modes, limiting the overall probability of success. To address this, we propose FlexRouter, a routing framework that explicitly models model complementarity. FlexRouter optimizes for answer coverage, maximizing the probability that at least one selected model yields a correct response. This objective aligns with practical inference pipelines where multiple candidate outputs are generated and a downstream verifier or user selects the final one. We formulate routing as a coverage-oriented subset selection problem and model the routing policy using Determinantal Point Processes (DPPs), which naturally capture both model competence and redundancy. To directly optimize coverage without requiring a ground-truth target subset, we introduce a training objective based on marginalizing over failure sets. During inference, we employ a greedy strategy based on marginal log-determinant gains, enabling the router to adaptively determine subset sizes without a predefined budget. Extensive experiments on the large-scale RouterEval benchmark demonstrate that our proposed FlexRouter achieves higher coverage with lower redundancy across both in-domain and out-of-domain tasks than strong baselines while maintaining flexible inference cost.

We study the dynamics of token launched on Pump.fun, a Solana-based launchpad platform, to identify the determinants of the token success. Pump.fun employs a bonding curve mechanism to bootstrap initial liquidity possibly leading to graduation to the on-chain market, which can be seen as a token success. We build predictive models of the probability of graduation conditional on the current amount of Solana locked in the bonding curve and a set of explanatory variables that capture structural and behavioral aspects of the launch process. Conditioning the graduation probability on these variables significantly improves its predictive power, providing insights into early-stage market behavior, speculative and manipulative dynamics, and the informational efficiency of bonding-curve-based token launches.

Multi-agent debate (MAD) is widely used to improve large language model (LLM) performance through test-time scaling, yet recent work shows that vanilla MAD often underperforms simple majority vote despite higher computational cost. Studies show that, under homogeneous agents and uniform belief updates, debate preserves expected correctness and therefore cannot reliably improve outcomes. Drawing on findings from human deliberation and collective decision-making, we identify two key mechanisms missing from vanilla MAD: (i) diversity of initial viewpoints and (ii) explicit, calibrated confidence communication. We propose two lightweight interventions. First, a diversity-aware initialisation that selects a more diverse pool of candidate answers, increasing the likelihood that a correct hypothesis is present at the start of debate. Second, a confidence-modulated debate protocol in which agents express calibrated confidence and condition their updates on others' confidence. We show theoretically that diversity-aware initialisation improves the prior probability of MAD success without changing the underlying update dynamics, while confidence-modulated updates enable debate to systematically drift to the correct hypothesis. Empirically, across six reasoning-oriented QA benchmarks, our methods consistently outperform vanilla MAD and majority vote. Our results connect human deliberation with LLM-based debate and demonstrate that simple, principled modifications can substantially enhance debate effectiveness.

Essential to visual generation is efficient modeling of visual data priors. Conventional next-token prediction methods define the process as learning the conditional probability distribution of successive tokens. Recently, next-scale prediction methods redefine the process to learn the distribution over multi-scale representations, significantly reducing generation latency. However, these methods condition each scale on all previous scales and require each token to consider all preceding tokens, exhibiting scale and spatial redundancy. To better model the distribution by mitigating redundancy, we propose Markovian Visual AutoRegressive modeling (MVAR), a novel autoregressive framework that introduces scale and spatial Markov assumptions to reduce the complexity of conditional probability modeling. Specifically, we introduce a scale-Markov trajectory that only takes as input the features of adjacent preceding scale for next-scale prediction, enabling the adoption of a parallel training strategy that significantly reduces GPU memory consumption. Furthermore, we propose spatial-Markov attention, which restricts the attention of each token to a localized neighborhood of size k at corresponding positions on adjacent scales, rather than attending to every token across these scales, for the pursuit of reduced modeling complexity. Building on these improvements, we reduce the computational complexity of attention calculation from O(N^2) to O(Nk), enabling training with just eight NVIDIA RTX 4090 GPUs and eliminating the need for KV cache during inference. Extensive experiments on ImageNet demonstrate that MVAR achieves comparable or superior performance with both small model trained from scratch and large fine-tuned models, while reducing the average GPU memory footprint by 3.0x.

We propose a novel approach to multi-fingered grasp planning leveraging learned deep neural network models. We train a convolutional neural network to predict grasp success as a function of both visual information of an object and grasp configuration. We can then formulate grasp planning as inferring the grasp configuration which maximizes the probability of grasp success. We efficiently perform this inference using a gradient-ascent optimization inside the neural network using the backpropagation algorithm. Our work is the first to directly plan high quality multifingered grasps in configuration space using a deep neural network without the need of an external planner. We validate our inference method performing both multifinger and two-finger grasps on real robots. Our experimental results show that our planning method outperforms existing planning methods for neural networks; while offering several other benefits including being data-efficient in learning and fast enough to be deployed in real robotic applications.

Recent approaches to reinforcement learning (RL) post-training for large language models increasingly remove the critic to reduce training instability and memory overhead. Even where a critic is trained, it is discarded once training ends, although it has learned to predict outcomes. We revisit this trend and show that a pretrained critic's ability to predict future outcomes can make it a valuable asset for efficient long-horizon reasoning. First, we find that instability in critic-based RL for long chain-of-thought reasoning is largely an optimization artifact: keeping policy updates small and low in variance restores stable convergence. Second, a well-pretrained critic estimates the posterior probability of eventual success from later trajectory states and unfinished prefixes. Its predictions provide outcome-derived, dense, per-prefix learning signals that, during policy optimization, require neither completed rollouts, step-level annotations, nor external reward labels. Building on this insight, we introduce Reward-Free Policy Optimization (RFPO), which repurposes a single calibrated, frozen critic as a rollout-level reward, a value baseline for generalized advantage estimation, and a success forecaster for unfinished prefixes. We further show that binarizing the debiased score stops the policy from exploiting the critic's length bias. Binarized, RFPO matches supervised PPO without a single label in the training loop, while cutting compute and memory overhead. This makes RFPO well suited to long-horizon reasoning tasks, where outcomes arrive late and generation dominates cost: because rollouts can be rewarded before they finish, training no longer has to pay for waiting on every trajectory to complete. Our findings challenge the prevailing critic-free paradigm and establish critic-based, reward-free optimization as a scalable and computationally efficient path for LLM post-training.

Process supervision, i.e., evaluating each step, is critical for complex large language model (LLM) reasoning and test-time searching with increased inference compute. Existing approaches, represented by process reward models (PRMs), primarily focus on rewarding signals up to the current step, exhibiting a one-directional nature and lacking a mechanism to model the distance to the final target. To address this problem, we draw inspiration from the A* algorithm, which states that an effective supervisory signal should simultaneously consider the incurred cost and the estimated cost for reaching the target. Building on this key insight, we introduce BiRM, a novel process supervision model that not only evaluates the correctness of previous steps but also models the probability of future success. We conduct extensive experiments on mathematical reasoning tasks and demonstrate that BiRM provides more precise evaluations of LLM reasoning steps, achieving an improvement of 3.1% on Gaokao2023 over PRM under the Best-of-N sampling method. Besides, in search-based strategies, BiRM provides more comprehensive guidance and outperforms ORM by 5.0% and PRM by 3.8% respectively on MATH-500.

It is typically understood that the training of modern neural networks is a process of fitting the probability distribution of desired output. However, recent paradoxical observations in a number of language generation tasks let one wonder if this canonical probability-based explanation can really account for the empirical success of deep learning. To resolve this issue, we propose an alternative utility-based explanation to the standard supervised learning procedure in deep learning. The basic idea is to interpret the learned neural network not as a probability model but as an ordinal utility function that encodes the preference revealed in training data. In this perspective, training of the neural network corresponds to a utility learning process. Specifically, we show that for all neural networks with softmax outputs, the SGD learning dynamic of maximum likelihood estimation (MLE) can be seen as an iteration process that optimizes the neural network toward an optimal utility function. This utility-based interpretation can explain several otherwise-paradoxical observations about the neural networks thus trained. Moreover, our utility-based theory also entails an equation that can transform the learned utility values back to a new kind of probability estimation with which probability-compatible decision rules enjoy dramatic (double-digits) performance improvements. These evidences collectively reveal a phenomenon of utility-probability duality in terms of what modern neural networks are (truly) modeling: We thought they are one thing (probabilities), until the unexplainable showed up; changing mindset and treating them as another thing (utility values) largely reconcile the theory, despite remaining subtleties regarding its original (probabilistic) identity.

The remarkable predictive performance of deep neural networks (DNNs) has led to their adoption in service domains of unprecedented scale and scope. However, the widespread adoption and growing commercialization of DNNs have underscored the importance of intellectual property (IP) protection. Devising techniques to ensure IP protection has become necessary due to the increasing trend of outsourcing the DNN computations on the untrusted accelerators in cloud-based services. The design methodologies and hyper-parameters of DNNs are crucial information, and leaking them may cause massive economic loss to the organization. Furthermore, the knowledge of DNN's architecture can increase the success probability of an adversarial attack where an adversary perturbs the inputs and alter the prediction. In this work, we devise a two-stage attack methodology "DeepPeep" which exploits the distinctive characteristics of design methodologies to reverse-engineer the architecture of building blocks in compact DNNs. We show the efficacy of "DeepPeep" on P100 and P4000 GPUs. Additionally, we propose intelligent design maneuvering strategies for thwarting IP theft through the DeepPeep attack and proposed "Secure MobileNet-V1". Interestingly, compared to vanilla MobileNet-V1, secure MobileNet-V1 provides a significant reduction in inference latency (approx60%) and improvement in predictive performance (approx2%) with very-low memory and computation overheads.

The original Grover's algorithm suffers from the souffle problem, which means that the success probability of quantum search decreases dramatically if the iteration time is too small or too large from the right time. To overcome the souffle problem, the fixed-point quantum search with an optimal number of queries was proposed [Phys. Rev. Lett. 113, 210501 (2014)], which always finds a marked state with a high probability when a lower bound of the proportion of marked states is given. The fixed-point quantum search relies on a key lemma regarding the explicit formula of recursive quasi-Chebyshev polynomials, but its proof is not given explicitly. In this work, we give a detailed proof of this lemma, thus providing a sound foundation for the correctness of the fixed-point quantum search. This lemma may be of independent interest as well, since it expands the mathematical form of the recursive relation of Chebyshev polynomials of the first kind, and it also constitutes a key component in overcoming the souffle problem of quantum walk-based search algorithms, for example, robust quantum walk search on complete bipartite graphs [Phys. Rev. A 106, 052207 (2022)]. Hopefully, more applications of the lemma will be found in the future.

The widespread use of Text-to-Image (T2I) models in content generation requires careful examination of their safety, including their robustness to adversarial attacks. Despite extensive research on adversarial attacks, the reasons for their effectiveness remain underexplored. This paper presents an empirical study on adversarial attacks against T2I models, focusing on analyzing factors associated with attack success rates (ASR). We introduce a new attack objective - entity swapping using adversarial suffixes and two gradient-based attack algorithms. Human and automatic evaluations reveal the asymmetric nature of ASRs on entity swap: for example, it is easier to replace "human" with "robot" in the prompt "a human dancing in the rain." with an adversarial suffix, but the reverse replacement is significantly harder. We further propose probing metrics to establish indicative signals from the model's beliefs to the adversarial ASR. We identify conditions that result in a success probability of 60% for adversarial attacks and others where this likelihood drops below 5%.

Reinforcement Learning with Verifiable Rewards (RLVR) has significantly advanced the reasoning capabilities of Large Language Models (LLMs). However, methods such as GRPO and DAPO suffer from substantial computational cost, since they rely on sampling many rollouts for each prompt. Moreover, in RLVR the relative advantage is often sparse: many samples become nearly all-correct or all-incorrect, yielding low within-group reward variance and thus weak learning signals. In this paper, we introduce arrol (Accelerating RLVR via online Rollout Pruning), an online rollout pruning method that prunes rollouts during generation while explicitly steering the surviving ones more correctness-balanced to enhance learning signals. Specifically, arrol trains a lightweight quality head on-the-fly to predict the success probability of partial rollouts and uses it to make early pruning decisions. The learned quality head can further weigh candidates to improve inference accuracy during test-time scaling. To improve efficiency, we present a system design that prunes rollouts inside the inference engine and re-batches the remaining ones for log-probability computation and policy updates. Across GRPO and DAPO on Qwen-3 and LLaMA-3.2 models (1B-8B), arrol improves average accuracy by +2.30 to +2.99 while achieving up to 1.7x training speedup, and yielding up to +8.33 additional gains in average accuracy in test-time scaling. The code is available at https://github.com/Hsu1023/ARRoL.

