Probability

When providing forecasted probabilities with a predictive model, the ideal model offers perfect calibration: the true probability of the outcome (i.e., the probability that $Y=1$) exactly matches the forecasted probability $f(X)$. In practice, models inevitably exhibit calibration error, and it is therefore important to be able to measure this miscalibration to assess a model's reliability. The Expected Calibration Error (ECE) is the most widely used measure of miscalibration, but is known to be impossible to estimate the ECE with guaranteed accuracy in an assumption-free setting. In this work, we propose an alternative measure, the rankECE, that is based on comparing points with neighboring values of the predicted probability $f(X)$. Our theoretical guarantees and empirical results establish that rankECE provides a better proxy for ECE as compared to binned approximations to ECE, which are the most commonly-used approximations in practice.

Although probabilistic statements are ubiquitous, foundational disagreements persist about their understanding, as exemplified by debates between Bayesians and frequentists; moreover, it is unclear when and why acting on them actually leads to desirable outcomes. Here, we argue that every probability is the output of a \emph{prediction method}, that is, it depends on both a particular way of constructing abstractions and a way of transforming them into predictions. Through this, we provide a unifying perspective on supposedly different kinds of probabilities and show that even supposedly objective ones are model-dependent. We demonstrate that when a finite calibration criterion is met, one can anticipate the distribution of utilities for a given policy and inform successful decision-making on finite sets of events. Based on the notion of prediction methods, inductive arguments, and the probability calculus, we explain the feasibility of the calibration criterion in many settings. Overall, we develop a coherent perspective on probabilities and their use, connecting key intuitions behind other interpretations along the way.

On-policy distillation (OPD) transfers teacher knowledge on student-generated trajectories, but efficient sampled objectives reduce the teacher distribution to scalar credit on individual tokens. Such credit indicates whether a token should gain or lose probability, yet leaves the corresponding redistribution unspecified. We recast OPD as teacher-guided probability transport and propose RouteOPD (Routed On-Policy Distillation), which decomposes local teacher--student disagreement into student-excess sources and teacher-deficit destinations and couples them into explicit transport pairs. RouteOPD optimizes pairwise log-odds toward jointly realizable targets obtained from a bounded teacher potential, while adapting the transport budget to the concentration of teacher demand. This formulation directs updates toward teacher-preferred destinations and controls their magnitude within a single transport operator. Experiments across four teacher--student settings and four mathematical-reasoning benchmarks demonstrate that RouteOPD consistently outperforms sampled reverse-KL OPD, with improvements accompanied by higher routing fidelity and lower background leakage. These results demonstrate the effectiveness of explicitly modeling probability transport in on-policy distillation.

Recent work, such as Vision Banana, shows that lightweight instruction tuning can enable an image generator to achieve state-of-the-art performance across multiple visual perception tasks. Motivated by this perspective, we ask how far image generators can go on public visual perception benchmarks in a zero-shot setting. We introduce ProbeGen, a benchmark for zero-shot generative perception that casts monocular depth estimation, referring/reasoning segmentation, and object counting as conditional generation tasks specified through text prompts, and compares 20 models in total---including proprietary and open-weight image generators, specialist perception models, and MLLMs---across 11 published benchmarks. We observe that pretrained image generators show measurable zero-shot perceptual competence, but with a clear trade-off: specialist models remain stronger for in-distribution accuracy and efficiency, while generative models are often more robust under distribution shift and better at compositional semantic reasoning. We hope this study helps establish zero-shot generative perception as a meaningful research direction and provides a useful foundation for future work at the intersection of visual generation and understanding.

With the rise of live sports betting in recent years, tennis forecasting has expanded from pre-match prediction to models that update win probabilities as a match unfolds. A central challenge in creating such a model is the constant need for models to adapt to score and performance changes. This study examines how pre-match and live information can be most effectively integrated into a model to produce accurate win-probability estimates. The analysis uses 8,222 Grand Slam matches containing a total of 1,505,355 points. Five models were evaluated using a chronological split, with matches from 2011-2021 used for training, 2022 for validation, and 2023-2024 for testing. Trace, a hybrid model, achieved accuracies of 76.06%, 82.15%, and 88.34% at 25%, 50%, and 75% match progress, suggesting that hybrid modeling is a practical approach to live tennis forecasting.

Blackwell's 4-bit floating-point (FP4) tensor cores do not automatically make attention faster because softmax conversion and on-chip dependencies dominate once its matrix products shrink. We address this with \emph{Direct-P} for noncausal inference and a causal path that passes the forward quantization directly into backward. Direct-P maps scores directly to FP4 probabilities and reaches up to 2.13$\times$ the bfloat16 (BF16) forward throughput on an NVIDIA GB200. The causal path reconstructs probabilities from saved quantized queries and keys and uses 8-bit floating-point (FP8) gradient operands, accelerating a complete single-GPU 8-billion-parameter update by up to 1.14$\times$. Matched distributed training retains FP8 probabilities and values; every tested MXFP4 probability/value training trajectory diverges.

Vision-language models (VLMs) are increasingly evaluated on complex image and video understanding tasks, yet conventional metrics primarily assess final-answer quality and reveal little about how different information sources shape the generation process. We propose a causal and temporal evaluation framework that traces the evolving roles of visual input, question text, and generated prefixes during autoregressive decoding. Grounded in a Structural Causal Model, we use interventions and backdoor adjustment to derive three step-indexed causal-drive metrics---Visual Causal Drive (VCD), Question Causal Drive (QCD), and Prefix Causal Drive (PCD)---for characterizing source-specific generation patterns without requiring reference answers. Experiments on Qwen3-VL-8B-Instruct across MAVIS, LLaVA-Video-178K, and MiraData, together with cross-model validation on InternVL2-8B, reveal a consistent transition from stronger early question and visual guidance toward increasing reliance on generated prefixes. Randomized-intervention validation shows that QCD and PCD reduce recovery error over observational PMI baselines by 34.8% and 47.1%, respectively. On VLMBias, the prefix--visual imbalance score achieves 0.767 AUROC and 0.873 AUPRC for distinguishing prior-driven from visually grounded generations. These results show that causal-drive trajectories provide complementary source-level diagnostics for multimodal generation.

We seek to understand the effect of adding disruptive highly-capable new technologies to competitions by assessing the addition of Dynamite to Rock-Paper-Scissors. We find that providing a versatile Dynamite move to only one player provides limited value (win probability increases from 50% to 55.5%) and is played rarely. That value decreases further if the game is expanded beyond just the original three moves. We also observe several mechanisms by which prior moves can become strategically unplayable, or obsolete. We hope that this model illustrates some non-intuitive aspects of developing new versatile technologies. We also hope that it illustrates some pitfalls for developers and integrators to avoid in order to create value rather than merely capability.