Recently, an identified non-interacting black hole (BH) binary, Gaia ID 3425577610762832384 (hereafter G3425), contains a BH (sim3.6 M_{odot}) falling within the mass gap and has a nearly circular orbit, challenging the classical binary evolution and supernova theory. Here, we propose that G3425 originates from a triple through a triple common envelope (TCE) evolution. The G3425 progenitor originally may consist of three stars with masses of 1.49 M_{odot}, 1.05 M_{odot}, and 21.81 M_{odot}, and inner and outer orbital periods of 4.22 days and 1961.78 days, respectively. As evolution proceeds, the tertiary fills its Roche lobe, leading to a TCE. We find that the orbital energy generated by the inspiral of the inner binary serves as an additional energy imparted for ejecting the common envelope (CE), accounting for sim97\% of the binding energy in our calculations. This means that the outer orbit needs to expend only a small amount of the orbital energy to successfully eject CE. The outcome of the TCE is a binary consisting of a 2.54 M_odot merger produced by the inner binary merger and a 7.67 M_odot helium star whose CE successfully ejected, with an orbital period of 547.53 days. The resulting post-TCE binary (PTB) has an orbital period that is 1-2 orders of magnitude greater than the orbital period of a successfully ejected classical binary CE. In subsequent simulations, we find that the successfully ejected helium star has a 44.2\% probability of forming a BH. In the case of a non-complete fallback forming a BH, with an ejected mass of 2.6 M_{odot} and a relatively low natal kick (11^{+16}_{-5} {rm km/s} to 49^{+39}_{-39} {rm km/s}), this PTB can form G3425 in the Milky Way.

The modeling of probability distributions, specifically generative modeling and density estimation, has become an immensely popular subject in recent years by virtue of its outstanding performance on sophisticated data such as images and texts. Nevertheless, a theoretical understanding of its success is still incomplete. One mystery is the paradox between memorization and generalization: In theory, the model is trained to be exactly the same as the empirical distribution of the finite samples, whereas in practice, the trained model can generate new samples or estimate the likelihood of unseen samples. Likewise, the overwhelming diversity of distribution learning models calls for a unified perspective on this subject. This paper provides a mathematical framework such that all the well-known models can be derived based on simple principles. To demonstrate its efficacy, we present a survey of our results on the approximation error, training error and generalization error of these models, which can all be established based on this framework. In particular, the aforementioned paradox is resolved by proving that these models enjoy implicit regularization during training, so that the generalization error at early-stopping avoids the curse of dimensionality. Furthermore, we provide some new results on landscape analysis and the mode collapse phenomenon.

Large Language Models (LLMs) generate longform text by successively sampling the next token based on the probability distribution of the token vocabulary at each decoding step. Current popular truncation sampling methods such as top-p sampling, also known as nucleus sampling, often struggle to balance coherence and creativity in generating text, particularly when using higher temperatures. To address this issue, we propose min-p, a dynamic truncation sampling method, that establishes a minimum base percentage threshold for tokens, which the scales according to the probability of the top candidate token. Through experiments on several benchmarks, such as GPQA, GSM8K and AlpacaEval Creative Writing, we demonstrate that min-p improves the coherence and quality of generated text even at high temperatures, while also facilitating more creative and diverse outputs compared to top-p and other sampling methods. As of writing, min-p has been adopted by multiple open-source LLM implementations, and have been independently assessed by members of the open-source LLM community, further validating its practical utility and potential.

Score-based generative modeling with probability flow ordinary differential equations (ODEs) has achieved remarkable success in a variety of applications. While various fast ODE-based samplers have been proposed in the literature and employed in practice, the theoretical understandings about convergence properties of the probability flow ODE are still quite limited. In this paper, we provide the first non-asymptotic convergence analysis for a general class of probability flow ODE samplers in 2-Wasserstein distance, assuming accurate score estimates. We then consider various examples and establish results on the iteration complexity of the corresponding ODE-based samplers.

On-policy distillation (OPD) has become a core technique in the post-training of large language models, yet its training dynamics remain poorly understood. This paper provides a systematic investigation of OPD dynamics and mechanisms. We first identify that two conditions govern whether OPD succeeds or fails: (i) the student and teacher should share compatible thinking patterns; and (ii) even with consistent thinking patterns and higher scores, the teacher must offer genuinely new capabilities beyond what the student has seen during training. We validate these findings through weak-to-strong reverse distillation, showing that same-family 1.5B and 7B teachers are distributionally indistinguishable from the student's perspective. Probing into the token-level mechanism, we show that successful OPD is characterized by progressive alignment on high-probability tokens at student-visited states, a small shared token set that concentrates most of the probability mass (97%-99%). We further propose two practical strategies to recover failing OPD: off-policy cold start and teacher-aligned prompt selection. Finally, we show that OPD's apparent free lunch of dense token-level reward comes at a cost, raising the question of whether OPD can scale to long-horizon distillation.

While score-based generative models (SGMs) have achieved remarkable success in enormous image generation tasks, their mathematical foundations are still limited. In this paper, we analyze the approximation and generalization of SGMs in learning a family of sub-Gaussian probability distributions. We introduce a notion of complexity for probability distributions in terms of their relative density with respect to the standard Gaussian measure. We prove that if the log-relative density can be locally approximated by a neural network whose parameters can be suitably bounded, then the distribution generated by empirical score matching approximates the target distribution in total variation with a dimension-independent rate. We illustrate our theory through examples, which include certain mixtures of Gaussians. An essential ingredient of our proof is to derive a dimension-free deep neural network approximation rate for the true score function associated with the forward process, which is interesting in its own right.

Mini-batch optimal transport (m-OT) has been successfully used in practical applications that involve probability measures with a very high number of supports. The m-OT solves several smaller optimal transport problems and then returns the average of their costs and transportation plans. Despite its scalability advantage, the m-OT does not consider the relationship between mini-batches which leads to undesirable estimation. Moreover, the m-OT does not approximate a proper metric between probability measures since the identity property is not satisfied. To address these problems, we propose a novel mini-batch scheme for optimal transport, named Batch of Mini-batches Optimal Transport (BoMb-OT), that finds the optimal coupling between mini-batches and it can be seen as an approximation to a well-defined distance on the space of probability measures. Furthermore, we show that the m-OT is a limit of the entropic regularized version of the BoMb-OT when the regularized parameter goes to infinity. Finally, we carry out experiments on various applications including deep generative models, deep domain adaptation, approximate Bayesian computation, color transfer, and gradient flow to show that the BoMb-OT can be widely applied and performs well in various applications.

Deep sequence models typically degrade in accuracy when test sequences significantly exceed their training lengths, yet many critical tasks--such as algorithmic reasoning, multi-step arithmetic, and compositional generalization--require robust length extrapolation. We introduce PRISM, a Probabilistic Relative-position Implicit Superposition Model, a novel positional encoding mechanism that enables Transformers to extrapolate accurately up to 10x beyond their training length. PRISM learns continuous relative positions through a differentiable histogram-filter update, preserving position uncertainty via a probabilistic superposition rather than conventional deterministic embeddings. Empirically, PRISM achieves state-of-the-art length extrapolation, successfully generalizing to previously intractable sequence lengths across algorithmic benchmarks--including arithmetic (addition, multiplication), SCAN compositionality tasks, and complex copy variants derived from DeepMind's recent datasets. Our analysis demonstrates that PRISM's stochastic positional encoding maintains sharp and interpretable internal states, providing a theoretical basis for reliable length generalization. These results advance the goal of neural sequence models that remain algorithmically robust at lengths far exceeding their training horizon.

Agentic theorem provers combine a reasoning model, retrieval, search, and a proof assistant verifier, yet it remains unclear which components actually improve finite-budget proof success and why they help on real mathematical workloads. We study this question through statistical provability: the probability of reaching a verified proof within a budget on a specified stream of theorem instances. We model formal proof search as a finite-horizon reachability MDP with deterministic verifier dynamics, and show that under a faithful state abstraction the optimal success probability coincides with ordinary syntactic provability. We then analyze a simple but practically important pipeline: depth-wise offline action-value regression followed by greedy test-time proving. Our main theorem bounds the provability gap between the learned prover and the optimal prover by an occupancy-weighted sum of uniform action-value errors; in the common uniform-error reading, the leading complexity multiplier is the learned prover's average truncated proof length. The error decomposes into approximation error, geometric coverage of the training distribution, and Monte Carlo label noise, and improves to a fast rate under an action-gap margin condition. The result gives a component-sensitive account of why verifier feedback, retrieval, representation geometry, and proof-shortening mechanisms help on biased theorem workloads, without contradicting classical worst-case hardness.

We present a distributed data structure, which we call the rainbow skip graph. To our knowledge, this is the first peer-to-peer data structure that simultaneously achieves high fault tolerance, constant-sized nodes, and fast update and query times for ordered data. It is a non-trivial adaptation of the SkipNet/skip-graph structures of Harvey et al. and Aspnes and Shah, so as to provide fault-tolerance as these structures do, but to do so using constant-sized nodes, as in the family tree structure of Zatloukal and Harvey. It supports successor queries on a set of n items using O(log n) messages with high probability, an improvement over the expected O(log n) messages of the family tree.

Smart contract vulnerabilities have led to billions in losses, yet finding actionable exploits remains challenging. Traditional fuzzers rely on rigid heuristics and struggle with complex attacks, while human auditors are thorough but slow and don't scale. Large Language Models offer a promising middle ground, combining human-like reasoning with machine speed. Early studies show that simply prompting LLMs generates unverified vulnerability speculations with high false positive rates. To address this, we present A1, an agentic system that transforms any LLM into an end-to-end exploit generator. A1 provides agents with six domain-specific tools for autonomous vulnerability discovery, from understanding contract behavior to testing strategies on real blockchain states. All outputs are concretely validated through execution, ensuring only profitable proof-of-concept exploits are reported. We evaluate A1 across 36 real-world vulnerable contracts on Ethereum and Binance Smart Chain. A1 achieves a 63% success rate on the VERITE benchmark. Across all successful cases, A1 extracts up to \8.59 million per exploit and 9.33 million total. Using Monte Carlo analysis of historical attacks, we demonstrate that immediate vulnerability detection yields 86-89% success probability, dropping to 6-21% with week-long delays. Our economic analysis reveals a troubling asymmetry: attackers achieve profitability at \6,000 exploit values while defenders require 60,000 -- raising fundamental questions about whether AI agents inevitably favor exploitation over defense.

Shadow tomography for quantum states provides a sample efficient approach for predicting the properties of quantum systems when the properties are restricted to expectation values of 2-outcome POVMs. However, these shadow tomography procedures yield poor bounds if there are more than 2 outcomes per measurement. In this paper, we consider a general problem of learning properties from unknown quantum states: given an unknown d-dimensional quantum state rho and M unknown quantum measurements M_1,...,M_M with Kgeq 2 outcomes, estimating the probability distribution for applying M_i on rho to within total variation distance epsilon. Compared to the special case when K=2, we need to learn unknown distributions instead of values. We develop an online shadow tomography procedure that solves this problem with high success probability requiring O(Klog^2Mlog d/epsilon^4) copies of rho. We further prove an information-theoretic lower bound that at least Omega(min{d^2,K+log M}/epsilon^2) copies of rho are required to solve this problem with high success probability. Our shadow tomography procedure requires sample complexity with only logarithmic dependence on M and d and is sample-optimal for the dependence on K.

In the pharmaceutical industry, the use of artificial intelligence (AI) has seen consistent growth over the past decade. This rise is attributed to major advancements in statistical machine learning methodologies, computational capabilities and the increased availability of large datasets. AI techniques are applied throughout different stages of drug development, ranging from drug discovery to post-marketing benefit-risk assessment. Kolluri et al. provided a review of several case studies that span these stages, featuring key applications such as protein structure prediction, success probability estimation, subgroup identification, and AI-assisted clinical trial monitoring. From a regulatory standpoint, there was a notable uptick in submissions incorporating AI components in 2021. The most prevalent therapeutic areas leveraging AI were oncology (27%), psychiatry (15%), gastroenterology (12%), and neurology (11%). The paradigm of personalized or precision medicine has gained significant traction in recent research, partly due to advancements in AI techniques hamburg2010path. This shift has had a transformative impact on the pharmaceutical industry. Departing from the traditional "one-size-fits-all" model, personalized medicine incorporates various individual factors, such as environmental conditions, lifestyle choices, and health histories, to formulate customized treatment plans. By utilizing sophisticated machine learning algorithms, clinicians and researchers are better equipped to make informed decisions in areas such as disease prevention, diagnosis, and treatment selection, thereby optimizing health outcomes for each individual.

Evaluating defensive performance in soccer remains challenging, as effective defending is often expressed not through visible on-ball actions such as interceptions and tackles, but through preventing dangerous opportunities before they arise. Existing approaches have largely focused on valuing on-ball actions, leaving much of defenders' true impact unmeasured. To address this gap, we propose DEFCON (DEFensive CONtribution evaluator), a comprehensive framework that quantifies player-level defensive contributions for every attacking situation in soccer. Leveraging Graph Attention Networks, DEFCON estimates the success probability and expected value of each attacking option, along with each defender's responsibility for stopping it. These components yield an Expected Possession Value (EPV) for the attacking team before and after each action, and DEFCON assigns positive or negative credits to defenders according to whether they reduced or increased the opponent's EPV. Trained on 2023-24 and evaluated on 2024-25 Eredivisie event and tracking data, DEFCON's aggregated player credits exhibit strong positive correlations with market valuations. Finally, we showcase several practical applications, including in-game timelines of defensive contributions, spatial analyses across pitch zones, and pairwise summaries of attacker-defender interactions.