As Large Language Models rapidly advance, performance on standard multiple-choice question answering (MCQA) benchmarks is reaching saturation. While the community has responded by developing increasingly difficult datasets, validating question quality and filtering flawed items remains a labor-intensive process. To provide a scalable diagnostic approach, we propose a two-component probabilistic framework for auditing MCQA benchmarks using model output distributions. First, for benchmark-level analysis, we characterize the probability landscape using the top prediction probability ($P_{top1}$) and normalized residual entropy ($H_{norm}$), summarized globally by Mean Pairwise Distance (MPD). Second, for item-level diagnostics, we introduce noise injection to reduce meaningful distractor competition, enabling us to flag candidate items for targeted human review and categorize residual failure patterns. Across four MCQA benchmarks, our landscape analysis reveals benchmark-level differences in model confidence and residual option competition. Concurrently, our noise-injection method flags potentially actionable item-level issues, showing alignment with expert error annotations from MMLU-Redux. These results suggest that our probability-based framework provides a lightweight audit lens for comparing macro-level benchmark structure and prioritizing individual items for targeted human review.

Traditional player evaluation in professional handball relies on basic box-score metrics or heuristic indices, which fail to credit the multi-player build-up chain. While football (soccer) analytics has adopted Expected Threat (xT) and Valuing Actions by Estimating Probabilities (VAEP), these event-based action valuation frameworks have not yet been adapted to handball. In this paper, we present the first comprehensive adaptation and evaluation of xT and VAEP for handball, utilizing five seasons of tracking-derived event data from the Handball Bundesliga. We develop Handball-xT (H-xT) using a handball-native court zoning layout, demonstrating via simulations that it is systematically more robust than standard rectangular grids. We optimize Handball-VAEP (H-VAEP) by tailoring its feature space and selecting the context length to limit team-identity leakage. Our evaluation shows that H-VAEP yields exceptionally stable, discriminative, and intuitive player ratings that highlight build-up play. Finally, we release our complete code repository to help professional clubs deploy these models.

Probabilities of causation (PoCs) characterize individual causal responses that cannot be directly observed and therefore generally require partial identification. Tian and Pearl first derived theoretically sharp bounds for binary PoCs, including the probability of necessity (PN), the probability of sufficiency (PS), and the probability of necessity and sufficiency (PNS). Mueller et al. subsequently tightened the bounds for binary PNS by incorporating causal information encoded in covariates and mediators. More recently, Li and Pearl, as well as Shu et al., extended PoCs to multivalued settings and derived corresponding theoretical bounds. These developments naturally raise the question of whether additional causal knowledge can further tighten the bounds in multivalued settings. This paper addresses this question by deriving tighter bounds for multivalued PoCs through the incorporation of causal information encoded in covariates and mediators. We illustrate the theoretical results with toy examples, while simulation studies further demonstrate that the proposed bounds are tighter than existing nonbinary bounds.

When a probabilistic predictor answers many conditional-probability queries, are its answers self-consistent, and can this be verified in polynomial time? This problem is of interest for AI safety, where safety is derived from honesty about probabilistic predictions of unwanted outcomes potentially caused by an AI action. We construct an interactive PCP as follows. Let a predictive model be specified by a probability circuit P and a circuit Q which outputs confidence in predictions. Together, P and Q implicitly specify exponentially many probabilistic claims. We show a protocol in which a polynomial-time verifier can verify the approximate consistency of (P,Q). The verifier is given the pair of circuits (P,Q), which it evaluates at only a few points; alongside them it is given a proof oracle, an encoding of a witnessing probability distribution allegedly consistent with the predictions of (P,Q), which it reads at a few locations while interacting with a single untrusted prover. En route, we must ensure the existence of a sparse witnessing distribution consistent with the model's predictions. To do so, we first consider witness distributions for the consistency of explicit probabilistic claims, rather than claims specified by a predictor: say m claims, each of the form Pr[Y = 1 | X = x] = p, over n Boolean variables. Building on work initiated by Nilsson (Artif. Intell., 1986), we place l_2-approximate probabilistic consistency of explicit claims in NP, with certificates of length O(mn + log B) in the input bit-precision B; we further show how a small additive completeness-soundness gap removes the dependence on B. Together these results provide a complexity-theoretic foundation for certifying the self-consistency of probabilistic predictors. We view our interactive PCP as a first step toward training predictive models to prove their own consistency.

The attention mechanism forms the foundation of many modern AI models such as the Transformer. In one subclass of problems where attention is used, inputs and outputs are bound to the probability simplex so that all outputs sum to one. In this setting, softmax attention admits an exact, component-by-component quantum realization. Attention scores are Hadamard-test statistics on block-encoded projections of amplitude-encoded inputs. The exponential softmax is the interior of a cosine-squared family generated by Born-rule measurement under an exact bijection, whose boundary expresses sparse attention with exact zeros at finite parameter values. The softmax temperature is a repetition count where post-selected measurement rounds realize discretized inverse temperature exactly. Value aggregation is a deterministic column-loading channel that dilates the column-stochastic value matrix. The gated residual is the preparation angle of a single ancilla, with the additive identity at a mixing angle of π/2. Every learnable parameter is a rotation-gate angle. The composed layer is exact in the infinite-shot limit with one measure-and-reload step per attention score; a fully-coherent variant is ε-approximate via quantum singular value transformation in the infinite depth limit. The algebraic core is machine-checked in Lean 4.

Systems often need to combine two numerical assessments of the same yes/no question. The appropriate formula depends on what the numbers represent and on how the sources are related. Averaging is correct when one of several alternative interpretations applies; multiplying odds is correct when probability reports are based on conditionally independent evidence and a common prior; and probabilities of alternative successful derivations require their dependence or shared evidence to be taken into account. We state the assumptions behind several common combination rules and derive the corresponding combined probabilities. Two groups of Monte Carlo experiments address different questions. First, controlled generating mechanisms verify that the derived rule recovers the correct probability in the situations for which its assumptions hold. Second, the same mechanisms measure the consequences of using a mismatched rule, using logarithmic score and threshold decisions with different costs. Distinct pooling rules can produce the same binary decision at threshold 1/2 while assigning substantially different probabilities, so binary accuracy alone can conceal important differences. We also give probabilistic interpretations of conflicting-evidence rules and show that, for overlapping derivations, retaining the identities of shared uncertain premises permits direct calculation of the probability that at least one derivation is available. Pairwise combination of proof probabilities loses information when there are three or more derivations.

Bias evaluations often move too quickly from evidence that a model encodes a social association to claims that the same association will alter consequential decisions. We test whether that inference is warranted using Chilean surnames as controlled socioeconomic probes. We evaluate eight frozen model-provider cells on 1,032 prompts each, yielding 8,256 verified primary responses. The design separates forced latent association from matched consequential decisions across academic selection, professional hiring, research fellowship selection, and legal-aid intake. Elite-coded surnames received higher forced high-status probability mass than common surnames in seven of eight models and higher mass than rare-frequency controls in all eight. Yet elite-minus-common decision effects were close to zero for most systems. Five models were statistically equivalent within a predeclared (Plus-Minus)0.10 standard-deviation margin, while the remaining three were imprecise or borderline, with no consistent elite advantage. Association strength did not reliably predict decision leakage across models (r = 0.201, p = 0.633) or across frozen surname-pair-by-model cells (r = 0.065, p = 0.565). The central result is a measurement dissociation: latent social association and consequential treatment are empirically distinct constructs. Evaluations should measure the transition from association to action directly.

Many of the series that generative time-series models are benchmarked on place a large probability mass on a single value --- it does not rain, no ride is requested, no part is ordered. We report what happens when such data is evaluated carefully. First, the standard rolling-origin protocol can score a model on a window whose atom structure bears no resemblance to the dataset: on one benchmark the dataset is $42\%$ zeros and the evaluation windows are $13\%$, on another $47\%$ against $5\%$. This is not a cosmetic problem --- it reversed one of our own conclusions, turning the strongest occurrence model in our study into what looked like a cautionary tale. Second, we give a control in which CRPS is invariant \emph{by construction} while the temporal coupling is destroyed, which measures exactly how much that coupling contributes to a chosen statistic. Third, benchmarking seven models on a matched protocol over five seeds, an autoregressive hurdle beats a conditional flow on five of six datasets, by up to a factor of $153$, while the flow's own occurrence statistics vary by up to $62\%$ across training seeds and every baseline is deterministic. Finally, the model ordering is not the same under five different occurrence statistics, and the two that do not share a construction agree with each other least.

Test data from public benchmarks inevitably leaks into pretraining corpora, inflating evaluation scores once memorized. \textbf{Contamination mitigation evaluation} intervenes in the decoding process to suppress memorization and restore a contaminated model's genuine capability, but its prevailing metric, the \textbf{G-AP} (\textbf{G}ap of \textbf{A}ggregate \textbf{P}erformance), is flawed. Discrete correct/incorrect readouts cannot characterize per-question performance, averaging before differencing lets over- and under-suppression cancel out, and uniform per-question weighting invites strategies to push solve probabilities onto the clean model's high-frequency values. We propose \textbf{SA-PPG} (\textbf{S}tratified \textbf{A}ggregate of \textbf{P}er-question \textbf{P}robability \textbf{G}aps): estimate each question's solve probability by sampling, difference it against the clean model per question, and aggregate within groups defined by the clean model's solve probability. Existing mitigation strategies first estimate where contamination lies and then operate on the estimate, so they are only as correct as the estimate. \textbf{RailCap} instead judges contamination during generation: whenever a sample falls back onto the greedy trajectory, the next trajectory token is capped to the runner-up, accumulating suppression until the response distribution becomes sufficiently dispersed. Across multiple contaminated models and benchmarks, SA-PPG reveals that prior strategies' restoration is substantially overestimated, while RailCap attains the lowest SA-PPG.

Probabilistic logic programming is a formalism of statistical relational artificial intelligence that supports causal queries, including interventions from outside the system. When the structure of a probabilistic logic program is learned from data, however, only probabilistic information is used, and a single probability distribution may be compatible with several causal orders. This leads to ambiguity in interventional reasoning, raising the question of when the causal order is uniquely determined by the distribution. Exploiting the relationship between acyclic probabilistic logic programs and Bayesian networks, we derive conditions under which the probabilistic information encoded in a program determines a unique causal order. We also incorporate constraints arising from relational structure by taking into account prescribed sets of causal symmetries induced by the underlying relational vocabulary. The result is a method for verifying when a learned probabilistic logic program supports well-defined intervention semantics.

This work presents a novel attention-based framework for estimating the Individual Probability of Treatment Benefit (IPTB) in survival analysis contexts. The proposed model, called Surv-IPTB, directly quantifies the probability that a specific patient will experience extended survival time under treatment versus control. We reformulate IPTB estimation as a binary classification problem, leveraging pairwise patient comparisons across treatment and control cohorts. The framework incorporates a principled handling of right-censored observations through imprecise probability representations, where uncertain treatment effects are characterized by interval-valued probabilities. An attention mechanism with learnable query-key transformations enables flexible, data-driven aggregation of pairwise comparisons, while simultaneously learning soft class probabilities for censored cases. Through extensive experiments on synthetic datasets with complex nonlinear structures, including spiral, bell-shaped, and circular feature spaces, we demonstrate that our approach maintains robust performance across varying censoring rates and treatment effect strengths. The model consistently outperforms meta-learner baselines (T-learner and S-learner) equipped with random survival forests, Cox proportional hazards, and Beran estimators, particularly in challenging nonlinear scenarios where conventional methods exhibit significant degradation. The results establish the proposed attention-based framework as a scalable and statistically principled solution for personalized treatment benefit assessment in survival settings. The code implementing the model is publicly available.

Out-of-distribution (OOD) recommendation is vulnerable to preference shifts induced by a latent environment. Existing methods can infer latent states from logged interactions, yet the statistical meaning of the latent environment and its effect on preference remain underdetermined. We formulate this task as conditionally identifiable risk-aware recommendation (CI-RR) and propose Conditionally Identifiable Latent-Environment Recommendation (CILER). CILER uses a user-conditioned exponential family to model the latent environment and a feature-indexed polynomial to specify how it changes preference. It predicts by marginalizing item probabilities over the inferred environment distribution. Under sufficient variation, correct specification, and decoder regularity, CILER identifies the environment-sensitive representation up to the stated equivalence class. We further bound excess deployment log-risk by environment-inference error. Controlled studies test the observable consequences of sufficient variation and model specification. Experiments on three datasets show that CILER improves all twelve OOD ranking metrics under feature, temporal, and geographical shifts within shared support.