Input Diversity (DI), which applies random resizing and padding at each attack iteration, is a near-default ingredient of transfer-based adversarial attacks, widely assumed to improve transferability. We show this assumption is regime-dependent and, for robustly trained surrogates, often reversed. Varying only the surrogate, increasing the DI probability raises transfer success for standard surrogates but lowers it for robust ones: the two response curves separate like a pair of scissors, a pattern we call the Scissors Effect. The effect is strong and consistent on ImageNet, where blind DI costs the robust source 10.3% attack success on average across CNN, ViT, Swin, and ConvNeXt targets and across ten attacks spanning 2018-2024; it is smaller on CIFAR-10 unless DI is made aggressive. A controlled robustness-strength sweep that varies only the training budget shows the harm is graded rather than binary, crossing from beneficial to harmful already in the little-robustness regime. We trace it to gradient geometry: a resize/translation decomposition attributes roughly 67% of the harm to resize, and a direct source-target gradient-alignment measurement confirms the same resize operation improves alignment for standard surrogates but degrades it for robust ones. We summarize the regime with Local Gradient Consistency (LGC), a single input-space probe that separates the two surrogate types, and prove a bias-variance crossover theorem isolating where DI helps from where its resize bias dominates. A training-free rule (CG-DI) that disables diversity when LGC is high avoids the loss on robust surrogates while keeping DI's benefit on standard ones, positioning the Scissors Effect as a DI-specific manifestation of the broader robustness-transferability trade-off.

Pre-trained Large Language Models (LLMs) are an integral part of modern AI that have led to breakthrough performances in complex AI tasks. Major AI companies with expensive infrastructures are able to develop and train these large models with billions and millions of parameters from scratch. Third parties, researchers, and practitioners are increasingly adopting these pre-trained models and fine-tuning them on their private data to accomplish their downstream AI tasks. However, it has been shown that an adversary can extract/reconstruct the exact training samples from these LLMs, which can lead to revealing personally identifiable information. The issue has raised deep concerns about the privacy of LLMs. Differential privacy (DP) provides a rigorous framework that allows adding noise in the process of training or fine-tuning LLMs such that extracting the training data becomes infeasible (i.e., with a cryptographically small success probability). While the theoretical privacy guarantees offered in most extant studies assume learning models from scratch through many training iterations in an asymptotic setting, this assumption does not hold in fine-tuning scenarios in which the number of training iterations is significantly smaller. To address the gap, we present \ewtune, a DP framework for fine-tuning LLMs based on Edgeworth accountant with finite-sample privacy guarantees. Our results across four well-established natural language understanding (NLU) tasks show that while \ewtune~adds privacy guarantees to LLM fine-tuning process, it directly contributes to decreasing the induced noise to up to 5.6\% and improves the state-of-the-art LLMs performance by up to 1.1\% across all NLU tasks. We have open-sourced our implementations for wide adoption and public testing purposes.

Process Reward Models (PRMs) provide step-level feedback for reasoning, but current PRMs usually output only a single reward score for each step. Downstream methods must therefore treat imperfect step-level reward predictions as reliable decision signals, with no indication of when these predictions should be trusted. We propose BetaPRM, a distributional PRM that predicts both a step-level success probability and the reliability of that prediction. Given step-success supervision from Monte Carlo continuations, BetaPRM learns a Beta belief that explains the observed number of successful continuations through a Beta-Binomial likelihood, rather than regressing to the finite-sample success ratio as a point target. This learned reliability signal indicates when a step reward should be trusted, enabling downstream applications to distinguish reliable rewards from uncertain ones. As one application, we introduce Adaptive Computation Allocation (ACA) for PRM-guided Best-of-N reasoning. ACA uses the learned reliability signal to stop when a high-reward solution is reliable and to spend additional computation on uncertain candidate prefixes. Experiments across four backbones and four reasoning benchmarks show that BetaPRM improves PRM-guided Best-of-N selection while preserving standard step-level error detection. Built on this signal, ACA improves the accuracy--token tradeoff over fixed-budget Best-of-16, reducing token usage by up to 33.57% while improving final-answer accuracy.

Knowledge tracing (KT) models, e.g., the deep knowledge tracing (DKT) model, track an individual learner's acquisition of skills over time by examining the learner's performance on questions related to those skills. A practical limitation in most existing KT models is that all questions nested under a particular skill are treated as equivalent observations of a learner's ability, which is an inaccurate assumption in real-world educational scenarios. To overcome this limitation we introduce qDKT, a variant of DKT that models every learner's success probability on individual questions over time. First, qDKT incorporates graph Laplacian regularization to smooth predictions under each skill, which is particularly useful when the number of questions in the dataset is big. Second, qDKT uses an initialization scheme inspired by the fastText algorithm, which has found success in a variety of language modeling tasks. Our experiments on several real-world datasets show that qDKT achieves state-of-art performance on predicting learner outcomes. Because of this, qDKT can serve as a simple, yet tough-to-beat, baseline for new question-centric KT models.

Pass@k is widely used to report the reasoning performance of LLMs, but it often produces unstable and potentially misleading rankings, especially when the number of trials (samples) is limited and computational resources are constrained. We present a principled Bayesian evaluation framework that replaces Pass@k and average accuracy over N trials (avg@N) with posterior estimates of a model's underlying success probability and credible intervals, yielding stable rankings and a transparent decision rule for differences. Evaluation outcomes are modeled as categorical (not just 0/1) with a Dirichlet prior, giving closed-form expressions for the posterior mean and uncertainty of any weighted rubric and enabling the use of prior evidence when appropriate. Theoretically, under a uniform prior, the Bayesian posterior mean is order-equivalent to average accuracy (Pass@1), explaining its empirical robustness while adding principled uncertainty. Empirically, in simulations with known ground-truth success rates and on AIME'24/'25, HMMT'25, and BrUMO'25, the posterior-based procedure achieves faster convergence and greater rank stability than Pass@k and recent variants, enabling reliable comparisons at far smaller sample counts. The framework clarifies when observed gaps are statistically meaningful (non-overlapping credible intervals) versus noise, and it naturally extends to graded, rubric-based evaluations. Together, these results recommend replacing Pass@k for LLM evaluation and ranking with a posterior-based, compute-efficient protocol that unifies binary and non-binary evaluation while making uncertainty explicit. Source code is available at https://github.com/mohsenhariri/scorio

The generations of large language models are commonly controlled through prompting techniques, where a user's query to the model is prefixed with a prompt that aims to guide the model's behaviour on the query. The prompts used by companies to guide their models are often treated as secrets, to be hidden from the user making the query. They have even been treated as commodities to be bought and sold. However, there has been anecdotal evidence showing that the prompts can be extracted by a user even when they are kept secret. In this paper, we present a framework for systematically measuring the success of prompt extraction attacks. In experiments with multiple sources of prompts and multiple underlying language models, we find that simple text-based attacks can in fact reveal prompts with high probability.

The ability of an architecture to realize permutations is quite fundamental. For example, Large Language Models need to be able to correctly copy (and perhaps rearrange) parts of the input prompt into the output. Classical universal approximation theorems guarantee the existence of parameter configurations that solve this task but offer no insights into whether gradient-based algorithms can find them. In this paper, we address this gap by focusing on two-layer fully connected feed-forward neural networks and the task of learning permutations on nonzero binary inputs. We show that in the infinite width Neural Tangent Kernel (NTK) regime, an ensemble of such networks independently trained with gradient descent on only the k standard basis vectors out of 2^k - 1 possible inputs successfully learns any fixed permutation of length k with arbitrarily high probability. By analyzing the exact training dynamics, we prove that the network's output converges to a Gaussian process whose mean captures the ground truth permutation via sign-based features. We then demonstrate how averaging these runs (an "ensemble" method) and applying a simple rounding step yields an arbitrarily accurate prediction on any possible input unseen during training. Notably, the number of models needed to achieve exact learning with high probability (which we refer to as ensemble complexity) exhibits a linearithmic dependence on the input size k for a single test input and a quadratic dependence when considering all test inputs simultaneously.

Predicting properties of complex, large-scale quantum systems is essential for developing quantum technologies. We present an efficient method for constructing an approximate classical description of a quantum state using very few measurements of the state. This description, called a classical shadow, can be used to predict many different properties: order log M measurements suffice to accurately predict M different functions of the state with high success probability. The number of measurements is independent of the system size, and saturates information-theoretic lower bounds. Moreover, target properties to predict can be selected after the measurements are completed. We support our theoretical findings with extensive numerical experiments. We apply classical shadows to predict quantum fidelities, entanglement entropies, two-point correlation functions, expectation values of local observables, and the energy variance of many-body local Hamiltonians. The numerical results highlight the advantages of classical shadows relative to previously known methods.

Neural networks provide state-of-the-art results for most machine learning tasks. Unfortunately, neural networks are vulnerable to adversarial examples: given an input x and any target classification t, it is possible to find a new input x' that is similar to x but classified as t. This makes it difficult to apply neural networks in security-critical areas. Defensive distillation is a recently proposed approach that can take an arbitrary neural network, and increase its robustness, reducing the success rate of current attacks' ability to find adversarial examples from 95% to 0.5%. In this paper, we demonstrate that defensive distillation does not significantly increase the robustness of neural networks by introducing three new attack algorithms that are successful on both distilled and undistilled neural networks with 100% probability. Our attacks are tailored to three distance metrics used previously in the literature, and when compared to previous adversarial example generation algorithms, our attacks are often much more effective (and never worse). Furthermore, we propose using high-confidence adversarial examples in a simple transferability test we show can also be used to break defensive distillation. We hope our attacks will be used as a benchmark in future defense attempts to create neural networks that resist adversarial examples.

While Large Vision-Language Models (LVLMs) have significantly advanced GUI agents' capabilities in parsing textual instructions, interpreting screen content, and executing tasks, a critical challenge persists: the irreversibility of agent operations-where a single erroneous action can trigger catastrophic deviations. To address this, we propose the GUI Action Critic's Data Flywheel System (GAIA), a training framework that enables the models to have iterative critic capabilities, which are used to improve the Test-Time Scaling (TTS) of basic GUI agents' performance. Specifically, we train an Intuitive Critic Model (ICM) using positive and negative action examples from a base agent first. This critic evaluates the immediate correctness of the agent's intended actions, thereby selecting operations with higher success probability. Then, the initial critic guides agent actions to collect refined positive/negative samples, initiating the self-improving cycle. The augmented data then trains a second-round critic with enhanced discernment capability. We conduct experiments on various datasets and demonstrate that the proposed ICM can improve the test-time performance of various closed-source and open-source models, and the performance can be gradually improved as the data is recycled. The code, dataset, and accompanying datasheet will be publicly released at https://github.com/SeerRay-Lab/GAIA.

Reliable probability estimation is of crucial importance in many real-world applications where there is inherent (aleatoric) uncertainty. Probability-estimation models are trained on observed outcomes (e.g. whether it has rained or not, or whether a patient has died or not), because the ground-truth probabilities of the events of interest are typically unknown. The problem is therefore analogous to binary classification, with the difference that the objective is to estimate probabilities rather than predicting the specific outcome. This work investigates probability estimation from high-dimensional data using deep neural networks. There exist several methods to improve the probabilities generated by these models but they mostly focus on model (epistemic) uncertainty. For problems with inherent uncertainty, it is challenging to evaluate performance without access to ground-truth probabilities. To address this, we build a synthetic dataset to study and compare different computable metrics. We evaluate existing methods on the synthetic data as well as on three real-world probability estimation tasks, all of which involve inherent uncertainty: precipitation forecasting from radar images, predicting cancer patient survival from histopathology images, and predicting car crashes from dashcam videos. We also give a theoretical analysis of a model for high-dimensional probability estimation which reproduces several of the phenomena evinced in our experiments. Finally, we propose a new method for probability estimation using neural networks, which modifies the training process to promote output probabilities that are consistent with empirical probabilities computed from the data. The method outperforms existing approaches on most metrics on the simulated as well as real-world data.

Words of estimative probability (WEP) are expressions of a statement's plausibility (probably, maybe, likely, doubt, likely, unlikely, impossible...). Multiple surveys demonstrate the agreement of human evaluators when assigning numerical probability levels to WEP. For example, highly likely corresponds to a median chance of 0.90+-0.08 in Fagen-Ulmschneider (2015)'s survey. In this work, we measure the ability of neural language processing models to capture the consensual probability level associated to each WEP. Firstly, we use the UNLI dataset (Chen et al., 2020) which associates premises and hypotheses with their perceived joint probability p, to construct prompts, e.g. "[PREMISE]. [WEP], [HYPOTHESIS]." and assess whether language models can predict whether the WEP consensual probability level is close to p. Secondly, we construct a dataset of WEP-based probabilistic reasoning, to test whether language models can reason with WEP compositions. When prompted "[EVENTA] is likely. [EVENTB] is impossible.", a causal language model should not express that [EVENTA&B] is likely. We show that both tasks are unsolved by off-the-shelf English language models, but that fine-tuning leads to transferable improvement.