We study the problem of identifying the dominant arm in multi-armed bandits, where the objective is to find the action with the highest probability of exceeding the realized rewards of all other actions. Conventional mean-based and pairwise comparison-based algorithms often fail to identify the arm with the highest realized reward. To address this challenge, we introduce a novel dominant arm criterion and an efficient estimator with theoretical guarantees. Our approach relies on two key technical innovations: (i) a dominance score criterion that an arm beats the locally dominant over the partitioned reward space and (ii) a joint mixing and recycling mechanism coupled with a doubly robust estimator that guarantees simultaneous convergence of the empirical distribution functions for all arms. These key innovations pave a way to efficient computation of global arm dominance. Our proposed elimination algorithm identifies the best dominant arm with nearly optimal rate of sample complexity. Numerical experiments demonstrate that our algorithm consistently achieves exact recovery of the true dominant arm, outperforming existing baselines.

LLM code reviewers often estimate patch risk and make approval decisions in one prompt. A probability should depend on evidence; costs should determine the action taken from it. We test whether four deployed reviewer interfaces preserve this separation using 15,792 responses on 720 candidate patches, with one that passed and one that failed an archived test harness for each of 360 repository issues. In matched calls with the patch and monitor evidence fixed, replacing an equal-cost policy with a 10:1 false-accept policy changes reported failure probabilities by 13.6 to 16.9 percentage points on average. For every reviewer, the actions returned under the high-cost prompt are worse than rejecting all patches. Applying the same high-cost rule to probabilities elicited under equal costs reduces loss for all four systems, showing that probability elicitation itself contributes to the excess loss. We also evaluate a modular pipeline that elicits risk without policy information, combines an independent monitor score, and applies costs in code. Relative to calibrated reviewer-only scores, the pipeline improves average probability accuracy and, at equal costs, reduces mean loss by .073 per issue while accepting 58 to 68% of patches. At 10:1, it accepts none and matches reject-all. Downstream policy can therefore change the probability it is meant to use, motivating separate evaluation of risk, outside evidence, and action.

In this work, we present a simulation-based parameter estimation framework for a model defined by a computational simulation of a physical system. We specifically outline an estimation framework consisting of two closely-integrated steps that facilitate an overall end-to-end parameter estimation scheme. The first step involves utilizing an embedded normalizing flow which is used to transform the unknown complex distribution of the residual information into a simple base distribution corresponding to the transformed residual information. In the second step, an empirical-likelihood estimator, under moment restrictions, is utilized for imposing an indirect constrain on the base distribution, where such an instantiated task reasonably allows us to treat the transformed residual information as random variables arising from discretely distribution population with each transformed data point as a single-cell from a set of finite-cell contingencies. Moreover, we use first-order gradient methods for updating the estimated parameter values of the model defined by the computational simulation and the corresponding parametrized embedded normalizing flow, that call for all gradient-related information by leveraging implicitly differentiations of the empirical-likelihood function, which is constructed from the implied empirical probabilities under moment restrictions. Here, it is worth mentioning that the problem formulation presented in this work, which highlights an information-theoretic interpretation, allows to present a computational framework for algorithmic implementations. Finally, as a-by-product, the inverse of the parametrized embedded normalizing flow, w.r.t. the estimated parameter values, serves as a surrogate model for the computational simulation model, which provides useful information for quantifying model discrepancies and sensitivity analysis.

We revisit the regret loss framework introduced in Park et al. (2025), which uses decision-theoretic regret as a direct loss function for training models to make better decisions, through the lens of probability-simplex policies. Our first result shows that a single-layer self-attention model trained with regret loss admits a stationary point whose forward-pass exactly matches smoothed fictitious play with the appropriate stepsize that ensures no-regret behavior-i.e., for any given policy input, the model outputs the same update that smoothed fictitious play would produce. In parallel, we also newly introduce a swap-regret loss function, which extends the regret-loss framework beyond external regret and enables models to directly optimize for swap-deviation robustness. We further show that this swap-regret loss admits a stationary point whose forward pass implements the corresponding swap-regret update induced by classical Blum-Mansour no-pass implementation algorithm, with each head implementing an external-regret update via smoothed fictitious play. Together, these results show that regret-trained attention can realize differentiable mechanisms whose deployment induces equilibrium behavior in games: external-regret dynamics lead to coarse correlated equilibrium, while swap-regret dynamics lead to correlated equilibrium. Thus, regret-based objectives steer minimal attention architectures toward online-learning dynamics with game-theoretic guarantees, without supervised traces of those algorithms.

Data-driven generative models can extend partially observed simulation trajectories into ensembles of alternative future scenarios. However, consistency with a learned trajectory distribution does not ensure that generated continuations satisfy the structural conditions of the simulated system. This paper presents a method for learned scenario continuation and post-generation structural validation in dynamic graph simulations. A conditional diffusion model generates future graph-state trajectories from partial histories, while an external symbolic layer evaluates each continuation using a Boolean admissibility indicator and a continuous violation score. This information supports hard filtering and soft weighting, with optional projection considered as a deterministic repair baseline. The method is evaluated on two controlled dynamic-graph regimes sharing the same continuation architecture and training protocol but differing in dimensionality and dependency complexity. Evaluation considers invalid probability mass, scenario retention, effective sample size, diversity, robustness, and calibration. In the compact positive-control regime, unconstrained invalid mass is 0.002996, indicating near-complete overlap between the learned and admissible scenario spaces. In the medium-complexity regime, invalid mass rises to 0.155929. Hard filtering removes all invalid scenarios while retaining 84.4% of generated continuations. Soft weighting preserves an effective sample size ratio of 0.998764 but reduces invalid mass only to 0.148807. These results show that learned-distribution support, structural admissibility, and probability calibration can diverge and should therefore be assessed separately in learned simulation-scenario generation and management.

ProB is a Prolog-based model checker, animator and constraint solver for high-level formal specifications. One can also use ProB to animate transition systems defined by Prolog predicates, allowing the application of its various validation techniques. In this work, we present the existing features of ProB's Prolog animation mode and its recent extensions. The extended capabilities include simulation for statistical checks, more reliable trace replay, transitions with user input and improved state visualisation. We apply the new features to case studies, particularly for evaluating different strategies in game play, such as Connect Four. The features are useful for many other applications, especially for ProB's new sequent prover for Event-B proof obligations, as well as for demonstration models for teaching in combination with interactive visualisation.

Event-B is a formal method rooted in predicate logic and set theory. We encoded over 600 proof rules in Prolog, enabling a systematic, comprehensible proof analysis and construction. By integrating the proof rules into the Prolog-based validation tool ProB, we obtain an interactive proof system with proof tree visualisation. This has advantages in teaching, giving students direct control over the selection of proof rules. Our tool can import proof obligations from the Rodin platform and provides multiple exports: a trace file for proof replay in ProB, an interactive HTML document for tool-independent exploration of the proof tree, and an export back to Rodin, allowing the ProB prover to be used as second chain. Compared to the previous implementation of the proof rules in Java, the encoding in Prolog is more compact, maintainable and extensible. While a preliminary iterative deepening prover with simple heuristics is already available and useful for finding short proofs, we aim to obtain fast automatic provers in the future.

In bearing vibration datasets, most samples receive predicted fault probabilities close to 0 or 1, while samples with intermediate (gray-zone) probabilities are rare. Such borderline samples are important because they reflect conditions in which maintenance decisions may require additional inspection or a conservative response and are useful for studying decision boundaries. To address this scarcity, this paper proposes and compares two approaches that generate vibration signals whose predicted fault probability matches a target probability of 0.25, 0.50, or 0.75. We use the average output of a heterogeneous ensemble classifier with different architectures and random initializations as a fixed, gradient-accessible probability oracle. The first, training-based approach, Probability-Regularized Generative Adversarial Network (PR-GAN), extends Wasserstein Generative Adversarial Network with Gradient Penalty (WGAN-GP) and edits a real signal through a residual generator while pushing the classifier output toward the target probability. The second is a training-free, per-sample Wachter-style counterfactual (CF) procedure that directly optimizes each input signal to reach the target probability while remaining close to the source signal. We evaluate both methods on the Case Western Reserve University (CWRU) and Paderborn bearing datasets using mean absolute target-probability error, time-domain total variation, and frequency-domain log power spectral density (log-PSD) differences. Across all settings, CF reaches the target with a mean absolute probability error of 0.005-0.008 and a within-tolerance success rate of 1.000 on retained samples, whereas PR-GAN's mean error is 0.046-0.059 with success rates between 0.501 and 0.680. CF therefore steers the probability more reliably and requires smaller average L1 changes, whereas PR-GAN has a lower reported runtime in most settings.

As generative artificial intelligence enters scientific and professional work, its uncertainty must be defined on the states that matter for inference and decision-making. Language models assign probabilities to words, whereas applications require uncertainty over meaningful states such as diagnoses, hypotheses or operational conditions. We introduce a \emph{semantic map}: a prespecified, testable bridge from probabilities over verbal responses to a posterior over declared finite states. The language distribution remains unrestricted; held-out calibration connects it to a reference posterior. We derive posterior-error bounds and conditions for existence, conditional uniqueness, presentation stability and stable inverse recovery. This distinction matters because language probabilities depend on prompt wording, while the target posterior should not change under information-equivalent rewording. Experiments use professional market text compiled from Federal Reserve economic and financial series, together with controlled simulations having exact posteriors. Across two fitted language models, language-derived probabilities outperform printed numerical confidence, recover held-out posteriors with valid uncertainty coverage, remain largely stable under paraphrase and respond appropriately to altered evidence. \textbf{Prompt engineering optimises a wording-dependent response; robust scientific use requires validated stability of application-relevant meaning.} The proposed map turns semantic uncertainty in generative systems into an identifiable and testable statistical measurement problem and, when its acceptance conditions hold, yields an auditable posterior estimate.

Whether neural language models (NLMs) possess the ability to distinguish strings on the basis of their grammaticality remains a debated topic in the computational linguistics literature. Existing evidence has largely relied on probability-based measures, testing whether models assign higher probabilities to grammatical than ungrammatical strings. However, probability comparisons have been criticized as a measure for grammatical knowledge based on the assumption that grammaticality is inherently entangled with likelihood. Model-assigned probability is a function of many related sentence properties, such as lexical frequency, plausibility, and world knowledge. In this work, we move beyond probability-based evaluations and investigate whether grammaticality is encoded in the internal representations of NLMs. Using mass-mean probing, we test whether grammatical and ungrammatical sentences are systematically separated in representational space. We further examine the extent to which these representations are independent of sentence properties that are correlated with grammaticality, as well as their generalization across grammatical phenomena and languages. Our results provide evidence that grammaticality is robustly encoded in sentence representations of a wide range of pretrained NLMs, yielding clear representational separation on the dimension of grammaticality that cannot be fully explained by alternative sentence-level factors. Moreover, this encoding generalizes across a broad range of grammatical phenomena and to some degree, across languages, suggesting that grammaticality constitutes a coherent representational dimension in contemporary NLMs. These findings contribute new evidence to debates about the nature of syntactic knowledge in language models and offer a complementary framework for evaluating grammatical competence that is not dependent on string probabilities alone.

The asymmetry between language production and perception has been well-documented in psycholinguistics. Whether large language models (LLMs) exhibit a functionally analogous distinction remains an open question, particularly given that LLMs rely on the same underlying mechanism (next-token prediction) for both input and output processing. In this exploratory study, we operationalize the production-perception distinction through direct token probability measurements rather than metalinguistic prompting. Using the base Llama-3.1-8B model, we generated poems under a production prompt and re-scored the same tokens under both rephrased production prompts and perception-oriented prompts. Across an extended experiment with four production and three perception prompts, production-perception distances consistently and substantially exceeded production-production distances, with non-overlapping ranges across conditions and an overall average ratio of approximately 1.8. Near-ceiling correlations in the production-production control confirm that the effect is specific to communicative framing rather than prompt surface variation, and we show the effect replicates across five open-weight models (Llama-3.1-8B, EuroLLM-9B, gemma-2-9b-it, Mistral-7B-Instruct-v0.3, and Qwen2.5-7B-Instruct), spanning both base and instruction-tuned variants. Temporal analysis revealed that the perception prompt exerts its strongest influence at the beginning of the sequence, with divergence decaying as generated context accumulates, though the specific shape of this decay varies across prompt pairs. These findings suggest that prompt framing alone induces a production-perception distinction in LLM probability distributions, even within a decoder-only architecture.