Probability estimation models play an important role in various fields, such as weather forecasting, recommendation systems, and sports analysis. Among several models estimating probabilities, it is difficult to evaluate which model gives reliable probabilities since the ground-truth probabilities are not available. The win probability estimation model for esports, which calculates the win probability under a certain game state, is also one of the fields being actively studied in probability estimation. However, most of the previous works evaluated their models using accuracy, a metric that only can measure the performance of discrimination. In this work, we firstly investigate the Brier score and the Expected Calibration Error (ECE) as a replacement of accuracy used as a performance evaluation metric for win probability estimation models in esports field. Based on the analysis, we propose a novel metric called Balance score which is a simple yet effective metric in terms of six good properties that probability estimation metric should have. Under the general condition, we also found that the Balance score can be an effective approximation of the true expected calibration error which has been imperfectly approximated by ECE using the binning technique. Extensive evaluations using simulation studies and real game snapshot data demonstrate the promising potential to adopt the proposed metric not only for the win probability estimation model for esports but also for evaluating general probability estimation models.

The robot learning community has made great strides in recent years, proposing new architectures and showcasing impressive new capabilities; however, the dominant metric used in the literature, especially for physical experiments, is "success rate", i.e. the percentage of runs that were successful. Furthermore, it is common for papers to report this number with little to no information regarding the number of runs, the initial conditions, and the success criteria, little to no narrative description of the behaviors and failures observed, and little to no statistical analysis of the findings. In this paper we argue that to move the field forward, researchers should provide a nuanced evaluation of their methods, especially when evaluating and comparing learned policies on physical robots. To do so, we propose best practices for future evaluations: explicitly reporting the experimental conditions, evaluating several metrics designed to complement success rate, conducting statistical analysis, and adding a qualitative description of failures modes. We illustrate these through an evaluation on physical robots of several learned policies for manipulation tasks.

In economy, markets are denoted as efficient when it is impossible to systematically generate profits which outperform the average. In the past years, the concept has been tested in other domains such as the growing sports betting market. Surprisingly, despite its large size and its level of maturity, sports betting shows traits of inefficiency. The anomalies indicate the existence of strategies which shift betting from a game of chance towards a game of skill. This article shows an example for an inefficiency detected in the German soccer betting TOTO 13er Wette, which is operated by state-run lottery agencies. Gamblers have to guess the outcome (win, draw, loss) of 13 soccer matches listed on a lottery tip. Applying stochastic methods, a recipe is presented to determine hit rates for single match outcomes. More important, the recipe provides the number of lottery tips required to achieve a specific number of strikes (number of correct match forecasts per lottery tip) for any given level of safety. An approximation is derived to cope with large numbers in hypergeometric distributions, valid under certain constraints. Overall, the strategy does lead to returns exceeding the aggregated lottery fees, resulting in moderate, but consistent profits. It is briefly discussed if lessions learned from soccer betting can be transferred back to financial markets, because gamblers and retail investors face similar challenges and opportunities.

We propose a general two-stage algorithm that enjoys a provable scaling law for the test-time compute of large language models (LLMs). Given an input problem, the proposed algorithm first generates N candidate solutions, and then chooses the best one via a multiple-round knockout tournament where each pair of candidates are compared for K times and only the winners move on to the next round. In a minimalistic implementation, both stages can be executed with a black-box LLM alone and nothing else (e.g., no external verifier or reward model), and a total of N times (K + 1) highly parallelizable LLM calls are needed for solving an input problem. Assuming that a generated candidate solution is correct with probability p_{gen} > 0 and a comparison between a pair of correct and incorrect solutions identifies the right winner with probability p_{comp} > 0.5 (i.e., better than a random guess), we prove theoretically that the failure probability of the proposed algorithm decays to zero exponentially with respect to N and K: $P(final output is incorrect) le (1 - p_{gen})^N + lceil log_2 N rceil e^{-2 K (p_{comp} - 0.5)^2}.$ Our empirical results with the challenging MMLU-Pro benchmark validate the technical assumptions, as well as the efficacy of the proposed algorithm and the gains from scaling up its test-time compute.

A probability forecast or probabilistic classifier is reliable or calibrated if the predicted probabilities are matched by ex post observed frequencies, as examined visually in reliability diagrams. The classical binning and counting approach to plotting reliability diagrams has been hampered by a lack of stability under unavoidable, ad hoc implementation decisions. Here we introduce the CORP approach, which generates provably statistically Consistent, Optimally binned, and Reproducible reliability diagrams in an automated way. CORP is based on non-parametric isotonic regression and implemented via the Pool-adjacent-violators (PAV) algorithm - essentially, the CORP reliability diagram shows the graph of the PAV- (re)calibrated forecast probabilities. The CORP approach allows for uncertainty quantification via either resampling techniques or asymptotic theory, furnishes a new numerical measure of miscalibration, and provides a CORP based Brier score decomposition that generalizes to any proper scoring rule. We anticipate that judicious uses of the PAV algorithm yield improved tools for diagnostics and inference for a very wide range of statistical and machine learning methods.

Can AI agents predict whether they will succeed at a task? We study agentic uncertainty by eliciting success probability estimates before, during, and after task execution. All results exhibit agentic overconfidence: some agents that succeed only 22% of the time predict 77% success. Counterintuitively, pre-execution assessment with strictly less information tends to yield better discrimination than standard post-execution review, though differences are not always significant. Adversarial prompting reframing assessment as bug-finding achieves the best calibration.

The handling of probabilities in the form of uncertainty or partial information is an essential task for LLMs in many settings and applications. A common approach to evaluate an LLM's probabilistic reasoning capabilities is to assess its ability to answer questions pertaining to probability through the use of multiple-choice questions (MCQs). However, this paradigm, which we refer to as explicit probabilistic reasoning, has been shown in the literature to yield significant limitations (e.g., sensitivity to answer ordering). In this work, we introduce an alternative approach, named implicit probabilistic reasoning, which evaluates the models' ability to integrate probabilistic reasoning into their text generation process. To achieve this, we rephrase MCQs as text-completion scenarios with a determined set of outcomes and compare the model's next-token probability assignments to the true likelihood of the outcomes. In line with previous work, we find that models exhibit solid performance in their explicit probabilistic reasoning (i.e., answers to MCQs). However, during text completion (i.e., implicit probabilistic reasoning), where the same information must be taken into account to generate text, the models' predictions often significantly diverge from the known ground truth. For instance, our evaluation method reveals that implicit probabilistic reasoning is improperly influenced by many factors, such as independent prior events, partial observations about a result, or statistical background information. All of these issues can cause erroneous results to be produced in text generation, which are not detected by conventional MCQ-based evaluation.

Uncertainty in perception, actuation, and the environment often require multiple attempts for a robotic task to be successful. We study a class of problems providing (1) low-entropy indicators of terminal success / failure, and (2) unreliable (high-entropy) data to predict the final outcome of an ongoing task. Examples include a robot trying to connect with a charging station, parallel parking, or assembling a tightly-fitting part. The ability to restart after predicting failure early, versus simply running to failure, can significantly decrease the makespan, that is, the total time to completion, with the drawback of potentially short-cutting an otherwise successful operation. Assuming task running times to be Poisson distributed, and using a Markov Jump process to capture the dynamics of the underlying Markov Decision Process, we derive a closed form solution that predicts makespan based on the confusion matrix of the failure predictor. This allows the robot to learn failure prediction in a production environment, and only adopt a preemptive policy when it actually saves time. We demonstrate this approach using a robotic peg-in-hole assembly problem using a real robotic system. Failures are predicted by a dilated convolutional network based on force-torque data, showing an average makespan reduction from 101s to 81s (N=120, p<0.05). We posit that the proposed algorithm generalizes to any robotic behavior with an unambiguous terminal reward, with wide ranging applications on how robots can learn and improve their behaviors in the wild.

The aim of this paper is to present an elementary computable theory of probability, random variables and stochastic processes. The probability theory is baed on existing approaches using valuations and lower integrals. Various approaches to random variables are discussed, including the approach based on completions in a Polish space. We apply the theory to the study of stochastic dynamical systems in discrete-time, and give a brief exposition of the Wiener process as a foundation for stochastic differential equations. The theory is based within the framework of type-two effectivity, so has an explicit direct link with Turing computation, and is expressed in a system of computable types and operations, so has a clean mathematical description.

The game show Lucky 13 differs from other television game shows in that contestants are required to place a bet on their own knowledge of trivia by selecting a range that contains the number of questions that they answered correctly. We present a model for this game show using binomial random variables and generate tables outlining the optimal range the player should select based on maximization of two different utility functions. After analyzing the decisions made by some actual contestants on this show, we present a numerical simulation for how many questions an average player is expected to answer correctly based on question categories observed for two sample contestants.

The widespread adoption of large language models (LLMs) makes it important to recognize their strengths and limitations. We argue that in order to develop a holistic understanding of these systems we need to consider the problem that they were trained to solve: next-word prediction over Internet text. By recognizing the pressures that this task exerts we can make predictions about the strategies that LLMs will adopt, allowing us to reason about when they will succeed or fail. This approach - which we call the teleological approach - leads us to identify three factors that we hypothesize will influence LLM accuracy: the probability of the task to be performed, the probability of the target output, and the probability of the provided input. We predict that LLMs will achieve higher accuracy when these probabilities are high than when they are low - even in deterministic settings where probability should not matter. To test our predictions, we evaluate two LLMs (GPT-3.5 and GPT-4) on eleven tasks, and we find robust evidence that LLMs are influenced by probability in the ways that we have hypothesized. In many cases, the experiments reveal surprising failure modes. For instance, GPT-4's accuracy at decoding a simple cipher is 51% when the output is a high-probability word sequence but only 13% when it is low-probability. These results show that AI practitioners should be careful about using LLMs in low-probability situations. More broadly, we conclude that we should not evaluate LLMs as if they are humans but should instead treat them as a distinct type of system - one that has been shaped by its own particular set of pressures.

Although the existing causal inference literature focuses on the forward-looking perspective by estimating effects of causes, the backward-looking perspective can provide insights into causes of effects. In backward-looking causal inference, the probability of necessity measures the probability that a certain event is caused by the treatment given the observed treatment and outcome. Most existing results focus on binary outcomes. Motivated by applications with ordinal outcomes, we propose a general definition of the probability of necessity. However, identifying the probability of necessity is challenging because it involves the joint distribution of the potential outcomes. We propose a novel assumption of monotonic incremental treatment effect to identify the probability of necessity with ordinal outcomes. We also discuss the testable implications of this key identification assumption. When it fails, we derive explicit formulas of the sharp large-sample bounds on the probability of necessity.

Serial verification gates are a core reliability primitive in LLM harnesses: a candidate answer is returned only if k verifier calls all accept it. Under conditionally independent gates, the recent Odds Law (arXiv:2606.15712) shows that posterior log-odds grow linearly in k, so failure decays exponentially, and states that "a tight theory of partially correlated verifier cascades remains open." This note gives a minimal such theory. Modeling the per-instance false-accept rate on the generator's own errors as a latent variable αsim G (de Finetti), the exact cascade posterior is ell_k = ell_0 - ln m_k, with m_k the k-th moment of G. Then: (i) ell_k is concave in k for every non-degenerate G -- the Odds Law is its tangent at the first gate and an upper bound; (ii) for Beta(a,b) latents, failure decays polynomially, 1-r_k asymp k^{-b}, with correlation parameter ρ_v = 1/(a+b+1); (iii) a blind-spot atom of mass 1-π at α=1 caps the evidence extractable from any number of gates at -ln(1-π) nats, so reliability saturates below 1; (iv) letting the true-accept rate also vary (βsim H) yields a trichotomy -- gates eventually always help, plateau, or actively harm -- decided by the upper-tail exponents of G and H, with closed-form crossover k^dagger. The mechanism is survivorship: errors surviving gates are the high-α ones. The theory is measurable: R repeated verdicts per instance identify the first R moments of G, so two verdicts identify ρ_v; beta-binomial likelihood and NPMLE recover the reliability curve and the ill-posed ceiling. In synthetic tests, independence-based extrapolation underestimates failure by 20x at k=5 and ~3000x at k=10; the correlated fit at R=8 tracks held-out depths. The practical lever is decorrelation -- changing model family, modality, or evidence source -- not adding gates.