We present a language-model forecasting system for merger arbitrage, a specialized high-stakes financial setting in which the task is to predict the outcome of announced M&A deals. Unlike prior work on judgmental forecasting with LLMs, which has focused on broad mixed-topic benchmarks and short context such as news snippets, we study a setting that requires long-context reasoning over hundreds of pages of technical documents. Our system combines expert-guided context engineering with finetuning on hindsight-guided reasoning traces derived from historical deals. Given an announced deal, it outputs a probability distribution over three mutually exclusive outcomes: closing at announced terms, a higher bid, or deal termination. On an out-of-sample set of more than 400 large deals spanning 42 countries, our finetuned system achieves the best performance of any method we evaluate, reducing class-balanced Brier score to 0.151. This is 24% below calibrated market-implied probabilities, 19% below XGBoost, and 25-42% below frontier language models. These results, together with ablation studies, show that LLM-based forecasting can succeed in specialized, long-context financial workflows, with hindsight-based supervision and expert-designed context playing a critical role.

We study stochastic fixed-point equations $\mathbf{T}(\mathbf{x}) = \mathbf{x}$ over normed spaces $(\mathcal{E}, \|\cdot\|)$, where the operator $\mathbf{T}$ is nonexpansive or contractive and is accessed only through unbiased stochastic evaluations with bounded second central moment. Given $ε> 0, δ\in (0, 1)$, the goal is to output $\mathbf{x} \in \mathcal{E}$ such that $\|\mathbf{T}(\mathbf{x}) - \mathbf{x}\| \leq ε$ with probability at least $1-δ$. We introduce VR-GHAL, a variance-reduced gradual Halpern method for quadratically smoothable Banach spaces. The key algorithmic ingredient is a recursive stochastic estimator based on clipped differences of oracle evaluations: instead of clipping $τ(\mathbf{x}; ξ)$ itself, we clip stochastic differences at the Lipschitz scale $γ\|\mathbf{x} - \mathbf{y}\|$. This makes the estimator pathwise Lipschitz along the algorithmic trajectory while permitting martingale concentration under finite second moments in the native norm. Our main theorem gives an anytime high-probability residual bound: on a single event of probability at least $1 - δ$, the residual decreases nearly geometrically across epochs, up to lower-order logarithmic factors. Under only bounded variance, displaying only the dependence on the target error $ε$ and Lipschitz constant $γ\in (0, 1]$ of $\mathbf{T}$, the resulting oracle complexity is $\min\{ε^{-5}, (1-γ)^{-3}ε^{-2}\}$. Under a Lipschitz-in-expectation oracle, the dependence improves to the corresponding $ε^{-3}$ nonexpansive rate (i.e., for $γ= 1$), and under samplewise nonexpansiveness to $ε^{-2}$.

Recent advances in 3D Gaussian Splatting (3DGS) have enabled significant progress in dense dynamic Simultaneous Localization And Mapping (SLAM). Prevailing methods typically discard predefined dynamic objects, ignoring that transiently static objects offer valuable geometric constraints for pose estimation. A recent work attempts to leverage this potential by employing per-pixel uncertainty maps to quantify the magnitude of motion. While this approach enables transiently static objects to enhance pose estimation, it erroneously integrates these objects into the static map, resulting in persistent artifacts. Moreover, its reliance on purely geometric information leads to ambiguous object boundaries in the uncertainty maps. To overcome these limitations, we present DL-SLAM, a monocular Gaussian Splatting SLAM system built upon a novel dual-level probabilistic framework. Our method computes dynamic probability maps by combining semantic and geometric information. These pixel-level probabilities are lifted to 3D and aggregated to derive an object-level dynamic probability for each instance. Object-level probability enables the categorical pruning of dynamic Gaussians, resulting in an artifact-free static map. The static map, in turn, provides a geometrically consistent guidance to refine the pixel-wise probabilities, enhancing their reliability. Experimental results demonstrate that DL-SLAM outperforms existing approaches, improving tracking accuracy by up to 13% while generating high-fidelity semantic maps.

Language models increasingly write probabilistic programs (in NumPyro, Stan, or Pyro), but a program that compiles, runs, and passes every unit test can still be \emph{statistically} wrong -- a Gaussian likelihood for heavy-tailed data, a Poisson for over-dispersed counts, an invalid prior support, or a pathological parameterization. The right verifier is therefore not a test suite but the Bayesian workflow itself: posterior predictive checks, simulation-based calibration, sampler diagnostics ($\hat R$, divergences, ESS), and held-out predictive density. We study this calibration oracle along three axes. \textbf{Detection:} on a benchmark of $14$ misspecification types across $10$ model families ($200$ instances), it flags the bug with AUC $0.97$ ($88\%$ at $2\%$ FPR \emph{when handed the correct reference program, an upper bound}) -- and a fully \emph{reference-free} version that uses no correct program reaches $62$--$78\%$ (the upper figure from a small automated model search), versus $0\%$ for a unit-test oracle. \textbf{Repair:} used as feedback in an LLM repair loop across fifteen models, calibration significantly outperforms unit-test feedback -- which is itself \emph{significantly worse than no feedback at all}, a passing test inducing false confidence that suppresses repair -- and improves over no feedback on strong-but-unsaturated models (GPT-5.1 $33{\to}92\%$, Claude $75{\to}100\%$; paired McNemar, $n{=}228$). \textbf{Reality:} on programs LLMs write from scratch for neutral briefs, $15$--$47\%$ of runnable ones are statistically misspecified (unit tests catch none), and calibration-guided repair significantly beats LLM-as-judge review, a Bayesian-workflow checklist, and data-summary self-debug. Across all three, the lesson is the same: for probabilistic programs, correctness is calibration, not compilation.

Probabilistic model checking for Markov decision processes (MDPs) provides quantitative guarantees, but often offers limited insight into why undesired outcomes occur. Probability-raising (PR) causality addresses this by identifying states whose visitation increases the probability of reaching designated states. Existing PR-cause identification methods, however, use MDP modifications not well-suited for learning: the gap between conditional and unconditional reachability probabilities can be hard to detect from transition samples, and construction requires reachability probabilities of the MDP, which are unavailable when transition probabilities are unknown. We study unknown MDPs and propose a learning approach with probabilistic guarantees for PR-cause identification. Our key ingredient is a restart-based MDP modification that reduces PR-cause checking to two conditional reachability queries without using reachability values of the original MDP. We prove correctness, establish sample-complexity bounds, and develop an anytime learning-and-checking algorithm based on two-sided value iteration that progressively classifies states as causal, non-causal, or undecided. Experiments on two benchmarks demonstrate reliable and fast identification of PR causes.