Many poker systems, whether created with heuristics or machine learning, rely on the probability of winning as a key input. However calculating the precise probability using combinatorics is an intractable problem, so instead we approximate it. Monte Carlo simulation is an effective technique that can be used to approximate the probability that a player will win and/or tie a hand. However, without the use of a memory-intensive lookup table or a supercomputer, it becomes infeasible to run millions of times when training an agent with self-play. To combat the space-time tradeoff, we use deep learning to approximate the probabilities obtained from the Monte Carlo simulation with high accuracy. The learned model proves to be a lightweight alternative to Monte Carlo simulation, which ultimately allows us to use the probabilities as inputs during self-play efficiently. The source code and optimized neural network can be found at https://github.com/brandinho/Poker-Probability-Approximation

Mehta and Panigrahi (2012) proposed Online Matching with Stochastic Rewards, which generalizes the Online Bipartite Matching problem of Karp, Vazirani, and Vazirani (1990) by associating the edges with success probabilities. This new feature captures the pay-per-click model in online advertising. Recently, Huang and Zhang (2020) studied this problem under the online primal dual framework using the Configuration Linear Program (LP), and got the best known competitive ratios of the Stochastic Balance algorithm. Their work suggests that the more expressive Configuration LP is more suitable for this problem than the Matching LP. This paper advances the theory of Configuration LP in two directions. Our technical contribution includes a characterization of the joint matching outcome of an offline vertex and all its neighbors. This characterization may be of independent interest, and is aligned with the spirit of Configuration LP. By contrast, previous analyses of Ranking generally focus on only one neighbor. Second, we designed a Stochastic Configuration LP that captures a stochastic benchmark proposed by Goyal and Udwani (2020), who used a Path-based LP. The Stochastic Configuration LP is smaller and simpler than the Path-based LP. Moreover, using the new LP we improved the competitive ratio of Stochastic Balance from 0.596 to 0.611 when the success probabilities are infinitesimal, and to 0.613 when the success probabilities are further equal.

Large Language Models (LLMs) have demonstrated remarkable capabilities across various applications, fundamentally reshaping the landscape of natural language processing (NLP) research. However, recent evaluation frameworks often rely on the output probabilities of LLMs for predictions, primarily due to computational constraints, diverging from real-world LLM usage scenarios. While widely employed, the efficacy of these probability-based evaluation strategies remains an open research question. This study aims to scrutinize the validity of such probability-based evaluation methods within the context of using LLMs for Multiple Choice Questions (MCQs), highlighting their inherent limitations. Our empirical investigation reveals that the prevalent probability-based evaluation method inadequately aligns with generation-based prediction. Furthermore, current evaluation frameworks typically assess LLMs through predictive tasks based on output probabilities rather than directly generating responses, owing to computational limitations. We illustrate that these probability-based approaches do not effectively correspond with generative predictions. The outcomes of our study can enhance the understanding of LLM evaluation methodologies and provide insights for future research in this domain.

Imitation-learning policies for robot manipulation inherit the quality of the success labels attached to their training episodes, and those labels are usually produced by the robot's own success check. A particularly damaging error is the false success: an episode the robot logs as a success when the task outcome was actually wrong. We ask a narrow but practical question about these episodes. Once an episode has already been flagged as a success, how much of the information needed to overturn that label is present in proprioception, and how much requires vision? We build a simulated testbed on two bimanual ALOHA tasks, induce failures through environment perturbations rather than label edits, label every episode by privileged simulator state that the detector never sees, and keep only episodes the robot flagged as successful. We then compare detectors restricted to proprioception against a vision-based detector. We find that recoverability spans a wide range: in cube transfer the false successes are almost fully recoverable from joint data alone, while in peg insertion proprioception recovers only part of them and a vision detector closes most of the gap. We also show that the proprioceptive separability we measure rests on velocity differences far below any realistic sensor noise floor, so it is best read as an optimistic upper bound that a noiseless simulator inflates. We release the generation and evaluation pipeline.

Sports betting's recent federal legalisation in the USA coincides with the golden age of machine learning. If bettors can leverage data to reliably predict the probability of an outcome, they can recognise when the bookmaker's odds are in their favour. As sports betting is a multi-billion dollar industry in the USA alone, identifying such opportunities could be extremely lucrative. Many researchers have applied machine learning to the sports outcome prediction problem, generally using accuracy to evaluate the performance of predictive models. We hypothesise that for the sports betting problem, model calibration is more important than accuracy. To test this hypothesis, we train models on NBA data over several seasons and run betting experiments on a single season, using published odds. We show that using calibration, rather than accuracy, as the basis for model selection leads to greater returns, on average (return on investment of +34.69% versus -35.17%) and in the best case (+36.93% versus +5.56%). These findings suggest that for sports betting (or any probabilistic decision-making problem), calibration is a more important metric than accuracy. Sports bettors who wish to increase profits should therefore select their predictive model based on calibration, rather than accuracy.

The notions of disintegration and Bayesian inversion are fundamental in conditional probability theory. They produce channels, as conditional probabilities, from a joint state, or from an already given channel (in opposite direction). These notions exist in the literature, in concrete situations, but are presented here in abstract graphical formulations. The resulting abstract descriptions are used for proving basic results in conditional probability theory. The existence of disintegration and Bayesian inversion is discussed for discrete probability, and also for measure-theoretic probability --- via standard Borel spaces and via likelihoods. Finally, the usefulness of disintegration and Bayesian inversion is illustrated in several examples.

Language models (LM) are capable of remarkably complex linguistic tasks; however, numerical reasoning is an area in which they frequently struggle. An important but rarely evaluated form of reasoning is understanding probability distributions. In this paper, we focus on evaluating the probabilistic reasoning capabilities of LMs using idealized and real-world statistical distributions. We perform a systematic evaluation of state-of-the-art LMs on three tasks: estimating percentiles, drawing samples, and calculating probabilities. We evaluate three ways to provide context to LMs 1) anchoring examples from within a distribution or family of distributions, 2) real-world context, 3) summary statistics on which to base a Normal approximation. Models can make inferences about distributions, and can be further aided by the incorporation of real-world context, example shots and simplified assumptions, even if these assumptions are incorrect or misspecified. To conduct this work, we developed a comprehensive benchmark distribution dataset with associated question-answer pairs that we will release publicly.

Hamiltonian Monte Carlo has proven a remarkable empirical success, but only recently have we begun to develop a rigorous understanding of why it performs so well on difficult problems and how it is best applied in practice. Unfortunately, that understanding is confined within the mathematics of differential geometry which has limited its dissemination, especially to the applied communities for which it is particularly important. In this review I provide a comprehensive conceptual account of these theoretical foundations, focusing on developing a principled intuition behind the method and its optimal implementations rather of any exhaustive rigor. Whether a practitioner or a statistician, the dedicated reader will acquire a solid grasp of how Hamiltonian Monte Carlo works, when it succeeds, and, perhaps most importantly, when it fails.

Many people have flipped coins but few have stopped to ponder the statistical and physical intricacies of the process. We collected 350{,}757 coin flips to test the counterintuitive prediction from a physics model of human coin tossing developed by Diaconis, Holmes, and Montgomery (DHM; 2007). The model asserts that when people flip an ordinary coin, it tends to land on the same side it started -- DHM estimated the probability of a same-side outcome to be about 51\%. Our data lend strong support to this precise prediction: the coins landed on the same side more often than not, Pr(same side) = 0.508, 95\% credible interval (CI) [0.506, 0.509], BF_{same-side bias} = 2359. Furthermore, the data revealed considerable between-people variation in the degree of this same-side bias. Our data also confirmed the generic prediction that when people flip an ordinary coin -- with the initial side-up randomly determined -- it is equally likely to land heads or tails: Pr(heads) = 0.500, 95\% CI [0.498, 0.502], BF_{heads-tails bias} = 0.182. Furthermore, this lack of heads-tails bias does not appear to vary across coins. Additional analyses revealed that the within-people same-side bias decreased as more coins were flipped, an effect that is consistent with the possibility that practice makes people flip coins in a less wobbly fashion. Our data therefore provide strong evidence that when some (but not all) people flip a fair coin, it tends to land on the same side it started.

The surprisingly likely criterion in the seminal work of Prelec (the Bayesian Truth Serum) guarantees truthfulness in a game-theoretic multi-agent setting, by rewarding rational agents to maximise the expected information gain with their answers w.r.t. their probabilistic beliefs. We investigate the relevance of a similar criterion for responses of LLMs. We hypothesize that if the surprisingly likely criterion works in LLMs, under certain conditions, the responses that maximize the reward under this criterion should be more accurate than the responses that only maximize the posterior probability. Using benchmarks including the TruthfulQA benchmark and using openly available LLMs: GPT-2 and LLaMA-2, we show that the method indeed improves the accuracy significantly (for example, upto 24 percentage points aggregate improvement on TruthfulQA and upto 70 percentage points improvement on individual categories of questions).

Survival analysis is the problem of estimating probability distributions for future event times, which can be seen as a problem in uncertainty quantification. Although there are fundamental theories on strictly proper scoring rules for uncertainty quantification, little is known about those for survival analysis. In this paper, we investigate extensions of four major strictly proper scoring rules for survival analysis and we prove that these extensions are proper under certain conditions, which arise from the discretization of the estimation of probability distributions. We also compare the estimation performances of these extended scoring rules by using real datasets, and the extensions of the logarithmic score and the Brier score performed the best.

Fine-tuning large language models (LLMs) on reasoning benchmarks via reinforcement learning requires a specific reward function, often binary, for each benchmark. This comes with two potential limitations: the need to design the reward, and the potentially sparse nature of binary rewards. Here, we systematically investigate rewards derived from the probability or log-probability of emitting the reference answer (or any other prompt continuation present in the data), which have the advantage of not relying on specific verifiers and being available at scale. Several recent works have advocated for the use of similar rewards (e.g., VeriFree, JEPO, RLPR, NOVER). We systematically compare variants of likelihood-based rewards with standard baselines, testing performance both on standard mathematical reasoning benchmarks, and on long-form answers where no external verifier is available. We find that using the log-probability of the reference answer as the reward for chain-of-thought (CoT) learning is the only option that performs well in all setups. This reward is also consistent with the next-token log-likelihood loss used during pretraining. In verifiable settings, log-probability rewards bring comparable or better success rates than reinforcing with standard binary rewards, and yield much better perplexity. In non-verifiable settings, they perform on par with SFT. On the other hand, methods based on probability, such as VeriFree, flatline on non-verifiable settings due to vanishing probabilities of getting the correct answer. Overall, this establishes log-probability rewards as a viable method for CoT fine-tuning, bridging the short, verifiable and long, non-verifiable answer settings.

With an increasing number of replication studies performed in psychological science, the question of how to evaluate the outcome of a replication attempt deserves careful consideration. Bayesian approaches allow to incorporate uncertainty and prior information into the analysis of the replication attempt by their design. The Replication Bayes Factor, introduced by Verhagen & Wagenmakers (2014), provides quantitative, relative evidence in favor or against a successful replication. In previous work by Verhagen & Wagenmakers (2014) it was limited to the case of t-tests. In this paper, the Replication Bayes Factor is extended to F-tests in multi-group, fixed-effect ANOVA designs. Simulations and examples are presented to facilitate the understanding and to demonstrate the usefulness of this approach. Finally, the Replication Bayes Factor is compared to other Bayesian and frequentist approaches and discussed in the context of replication attempts. R code to calculate Replication Bayes factors and to reproduce the examples in the paper is available at https://osf.io/jv39h/.

Prompt optimization in compound AI systems is statistically indistinguishable from a coin flip: across 72 optimization runs on Claude Haiku 4.5 (6 methods times 4 tasks times 3 repeats), 49% score below zero-shot; on Amazon Nova Lite, the failure rate is even higher. Yet on one task, all six methods improve over zero-shot by up to +6.8 points. What distinguishes success from failure? We investigate with 18,000 grid evaluations and 144 optimization runs, testing two assumptions behind end-to-end optimization tools like TextGrad and DSPy, in the order they must be answered: (A) agent prompts interact, requiring joint rather than independent optimization, and (B) individual prompts are worth optimizing at all. Interaction effects are never significant (p > 0.52, all F < 1.0), and optimization helps only when the task has exploitable output structure: a format the model can produce but does not default to. We further give a mechanistic account: instruction-tuning compresses input phrasing into a narrow output distribution, eliminating the very phrasing-sensitivity that joint optimization assumes. We provide a two-stage diagnostic: an \$80 ANOVA pre-test for agent coupling, and a 10-minute headroom test that predicts whether optimization is worthwhile, turning a coin flip into an informed decision.

In this article, we propose a new counter-based implementation of John von Neumann's middle-square random number generator (RNG). Several rounds of squaring are applied to a counter to produce a random output. We discovered that four rounds are sufficient to provide satisfactory data. Two versions of the RNG are presented, a 4-round version with 32-bit output and a 5-round version with 64-bit output. Both pass stringent tests of randomness and may be the fastest counter-based generators.

We study whether language models can evaluate the validity of their own claims and predict which questions they will be able to answer correctly. We first show that larger models are well-calibrated on diverse multiple choice and true/false questions when they are provided in the right format. Thus we can approach self-evaluation on open-ended sampling tasks by asking models to first propose answers, and then to evaluate the probability "P(True)" that their answers are correct. We find encouraging performance, calibration, and scaling for P(True) on a diverse array of tasks. Performance at self-evaluation further improves when we allow models to consider many of their own samples before predicting the validity of one specific possibility. Next, we investigate whether models can be trained to predict "P(IK)", the probability that "I know" the answer to a question, without reference to any particular proposed answer. Models perform well at predicting P(IK) and partially generalize across tasks, though they struggle with calibration of P(IK) on new tasks. The predicted P(IK) probabilities also increase appropriately in the presence of relevant source materials in the context, and in the presence of hints towards the solution of mathematical word problems. We hope these observations lay the groundwork for training more honest models, and for investigating how honesty generalizes to cases where models are trained on objectives other than the imitation of human writing.