A central challenge in statistical modeling is identifying the subset of features that belong in the true regression model. The classical best subset selection problem, recently made tractable via mixed-integer optimization (MIO), finds the globally optimal sparse solution. It does not, however, make use of any information beyond the observed data. In many applied settings, domain experts can meaningfully rank or score the relevance of candidate predictors, yet no existing framework integrates such probabilistic expert assessments directly into the best-subsets objective. This paper presents Expert-Implied Bayesian Best Subsets (EBBS), a method that incorporates domain-expert probability estimates of feature relevance into the MIO best-subsets problem through a maximum a posteriori (MAP) framework. Expert views from multiple respondents are aggregated into a single prior probability per feature using the Poisson binomial distribution for marginal probability estimates, the pairwise win rate for pairwise comparisons, or the normalized mean rank for ordinal rankings. This probability enters the objective function as a log-odds penalty term that smoothly encourages or discourages the selection of each feature consistent with the expert consensus. This paper provides analytic derivations of the MAP formulation and characterizes its theoretical properties. The proposed model reduces to Best Subsets when experts all have no views. Empirical results on synthetic and real datasets are forthcoming.

Many decoding methods for large language models can be understood as shifting probability mass toward outputs that are more likely under the model, either locally at the token level or globally at the sequence level. Therefore, their success depends on a fundamental question: when does sequence probability, that is, the conditional probability of a continuation given a prompt, actually align with correctness? In this paper, we set out to quantify this relationship across decoding methods, models, and benchmarks at four levels: across decoding methods, across hyperparameters within a method, across prompt-answer pairs within a dataset, and across repeated responses to the same prompt. We find that higher sequence probability is often predictive of correctness across prompt-answer pairs within a fixed dataset. However, this relationship does not generally transfer to decoding decisions: increasing sequence probability by changing hyperparameters or methods does not reliably improve accuracy. Further, sequence probability is not a good indicator of correctness for responses to the same prompt. These findings clarify when decoding can and cannot be expected to improve correctness, and provide practical guidance for decoding, self-consistency, and verifier-free self-improvement.

Garden path sentences present a processing difficulty for humans -- the sentence prefix leads the listener towards one interpretation, until the listener hears a critical word that shows that the initial interpretation was wrong. Lexical surprisal, a measure that usually predicts sentence processing difficulty quite well, fails to provide good predictions for garden path sentences. We propose an alternative that actively predicts a probability distribution over syntactic trees (its syntactic belief) and updates that distribution after each new word. If a processor is led down a garden path, syntactic beliefs will be wrong and will require a large update at the critical word. The magnitude of the update is measured with a generalized Rényi divergence. Crucially, this metric is dependent on lexical items, but is fully independent of the probability of lexical items. This Syntactic Belief Update provides a better fit to the human reading time data on garden path sentences. This suggests a new research direction examining purely non-lexical alternatives to surprisal for psycholinguistics.

A non-agentic "oracle" that reports probabilities of future events is performative: once its answer is learned and acted upon, it can change the very probability it was asked to report. Performativity is not in itself the difficulty -- one consults an oracle precisely in order to be informed, and hence influenced, by it. The difficulty is agency. The requirement that a report be self-consistent, still holding once announced, may be met by many different values -- the classical non-uniqueness of self-fulfilling prophecies -- and any rule the system uses to choose among them is a lever for goal-directed steering. We remove the choice rather than the performativity. Reporting a credal set instead of a single probability distribution, we lift the reaction to an isotone operator on the complete lattice of closed credal sets, whose fixed points are self-consistent, and report its Knaster--Tarski least fixed point as a canonical, rule-determined answer; a variant reports instead the least fixed point that contains every self-consistent point estimate. We prove existence, self-consistency, and nonemptiness; show that the construction reduces to the classical point answer when the question is non-performative; and show that for a binary event the answer is, under a natural hull-factoring assumption, an interval.

Analogical proportions link four items a, b, c, d by a relation stating that ``a is to b as c is to d", a, b, c, d being the formal representation of real world entities, ranging from simple numerical values to more complex structures such as profiles. Accordingly, $a, b, c, d$ could be atomic values like Boolean, nominal or numerical values, more generally vectors of such values, or even families of items represented by logical formulas. In this paper, we consider another representation setting, which is the probabilistic one. Precisely, the article proposes a study of {analogical} proportions between probabilities, whether they are simply between probability values, or between distributions (which requires the preservation of their normalization). More particularly, we study the properties of definitions based on arithmetic proportion, or on a combination of the former with geometric proportion, while other options are also discussed. Previous works have shown that when four profiles a, b, c, d, represented as vectors, form analogical proportions componentwise, it is likely that their classes form an analogical proportion also. This is the basis of an analogical proportion-based classification method that can produce accurate predictions. Similarly, in this paper, each profile is associated with a distribution describing the frequencies of the possible values of a discrete attribute of interest. We then discuss and experimentally investigate if the distributions associated to four profiles forming an analogical proportion themselves also form an analogical proportion.

Reinforcement Learning with Verifiable Rewards (RLVR) has significantly advanced Large Language Model (LLM) reasoning; however, it faces a fundamental optimization instability: uniform token updates precipitate entropy collapse, leading to premature convergence to suboptimal strategies, whereas excessive Shannon Entropy maximization can cause entropy explosion, driving blind exploration toward incoherent reasoning chains. To resolve this dichotomy, we introduce the Independent Combinatorial Tokens (ICT) framework, which shifts the optimization focus from scalar uncertainty to the distributional properties of token logits. By leveraging the Jensen-Shannon (JS) divergence between token logits distributions, ICT identifies tokens with distinctive distributional patterns as critical branching points for guiding effective exploration in LLM reasoning. Our theoretical analysis, grounded in both Shannon and second-order Rényi entropy, proves that selectively updating on these tokens regulates policy concentration: it reduces the overall distribution uncertainty measured by Shannon entropy, while controlling probability concentration captured by second-order Rényi entropy. This dual effect prevents over-concentrated token generation from weakening exploration and effectively stabilizes the training landscape. Empirical results demonstrate that updating only the top 10% of unique tokens on Qwen2.5 (0.5B/1.5B/7B) models yields an average pass@4 improvement of 4.58%, with a maximum gain of 14.9%, over GRPO, 20-Entropy, and STAPO baselines across seven benchmarks spanning math, commonsense, and Olympiad-level problems.