In this paper we argue for the fundamental importance of the value distribution: the distribution of the random return received by a reinforcement learning agent. This is in contrast to the common approach to reinforcement learning which models the expectation of this return, or value. Although there is an established body of literature studying the value distribution, thus far it has always been used for a specific purpose such as implementing risk-aware behaviour. We begin with theoretical results in both the policy evaluation and control settings, exposing a significant distributional instability in the latter. We then use the distributional perspective to design a new algorithm which applies Bellman's equation to the learning of approximate value distributions. We evaluate our algorithm using the suite of games from the Arcade Learning Environment. We obtain both state-of-the-art results and anecdotal evidence demonstrating the importance of the value distribution in approximate reinforcement learning. Finally, we combine theoretical and empirical evidence to highlight the ways in which the value distribution impacts learning in the approximate setting.

We revisit the noisy binary search model of Karp and Kleinberg, in which we have n coins with unknown probabilities p_i that we can flip. The coins are sorted by increasing p_i, and we would like to find where the probability crosses (to within varepsilon) of a target value tau. This generalized the fixed-noise model of Burnashev and Zigangirov , in which p_i = 1{2} pm varepsilon, to a setting where coins near the target may be indistinguishable from it. Karp and Kleinberg showed that Theta(1{varepsilon^2} log n) samples are necessary and sufficient for this task. We produce a practical algorithm by solving two theoretical challenges: high-probability behavior and sharp constants. We give an algorithm that succeeds with probability 1-delta from \[ 1{C_{\tau, \varepsilon}} \cdot \left(\lg n + O(\log^{2/3} n \log^{1/3} 1{\delta} + \log 1{\delta})\right) \] samples, where C_{tau, varepsilon} is the optimal such constant achievable. For delta > n^{-o(1)} this is within 1 + o(1) of optimal, and for delta ll 1 it is the first bound within constant factors of optimal.

We study a model of a branching process subject to selection, modeled by giving each family an individual fitness acting as a branching rate, and mutation, modeled by resampling the fitness of a proportion of offspring in each generation. For two large classes of fitness distributions of Gumbel type we determine the growth of the population, almost surely on survival. We then study the empirical fitness distribution in a simplified model, which is numerically indistinguishable from the original model, and show the emergence of a Gaussian travelling wave.

The rapid proliferation of Large Language Models (LLMs) and diverse specialized benchmarks necessitates a shift from fragmented, task-specific metrics to a holistic, competitive ranking system that effectively aggregates performance across multiple ability dimensions. Primarily using static scoring, current evaluation methods are fundamentally limited. They struggle to determine the proper mix ratio across diverse benchmarks, and critically, they fail to capture a model's dynamic competitive fitness or its vulnerability when confronted with sequential, high-stakes tasks. To address this, we introduce the novel Competitive Swiss-System Dynamics (CSD) framework. CSD simulates a multi-round, sequential contest where models are dynamically paired across a curated sequence of benchmarks based on their accumulated win-loss record. And Monte Carlo Simulation (N=100,000 iterations) is used to approximate the statistically robust Expected Win Score (E[S_m]), which eliminates the noise of random pairing and early-round luck. Furthermore, we implement a Failure Sensitivity Analysis by parameterizing the per-round elimination quantity (T_k), which allows us to profile models based on their risk appetite--distinguishing between robust generalists and aggressive specialists. We demonstrate that CSD provides a more nuanced and context-aware ranking than traditional aggregate scoring and static pairwise models, representing a vital step towards risk-informed, next-generation LLM evaluation.

We present a state-of-the-art model for fine-grained probability estimation of propositions conditioned on context. Recent advances in large language models (LLMs) have significantly enhanced their reasoning capabilities, particularly on well-defined tasks with complete information. However, LLMs continue to struggle with making accurate and well-calibrated probabilistic predictions under uncertainty or partial information. While incorporating uncertainty into model predictions often boosts performance, obtaining reliable estimates of that uncertainty remains understudied. In particular, LLM probability estimates tend to be coarse and biased towards more frequent numbers. Through a combination of human and synthetic data creation and assessment, scaling to larger models, and better supervision, we propose a set of strong and precise probability estimation models. We conduct systematic evaluations across tasks that rely on conditional probability estimation and show that our approach consistently outperforms existing fine-tuned and prompting-based methods by a large margin.

Consistently checking the statistical significance of experimental results is one of the mandatory methodological steps to address the so-called "reproducibility crisis" in deep reinforcement learning. In this tutorial paper, we explain how the number of random seeds relates to the probabilities of statistical errors. For both the t-test and the bootstrap confidence interval test, we recall theoretical guidelines to determine the number of random seeds one should use to provide a statistically significant comparison of the performance of two algorithms. Finally, we discuss the influence of deviations from the assumptions usually made by statistical tests. We show that they can lead to inaccurate evaluations of statistical errors and provide guidelines to counter these negative effects. We make our code available to perform the tests.

As browser-use agents improve, benchmarks keep pace by collecting new tasks, websites, and applications, often making tasks longer or more novel. This makes difficulty expensive to refresh and difficult to control: when many aspects change at once, it is unclear what actually makes a task challenging. We instead construct challenging instances from tasks agents already solve, turning difficulty into a programmable property of the environment. BreakingWeb pairs every base task with an intervention condition that preserves the user instruction, latent target, and backend success criterion while changing the environment at different web stack layers. Each intervention is deterministic, detectable, and recoverable, and is annotated with the cognitive primitive it primarily loads. The benchmark contains 519 clean/intervention task pairs across seven self-hosted websites and 29 intervention families, all graded against outcomes. We evaluate six strong browser-use agents, three GUI-only agents that see only screenshots, and humans. The construction is effective: interventions cut agent pass rate by 22.9% on average and overturn nearly half of the tasks each agent solves cleanly, whereas humans lose 10.0% on a first attempt and 5.7% after one familiarisation attempt. The dominant failure is belief failure: 75% of the six agents' failures end with a declared success although the required change never happened. Our code, data and environment are publicly available at www.breakingweb.app.

Training classifiers is difficult with severe class imbalance, but many rare events are the culmination of a sequence with much more common intermediate outcomes. For example, in online marketing a user first sees an ad, then may click on it, and finally may make a purchase; estimating the probability of purchases is difficult because of their rarity. We show both theoretically and through data experiments that the more abundant data in earlier steps may be leveraged to improve estimation of probabilities of rare events. We present PRESTO, a relaxation of the proportional odds model for ordinal regression. Instead of estimating weights for one separating hyperplane that is shifted by separate intercepts for each of the estimated Bayes decision boundaries between adjacent pairs of categorical responses, we estimate separate weights for each of these transitions. We impose an L1 penalty on the differences between weights for the same feature in adjacent weight vectors in order to shrink towards the proportional odds model. We prove that PRESTO consistently estimates the decision boundary weights under a sparsity assumption. Synthetic and real data experiments show that our method can estimate rare probabilities in this setting better than both logistic regression on the rare category, which fails to borrow strength from more abundant categories, and the proportional odds model, which is too inflexible.

We propose a new test to measure a text model's multitask accuracy. The test covers 57 tasks including elementary mathematics, US history, computer science, law, and more. To attain high accuracy on this test, models must possess extensive world knowledge and problem solving ability. We find that while most recent models have near random-chance accuracy, the very largest GPT-3 model improves over random chance by almost 20 percentage points on average. However, on every one of the 57 tasks, the best models still need substantial improvements before they can reach expert-level accuracy. Models also have lopsided performance and frequently do not know when they are wrong. Worse, they still have near-random accuracy on some socially important subjects such as morality and law. By comprehensively evaluating the breadth and depth of a model's academic and professional understanding, our test can be used to analyze models across many tasks and to identify important shortcomings.

Predictive models often need to work with incomplete information in real-world tasks. Consequently, they must provide reliable probability or confidence estimation, especially in large-scale decision-making and planning tasks. Current large language models (LLMs) are insufficient for accurate estimations, but they can generate relevant factors that may affect the probabilities, produce coarse-grained probabilities when the information is more complete, and help determine which factors are relevant to specific downstream contexts. In this paper, we make use of these capabilities of LLMs to provide a significantly more accurate probabilistic estimation. We propose BIRD, a novel probabilistic inference framework that aligns a Bayesian network with LLM abductions and then estimates more accurate probabilities in a deduction step. We show BIRD provides reliable probability estimations that are 30% better than those provided directly by LLM baselines. These estimates further contribute to better and more trustworthy decision making.

Long-horizon LLM agents depend on intermediate information-gathering turns, yet training feedback is usually observed only at the final answer, because process-level rewards require high-quality human annotation. Existing turn-level shaping methods reward turns that increase the likelihood of a gold answer, but they require answer supervision or stable task-specific verifiers. Conversely, label-free RL methods extract self-signals from output distributions, but mainly at the answer or trajectory level and therefore cannot assign credit to intermediate turns. We propose Self-Induced Outcome Potential (SIOP), which treats semantic clusters of final answers as latent future outcome states for potential-based turn-level credit assignment. For each query, SIOP samples multiple rollouts, clusters final answers into semantic outcome modes, and builds a reliability-aware target distribution over these states. It then rewards turns for increasing posterior support for reliable future states using a tractable cluster-level approximation. The objective generalizes information-potential shaping from gold-answer supervision to settings without task-specific gold verifiers while avoiding the broadcasted rollout-level advantages used by standard GRPO. We formalize the framework, characterize its supervised gold-answer limit, and show that SIOP improves average performance over verifier-free outcome-level baselines on seven search-augmented agentic reasoning benchmarks while approaching a gold-supervised outcome baseline. Code is available at https://github.com/dl-m9/SIOP.git.

Video world models approximate the stochastic distribution of physical outcomes through generative sampling, but existing benchmarks score individual generations or compare distributions coarsely over a whole dataset, leaving the fine-grained aleatoric uncertainty of specific phenomena untested. We introduce CaliBench, which scores outcomes in a physically interpretable discrete space - a bin index, a die face, a suit, a colour - rather than a learned feature space such as in FID, so the distance from a known reference distribution is measured directly. We curate outcome spaces whose reference is known in closed form (binomial Galton boards, Bernoulli forks, uniform dice/cards/lottery, a skewed European-roulette colour), enabling an exact calibration test. We decompose performance into two orthogonal axes that a single accuracy metric conflates: scorability, the fraction of generations yielding a scoreable outcome, and calibration, the total variation distance from the reference on that sample. A chi-squared test assesses significance; as calibration is its null hypothesis it can evidence only miscalibration, and at N=32 per cell detects only large deviations. We apply it to nine scenes and six image-to-video models (WAN-2.7, SeeDance-2.0, HappyHorse-1.0, Veo 3.1, Runway Gen-4.5, Cosmos3-Super), 32 generations each. Models consistently concentrate probability mass on a few outcomes rather than reproducing the reference. Most scene-model combinations are significantly miscalibrated, in the extreme collapsing to one outcome, as Veo 3.1 does on dice. On roulette, generations often leave the ball ambiguously placed, giving several models low scorability. Performance varies by scene: no model dominates all nine. We release the protocol and a metric (mean normalised total variation, mnTV) for comparing new models against our results.

Many events occur in the world. Some event types are stochastically excited or inhibited---in the sense of having their probabilities elevated or decreased---by patterns in the sequence of previous events. Discovering such patterns can help us predict which type of event will happen next and when. We model streams of discrete events in continuous time, by constructing a neurally self-modulating multivariate point process in which the intensities of multiple event types evolve according to a novel continuous-time LSTM. This generative model allows past events to influence the future in complex and realistic ways, by conditioning future event intensities on the hidden state of a recurrent neural network that has consumed the stream of past events. Our model has desirable qualitative properties. It achieves competitive likelihood and predictive accuracy on real and synthetic datasets, including under missing-data conditions.

Process reward models (PRMs) have become a key component for LLMs, as their step-level feedback supports both post-training and test-time reasoning. However, training strong PRMs remains costly: human step annotation is difficult to scale, while Monte Carlo estimation is computationally expensive and can drift from the intrinsic correctness of steps. To get effective PRMs at low cost, we introduce TIPS (Thinking-Induced Process Supervision), an outcome-only reinforcement learning (RL) framework for training generative PRMs. In TIPS, the model generates a chain-of-thought (CoT) followed by step-level labels and an outcome label. The reward depends solely on whether the predicted outcome matches the ground truth, and the resulting group-relative advantage is used to optimize the entire generated response. Intuitively, when checking intermediate steps helps determine the outcome, more accurate checks can lead to better outcome judgments and higher rewards. Outcome-only RL can therefore reinforce step-level verification without explicit process supervision. We validate the effectiveness of TIPS across math and agent benchmarks and four backbone families. Notably, TIPS-Qwen3-4B-Thinking-2507 reaches 85.2 F1 on ProcessBench with only 3.2K outcome-labeled trajectories, surpassing all evaluated trained PRMs and strong prompt-only judges such as GPT-5.4-Instruct and Claude-4.7-Opus, while still trailing o1-mini. Code and data are available at https://github.com/RUCBM/TIPS.

Combined electric power system and High-Altitude Electromagnetic Pulse (HEMP) models are being developed to determine the effect of a HEMP on the US power grid. The work relies primarily on deterministic methods; however, it is computationally untenable to evaluate the E1 HEMP response of large numbers of grid components distributed across a large interconnection. Further, the deterministic assessment of these components' failures are largely unachievable. E1 HEMP laboratory testing of the components is accomplished, but is expensive, leaving few data points to construct failure models of grid components exposed to E1 HEMP. The use of Bayesian priors, developed using the subject matter expertise, combined with the minimal test data in a Bayesian inference process, provides the basis for the development of more robust and cost-effective statistical component failure models. These can be used with minimal computational burden in a simulation environment such as sampling of Cumulative Distribution Functions (CDFs).

Collecting high-quality training examples for language model fine-tuning is expensive, with practical budgets limiting the amount of data that can be procured. We investigate whether example difficulty affects GRPO training effectiveness by comparing selection strategies (easy, medium, hard, random) across multiple models and reasoning tasks. Training on the hardest 10\% of examples (those where the base model fails most often) yields dramatic performance gains up to 47\%, while easy examples produce minimal improvements of 3-15\%. This occurs because GRPO requires outcome variance to generate learning signals; hard examples maintain mixed success/failure outcomes throughout training while easy examples quickly converge to consistent success, eliminating learning opportunities. Moreover, models trained on hard examples show superior out-of-distribution generalization, with only hard-trained models achieving meaningful gains on the AIME2025 benchmark. Our findings provide clear guidance: when budget-constrained, prioritize collecting and annotating examples where your base model struggles, as these drive nearly all learning value in GRPO fine-tuning

We consider the problem of performing Bayesian inference in probabilistic models where observations are accompanied by uncertainty, referred to as "uncertain evidence." We explore how to interpret uncertain evidence, and by extension the importance of proper interpretation as it pertains to inference about latent variables. We consider a recently-proposed method "distributional evidence" as well as revisit two older methods: Jeffrey's rule and virtual evidence. We devise guidelines on how to account for uncertain evidence and we provide new insights, particularly regarding consistency. To showcase the impact of different interpretations of the same uncertain evidence, we carry out experiments in which one interpretation is defined as "correct." We then compare inference results from each different interpretation illustrating the importance of careful consideration of uncertain evidence.

Organizations across finance, healthcare, transportation, content moderation, and critical infrastructure are rapidly deploying highly automated AI systems, yet they lack principled methods to quantify how increasing automation amplifies harm when failures occur. We propose a parsimonious Bayesian risk decomposition expressing expected loss as the product of three terms: the probability of system failure, the conditional probability that a failure propagates into harm given the automation level, and the expected severity of harm. This framework isolates a critical quantity -- the conditional probability that failures propagate into harm -- which captures execution and oversight risk rather than model accuracy alone. We develop complete theoretical foundations: formal proofs of the decomposition, a harm propagation equivalence theorem linking the harm propagation probability to observable execution controls, risk elasticity measures, efficient frontier analysis for automation policy, and optimal resource allocation principles with second-order conditions. We motivate the framework with an illustrative case study of the 2012 Knight Capital incident ($440M loss) as one instantiation of a broadly applicable failure pattern, and characterize the research design required to empirically validate the framework at scale across deployment domains. This work provides the theoretical foundations for a new class of deployment-focused risk governance tools for agentic and automated AI systems.

One of the most widely used methods to evaluate LLMs are Multiple Choice Question (MCQ) tests. MCQ benchmarks enable the testing of LLM knowledge on almost any topic at scale as the results can be processed automatically. To help the LLM answer, a few examples called few shots can be included in the prompt. Moreover, the LLM can be asked to answer the question directly with the selected option or to first provide the reasoning and then the selected answer, which is known as chain of thought. In addition to checking whether the selected answer is correct, the evaluation can look at the LLM-estimated probability of its response as an indication of the confidence of the LLM in the response. In this paper, we study how the LLM confidence in its answer depends on whether the model has been asked to answer directly or to provide the reasoning before answering. The results of the evaluation of questions on a wide range of topics in seven different models show that LLMs are more confident in their answers when they provide reasoning before the answer. This occurs regardless of whether the selected answer is correct. Our hypothesis is that this behavior is due to the reasoning that modifies the probability of the selected answer, as the LLM predicts the answer based on the input question and the reasoning that supports the selection made. Therefore, LLM estimated probabilities seem to have intrinsic limitations that should be understood in order to use them in evaluation procedures. Interestingly, the same behavior has been observed in humans, for whom explaining an answer increases confidence in its correctness.

A Neural Process (NP) estimates a stochastic process implicitly defined with neural networks given a stream of data, rather than pre-specifying priors already known, such as Gaussian processes. An ideal NP would learn everything from data without any inductive biases, but in practice, we often restrict the class of stochastic processes for the ease of estimation. One such restriction is the use of a finite-dimensional latent variable accounting for the uncertainty in the functions drawn from NPs. Some recent works show that this can be improved with more "data-driven" source of uncertainty such as bootstrapping. In this work, we take a different approach based on the martingale posterior, a recently developed alternative to Bayesian inference. For the martingale posterior, instead of specifying prior-likelihood pairs, a predictive distribution for future data is specified. Under specific conditions on the predictive distribution, it can be shown that the uncertainty in the generated future data actually corresponds to the uncertainty of the implicitly defined Bayesian posteriors. Based on this result, instead of assuming any form of the latent variables, we equip a NP with a predictive distribution implicitly defined with neural networks and use the corresponding martingale posteriors as the source of uncertainty. The resulting model, which we name as Martingale Posterior Neural Process (MPNP), is demonstrated to outperform baselines on various tasks.

In any ranking system, the retrieval model outputs a single score for a document based on its belief on how relevant it is to a given search query. While retrieval models have continued to improve with the introduction of increasingly complex architectures, few works have investigated a retrieval model's belief in the score beyond the scope of a single value. We argue that capturing the model's uncertainty with respect to its own scoring of a document is a critical aspect of retrieval that allows for greater use of current models across new document distributions, collections, or even improving effectiveness for down-stream tasks. In this paper, we address this problem via an efficient Bayesian framework for retrieval models which captures the model's belief in the relevance score through a stochastic process while adding only negligible computational overhead. We evaluate this belief via a ranking based calibration metric showing that our approximate Bayesian framework significantly improves a retrieval model's ranking effectiveness through a risk aware reranking as well as its confidence calibration. Lastly, we demonstrate that this additional uncertainty information is actionable and reliable on down-stream tasks represented via cutoff prediction.

We propose a new method to determine the shape of the gravitational potential of the dark matter (DM) halo of the Milky Way (MW) with the galactocentric tangential velocities of a sample of hypervelocity stars (HVSs). We compute the trajectories of different samples of HVSs in a MW where the baryon distribution is axisymmetric and the DM potential either is spherical or is spheroidal or triaxial with radial-dependent axis ratios. We determine the shape of the DM potential with the distribution of the latitudinal velocity |v_{vartheta}| in axisymmetric Galactic potentials, or with the distribution of |v_{vartheta}| and of a function bar v_{varphi} of the azimuthal velocity in non-axisymmetric Galactic potentials. We recover the correct shape of the DM potential by comparing the distribution of |v_{vartheta}| and bar v_{varphi} against the corresponding distributions of mock samples of HVSs that traveled in DM halos of different shapes. We use the largest possible sample of sim 800 HVSs of 4~M_odot ejected with the Hills mechanism at a rate sim 10^{-4} yr^{-1}, currently outgoing, and located at more than 10 kpc from the Galactic center. In our ideal case of galactocentric velocities with null uncertainties and no observational limitations, our method recovers the correct shape of the DM potential with a success rate Sgtrsim 89% in axisymmetric Galactic potentials, and S > 96% in the explored non-axisymmetric cases. The unsuccessful cases yield axis ratios of the DM potential that are off by pm 0.1. The success rate decreases with decreasing sample size: for example, for a spherical DM halo, S drops from sim 98% to sim 38% when the sample size decreases from sim 800 to sim 40 HVSs. A robust determination of the shape of the DM potential thus requires the measure of the galactocentric velocity of a few hundred genuine HVSs.

Commonly used evaluation measures including Recall, Precision, F-Measure and Rand Accuracy are biased and should not be used without clear understanding of the biases, and corresponding identification of chance or base case levels of the statistic. Using these measures a system that performs worse in the objective sense of Informedness, can appear to perform better under any of these commonly used measures. We discuss several concepts and measures that reflect the probability that prediction is informed versus chance. Informedness and introduce Markedness as a dual measure for the probability that prediction is marked versus chance. Finally we demonstrate elegant connections between the concepts of Informedness, Markedness, Correlation and Significance as well as their intuitive relationships with Recall and Precision, and outline the extension from the dichotomous case to the general multi-class case.

Large language models (LLMs) are increasingly used for tasks that require complex reasoning. Most benchmarks focus on final outcomes but overlook the intermediate reasoning steps - such as planning, revision, and decision making under resource constraints. We argue that measuring these internal processes is essential for understanding model behavior and improving reliability. We propose using strategic games as a natural evaluation environment: closed, rule-based systems with clear states, limited resources, and automatic feedback. We introduce a framework that evaluates LLMs along three core dimensions: planning, revision, and resource-constrained decision making. To operationalize this, we define metrics beyond win rate, including overcorrection risk rate, correction success rate, improvement slope, and over-budget ratio. In 4320 adversarial rounds across 12 leading models, ChatGPT-o3-mini achieves the top composite score, with a win rate of 74.7 percent, a correction success rate of 78.6 percent, and an improvement slope of 0.041. By contrast, Qwen-Plus, despite an overcorrection risk rate of 81.6 percent, wins only 25.6 percent of its matches - primarily due to excessive resource use. We also observe a negative correlation between overcorrection risk rate and correction success rate (Pearson r = -0.51, p = 0.093), suggesting that more frequent edits do not always improve outcomes. Our findings highlight the value of assessing not only what LLMs decide but how they arrive at those decisions

The study conducted by Shumailov et al. (2024) demonstrates that repeatedly training a generative model on synthetic data leads to model collapse. This finding has generated considerable interest and debate, particularly given that current models have nearly exhausted the available data. In this work, we investigate the effects of fitting a distribution (through Kernel Density Estimation, or KDE) or a model to the data, followed by repeated sampling from it. Our objective is to develop a theoretical understanding of the phenomenon observed by Shumailov et al. (2024). Our results indicate that the outcomes reported are a statistical phenomenon and may be unavoidable.

Information accounting provides a better foundation for hypothesis testing than does uncertainty quantification. A quantitative account of science is derived under this perspective that alleviates the need for epistemic bridge principles, solves the problem of ad hoc falsification criteria, and deals with verisimilitude by facilitating a general approach to process-level diagnostics. Our argument is that the well-known inconsistencies of both Bayesian and classical statistical hypothesis tests are due to the fact that probability theory is an insufficient logic of science. Information theory, as an extension of probability theory, is required to provide a complete logic on which to base quantitative theories of empirical learning. The organizing question in this case becomes not whether our theories or models are more or less true, or about how much uncertainty is associated with a particular model, but instead whether there is any information available from experimental data that might allow us to improve the model. This becomes a formal hypothesis test, provides a theory of model diagnostics, and suggests a new approach to building dynamical systems models.

Can we learn more from data than existed in the generating process itself? Can new and useful information be constructed from merely applying deterministic transformations to existing data? Can the learnable content in data be evaluated without considering a downstream task? On these questions, Shannon information and Kolmogorov complexity come up nearly empty-handed, in part because they assume observers with unlimited computational capacity and fail to target the useful information content. In this work, we identify and exemplify three seeming paradoxes in information theory: (1) information cannot be increased by deterministic transformations; (2) information is independent of the order of data; (3) likelihood modeling is merely distribution matching. To shed light on the tension between these results and modern practice, and to quantify the value of data, we introduce epiplexity, a formalization of information capturing what computationally bounded observers can learn from data. Epiplexity captures the structural content in data while excluding time-bounded entropy, the random unpredictable content exemplified by pseudorandom number generators and chaotic dynamical systems. With these concepts, we demonstrate how information can be created with computation, how it depends on the ordering of the data, and how likelihood modeling can produce more complex programs than present in the data generating process itself. We also present practical procedures to estimate epiplexity which we show capture differences across data sources, track with downstream performance, and highlight dataset interventions that improve out-of-distribution generalization. In contrast to principles of model selection, epiplexity provides a theoretical foundation for data selection, guiding how to select, generate, or transform data for learning systems.

Probabilistic forecasting estimates the likelihood of uncertain future events. To improve LLM forecasting, existing methods typically learn from binary outcomes to output verbalized forecasts. However, while aggregated human forecasts contain rich information in both the crowd probability estimate and the degree of agreement among forecasters, how to utilize these signals remains underexplored. To address this, we propose the Beta-Bernoulli Calibrator (BBC), which converts an initial point estimate forecast from any model into a distribution over event likelihood, using supervision from both binary outcomes and human forecasts. BBC models event likelihood p sim Beta(α, β) and outcome y sim Bernoulli(p), with the mean as the calibrated point forecast and the variance as the epistemic uncertainty. Our results show that BBC generally provides better calibrated and more accurate forecasts than both traditional post-hoc calibration methods and models fine-tuned specifically for forecasting, while remaining lightweight and having good generalization. We also show that the epistemic uncertainty captured by BBC is a more reliable predictor of forecasting error than verbalized confidence.

In physics and engineering literature, the distribution of the excursion-above-zero time distribution (exceedance distribution) for a stationary Gaussian process has been approximated by a stationary switching process with independently distributed switching times. The approach matched the covariance of the clipped Gaussian process with the one for the stationary switching process and the distribution of the latter was used as the so-called independent interval approximation (IIA). The approach successfully assessed the persistency exponent for many physically important processes but left an unanswered question when such an approach leads to a mathematically meaningful and proper exceedance distribution. Here we address this question by proposing an alternative matching of the expected values of the clipped Slepian process and the corresponding switched process initiated at the origin. The method has allowed resolving the mathematical correctness of the matching method for a large subclass of the Gaussian processes with monotonic covariance, for which we provide a sufficient condition for the validity of the IIA. Within this class, the IIA produces a valid distribution for the excursion time and is represented in an explicit stochastic form that connects directly to the covariance of the underlying Gaussian process. We compare the excursion level distributions as well as the corresponding persistency exponents obtained through the IIA method with numerically computed exact distributions, and the simulated distribution for several important Gaussian models. We also argue that for stationary Gaussian processes with a non-monotonic covariance, the IIA fails and should not be used.

In the past couple of years, various approaches to representing and quantifying different types of predictive uncertainty in machine learning, notably in the setting of classification, have been proposed on the basis of second-order probability distributions, i.e., predictions in the form of distributions on probability distributions. A completely conclusive solution has not yet been found, however, as shown by recent criticisms of commonly used uncertainty measures associated with second-order distributions, identifying undesirable theoretical properties of these measures. In light of these criticisms, we propose a set of formal criteria that meaningful uncertainty measures for predictive uncertainty based on second-order distributions should obey. Moreover, we provide a general framework for developing uncertainty measures to account for these criteria, and offer an instantiation based on the Wasserstein distance, for which we prove that all criteria are satisfied.

While reinforcement learning with verifiable rewards (RLVR) has advanced LLM reasoning in structured domains like mathematics and programming, its application to general-domain reasoning tasks remains challenging due to the absence of verifiable reward signals. To this end, methods like Reinforcement Learning with Reference Probability Reward (RLPR) have emerged, leveraging the probability of generating the final answer as a reward signal. However, these outcome-focused approaches neglect crucial step-by-step supervision of the reasoning process itself. To address this gap, we introduce Probabilistic Process Supervision (P2S), a novel self-supervision framework that provides fine-grained process rewards without requiring a separate reward model or human-annotated reasoning steps. During reinforcement learning, P2S synthesizes and filters a high-quality reference reasoning chain (gold-CoT). The core of our method is to calculate a Path Faithfulness Reward (PFR) for each reasoning step, which is derived from the conditional probability of generating the gold-CoT's suffix, given the model's current reasoning prefix. Crucially, this PFR can be flexibly integrated with any outcome-based reward, directly tackling the reward sparsity problem by providing dense guidance. Extensive experiments on reading comprehension and medical Question Answering benchmarks show that P2S significantly outperforms strong baselines.

Computer-use agents have rapidly improved on real-world tasks such as web navigation, desktop automation, and software interaction, in some cases surpassing human performance. Yet even when the task and model are unchanged, an agent that succeeds once may fail on a repeated execution of the same task. This raises a fundamental question: if an agent can succeed at a task once, what prevents it from doing so reliably? In this work, we study the sources of unreliability in computer-use agents through three factors: stochasticity during execution, ambiguity in task specification, and variability in agent behavior. We analyze these factors on OSWorld using repeated executions of the same task together with paired statistical tests that capture task-level changes across settings. Our analysis shows that reliability depends on both how tasks are specified and how agent behavior varies across executions. These findings suggest the need to evaluate agents under repeated execution, to allow agents to resolve task ambiguity through interaction, and to favor strategies that remain stable across runs.

This paper is an extended and reworked version of a short course given by the author at ''Uzbekistan-Ukrainian readings in stochastic processes'', Tashkent-Kyiv, 2022, and was prepared for a special issue of ''Theory of stochastic processes'', devoted to publishing lecture notes from the aforementioned workshop. The survey is devoted to operator splitting methods in the abstract formulation and their applications in probability. While the survey is focused on multiplicative methods, the BCH formula is used to discuss exponential splitting methods and a short informal introduction to additive splitting is presented. We introduce frameworks and available deterministic and probabilistic results and concentrate on constructing a wide picture of the field of operator splitting methods, providing a rigorous description in the setting of abstract Cauchy problems and an informal discussion for further and parallel advances. Some limitations and common difficulties are listed, as well as examples of works that provide solutions or hints. No new results are given. The bibliography contains illustrative deterministic examples and a selection of probability-related works.

We study the problem of predicting numeric labels that are constrained to the integers or to a subrange of the integers. For example, the number of up-votes on social media posts, or the number of bicycles available at a public rental station. While it is possible to model these as continuous values, and to apply traditional regression, this approach changes the underlying distribution on the labels from discrete to continuous. Discrete distributions have certain benefits, which leads us to the question whether such integer labels can be modeled directly by a discrete distribution, whose parameters are predicted from the features of a given instance. Moreover, we focus on the use case of output distributions of neural networks, which adds the requirement that the parameters of the distribution be continuous so that backpropagation and gradient descent may be used to learn the weights of the network. We investigate several options for such distributions, some existing and some novel, and test them on a range of tasks, including tabular learning, sequential prediction and image generation. We find that overall the best performance comes from two distributions: Bitwise, which represents the target integer in bits and places a Bernoulli distribution on each, and a discrete analogue of the Laplace distribution, which uses a distribution with exponentially decaying tails around a continuous mean.

We introduce UnpredictaBench, an evaluation that tests the ability of large language models (LLMs) to capture true underlying distributions. As LLMs are increasingly used as substitutes for other entities (e.g., for humans in economic simulations), the tendency of many models to collapse towards a single plausible answer means a failure to capture the unpredictability of real systems. Recent work on improving output diversity is insufficient for this setting: simulation requires samples that are calibrated to a target distribution, not merely varied outputs. UnpredictaBench isolates a simplified but fundamental version of this problem: sampling outcomes from individual target distributions, including canonical statistical distributions, distributions induced by stochastic programs, and natural-language scenarios that describe random processes. We introduce 448 such problems together with KS@N, a general-purpose evaluation metric that quantifies how well a model outputs approximate black-box target distributions via the Kolmogorov-Smirnov statistical test. This is the rate at which we fail to reject model samples of size N against ground-truth samples, with larger N indicating greater difficulty. Tested across open and proprietary models, we find a large spread in distributional capabilities. For instance, when models generate samples of size 100 (KS@100, our standard metric), scores range from near 0 to over 20%. No model is able to achieve over 40% at KS@100, showing significant headroom in distributional sampling as a capability. Although adding reasoning can somewhat increase scores, we find no immediate solution for this issue. UnpredictaBench shows that even simple distributional simulation remains challenging, making it a necessary first step toward using LLMs as stand-ins for complex systems.

Prospect Theory (PT) models human decision-making behaviour under uncertainty, among which linguistic uncertainty is commonly adopted in real-world scenarios. Although recent studies have developed some frameworks to test PT parameters for Large Language Models (LLMs), few have considered the fitness of PT itself on LLMs. Moreover, whether PT is robust under linguistic uncertainty perturbations, especially epistemic markers (e.g. "likely"), remains highly under-explored. To address these gaps, we design a three-stage workflow based on a classic behavioural economics experimental setup. We first estimate PT parameters with economics questions and evaluate PT's fitness with performance metrics. We then derive probability mappings for epistemic markers in the same context, and inject these mappings into the prompt to investigate the stability of PT parameters. Our findings suggest that modelling LLMs' decision-making with PT is not consistently reliable across models, and applying Prospect Theory to LLMs is likely not robust to epistemic uncertainty. The findings caution against the deployment of PT-based frameworks in real-world applications where epistemic ambiguity is prevalent, giving valuable insights in behaviour interpretation and future alignment direction for LLM decision-making.

Predictable behavior from scaling advanced AI systems is an extremely desirable property. Although a well-established literature exists on how pretraining performance scales, the literature on how particular downstream capabilities scale is significantly muddier. In this work, we take a step back and ask: why has predicting specific downstream capabilities with scale remained elusive? While many factors are certainly responsible, we identify a new factor that makes modeling scaling behavior on widely used multiple-choice question-answering benchmarks challenging. Using five model families and twelve well-established multiple-choice benchmarks, we show that downstream performance is computed from negative log likelihoods via a sequence of transformations that progressively degrade the statistical relationship between performance and scale. We then reveal the mechanism causing this degradation: downstream metrics require comparing the correct choice against a small number of specific incorrect choices, meaning accurately predicting downstream capabilities requires predicting not just how probability mass concentrates on the correct choice with scale, but also how probability mass fluctuates on specific incorrect choices with scale. We empirically study how probability mass on the correct choice co-varies with probability mass on incorrect choices with increasing compute, suggesting that scaling laws for incorrect choices might be achievable. Our work also explains why pretraining scaling laws are commonly regarded as more predictable than downstream capabilities and contributes towards establishing scaling-predictable evaluations of frontier AI models.

Autonomous agents increasingly perform bounded software tasks under an orchestrator that retries, resumes, and budgets them. The machinery such orchestrators reach for is the service mesh's: retry, timeout, and error-rate circuit breaking. We report a failure study of a production agentic software-delivery platform over 147 numbered incidents spanning 81 runs, each with a measured cost and, in most cases, a mutation proof reproducing the failure. All three assumptions those primitives rest on are violated in practice, and we quantify the consequences: a loop of fifty-four consecutive successful tool calls no error-rate breaker could see; a progress signal constant by construction, guaranteeing a false trip on the third repair round and driving one run from six of six components to three; twenty-one events accumulated across six invocations of one delegation, making a correct, idempotent component unwinnable; a misrouted failure that woke five components for a two-component fault, leaving three bystanders regressing working code; and twelve incidents in which the enforcement layer blocked correct work, the most expensive costing 107 agent turns and zero accepted writes. We find one cross-cutting cause and its dual. Identity adequacy: in five separate subsystems an identity that failed to discriminate produced a confident wrong answer, and two of them derived the corrective rule independently. Evidence adequacy: a reliability decision may be taken only on evidence capable of moving, attributable to what it measures, and deterministic under identical conditions. From the findings we derive seven reliability primitives whose enforcement unit is the delegation rather than the message, and specify the controlled evaluation the study motivates but does not constitute.

There has been increasing interest in modeling survival data using deep learning methods in medical research. In this paper, we proposed a Bayesian hierarchical deep neural networks model for modeling and prediction of survival data. Compared with previously studied methods, the new proposal can provide not only point estimate of survival probability but also quantification of the corresponding uncertainty, which can be of crucial importance in predictive modeling and subsequent decision making. The favorable statistical properties of point and uncertainty estimates were demonstrated by simulation studies and real data analysis. The Python code implementing the proposed approach was provided.

Large Language Models (LLMs) are employed across various high-stakes domains, where the reliability of their outputs is crucial. One commonly used method to assess the reliability of LLMs' responses is uncertainty estimation, which gauges the likelihood of their answers being correct. While many studies focus on improving the accuracy of uncertainty estimations for LLMs, our research investigates the fragility of uncertainty estimation and explores potential attacks. We demonstrate that an attacker can embed a backdoor in LLMs, which, when activated by a specific trigger in the input, manipulates the model's uncertainty without affecting the final output. Specifically, the proposed backdoor attack method can alter an LLM's output probability distribution, causing the probability distribution to converge towards an attacker-predefined distribution while ensuring that the top-1 prediction remains unchanged. Our experimental results demonstrate that this attack effectively undermines the model's self-evaluation reliability in multiple-choice questions. For instance, we achieved a 100 attack success rate (ASR) across three different triggering strategies in four models. Further, we investigate whether this manipulation generalizes across different prompts and domains. This work highlights a significant threat to the reliability of LLMs and underscores the need for future defenses against such attacks. The code is available at https://github.com/qcznlp/uncertainty_attack.

Most learning algorithms with formal regret guarantees assume that no mistake is irreparable and essentially rely on trying all possible behaviors. This approach is problematic when some mistakes are catastrophic, i.e., irreparable. We propose an online learning problem where the goal is to minimize the chance of catastrophe. Specifically, we assume that the payoff in each round represents the chance of avoiding catastrophe that round and aim to maximize the product of payoffs (the overall chance of avoiding catastrophe) while allowing a limited number of queries to a mentor. We first show that in general, any algorithm either constantly queries the mentor or is nearly guaranteed to cause catastrophe. However, in settings where the mentor policy class is learnable in the standard online learning model, we provide an algorithm whose regret and rate of querying the mentor both approach 0 as the time horizon grows. Conceptually, if a policy class is learnable in the absence of catastrophic risk, it is learnable in the presence of catastrophic risk if the agent can ask for help.