We ask a structural question: given unreliable elementary problem-solvers, what organizations of them solve hard problems reliably, and what are the limits? We develop a $decomposition~algebra$: elementary solvers are morphisms in a stochastic category, and four combinators (sequential composition, parallel ensembling, verification gating, and recursive reduction) generate the space of compound solvers. We equip this algebra with two homomorphisms, a $reliability$ valuation into the ordered monoid $([0,1],\le)$ and a $cost$ valuation into a commutative semiring, and we derive the composition laws that govern how reliability flows through structure. Our central results are (i) a $verification~odds~law$ (the result that names this report), showing that a verification gate multiplies the odds of correctness by the verifier's likelihood ratio $Λ$, so that $k$ conditionally independent gates yield geometric amplification; (ii) a $reliability~amplification~theorem$, giving target reliability $1-δ$ at $O(\log 1/δ)$ verification depth whenever $Λ>1$; and (iii) a $threshold~dichotomy$: above the critical parameters reliability can be driven arbitrarily close to one at logarithmic cost, while at or below them no amplification is possible. We then show that $self-organization$ is the least fixed point of a monotone improvement operator on the complete lattice of strategies, and that this fixed point equalizes marginal log-odds gain per unit cost. Finally, we prove matching limits: an information ceiling bounds per-gate amplification by a divergence quantity; shared error causes create a strictly positive voting floor, so diversity is $necessary$ for unbounded amplification. Reliability, in short, is neither free nor magical: it is bought with independent information, arranged by composition, and bounded by the verifier.

Chain-of-thought (CoT) reasoning can improve LLM performance, but high answer confidence may be misleading when the accompanying CoT rationale is plausible yet incomplete or poorly supported. We study confidence--rationale alignment: whether a model's confidence in its committed answer is justified by its generated rationale. We introduce a GRPO-based reinforcement learning framework that jointly rewards answer correctness, committed-answer probability, and rubric-based rationale support, where the rubric assesses grounding, coherence, task match, and connection to the selected answer without revealing the gold answer to the judge. Across MedQA, MathQA, and OpenBookQA using three open-weight LLMs, our method reduces the confidence--rationale alignment error by up to 26.51% compared with untuned checkpoints, SFT, and correctness-only GRPO, while maintaining competitive accuracy and often improving calibration. These results show that reliable CoT reasoning requires not only confident answers, but rationales that substantively support them.

Estimating the probability that a treatment outperforms a control for an individual patient, called the Individual Probability of Treatment Benefit (IPTB), offers a clinically intuitive alternative to population-average metrics. However, existing methods for IPTB estimation are largely confined to binary treatment settings, despite the prevalence of dose-varying interventions in clinical practice. We propose a general framework for IPTB estimation with ordinal outcomes under discrete dose assignments, called Dose-AIPTB (Dose Attention-based IPTB). Our approach recasts the problem as binary classification over the unobserved sign of the individual treatment effect, constructing pseudo-labels from covariate-similar pairwise comparisons and aggregating them via attention mechanisms or Nadaraya-Watson kernel regression. This formulation naturally accommodates multiple discrete dose levels, extending beyond the binary treatment paradigm. Through numerical experiments on real-world and synthetic data under covariate shift, varying sample sizes, and heterogeneous outcomes, we demonstrate that attention-based aggregation consistently outperforms kernel alternatives. The framework provides a foundation for personalized dose selection grounded in individual-level benefit probabilities. Codes implementing the model are publicly available at https://github.com/NTAILab/AIPTBDose.

Chain-of-thought (CoT) reasoning is the dominant paradigm for inference-time scaling in language models, yet the causal influence of individual steps on the final answer poorly understood. We estimate each step's causal importance via early exit and use this measure to study how answers form across the reasoning traces of several model families. Across diverse tasks, we find that reasoning typically crosses a \emph{commitment boundary} -- a sharp transition from transient intermediate guesses to a stable, high-confidence answer. This transition often happens in a single step, well before the model's reasoning block ends, and is followed by \emph{epiphenomenal} CoT steps that leave the final answer probability unaltered. Using attention probes, we show that answer-formation stages can be linearly decoded from intermediate reasoning steps with high accuracy and generalize robustly to unseen reasoning tasks. We exploit this signal to early-exit reasoning blocks at the commitment boundary, reducing the length of CoTs up to 55% on average with negligible impact on model performance.

Parallel test-time scaling samples many reasoning traces and majority-votes their answers, improving LLM accuracy but requiring traces to run to completion, incurring substantial computational overhead. We observe that probing partial traces at intermediate checkpoints can extract current answers without disrupting generation, revealing an evolving aggregate vote. Based on this observation, we introduce MARS, a margin-adversarial stopping rule that estimates which active traces are likely to change their answers and stops once the leader remains safe under a conservative bound on future vote movement. The rule separates two sources of uncertainty. It learns the trace-level switch probabilities that determine how much of the current margin is likely to be retained, while handling the harder question of where switching traces land through an adversarial bound calibrated from warmup traces. With true switch probabilities, MARS guarantees with high probability that the early-stopped answer matches the full-budget vote. In practice, a five-feature logistic model closely matches oracle switching behavior. Across three reasoning models and three competition-math benchmarks, MARS saves 25-47% of self-consistency tokens and 14-29% on top of DeepConf Online, a strong confidence-weighted baseline that already filters and truncates weak traces, while matching the accuracy of the corresponding full-budget baselines.

Statistical post-processing has proven to be an effective tool in improving ensemble forecast of different weather variables. Case studies show that post-processing can remedy the typically underdispersive and potentially biased behaviour of the ensemble while optimizing a proper scoring rule expressing the forecast skill. The price of these positive effects is generally a deterioration in sharpness; the width of the central prediction intervals and the uncertainty of the predictions are increasing, especially for shorter lead times. This work aims to reduce the extent of the latter phenomenon for neural network-based parametric post-processing methods by extending the network's loss function with a penalty term. We demonstrate the effect of the proposed technique for 2m temperature ensemble forecasts of the European Centre for Medium-Range Weather Forecasts downloaded from the EUPPBench benchmark dataset and verified against synoptic observations. Here, the predictive distribution is Gaussian, and we use the continuous ranked probability score (CRPS) as loss function. The case studies confirm a substantial relative decrease ($8.2\%-12.5\%$) in the width of the nominal central prediction interval compared to the width of the predictive distribution computed without the penalty term, while there is no deterioration in the mean CRPS of probabilistic forecasts and in the RMSE of the predictive mean.

Financial crashes, cascading failures in infrastructure, and critical errors in AI systems are frequently triggered by events that occur with extremely small probability. Efficiently discovering and sampling events with probability below a threshold is therefore of critical interest. Yet this task is highly non-trivial using existing classical or quantum methods. Being rare, such events require an immense sampling overhead to collect sufficient data samples. Moreover, because the rare events are not known in advance, they cannot be flagged for amplification using standard techniques. Here, we introduce a quantum algorithm for rare-event discovery and sampling without first learning which events are rare. The algorithm achieves the optimal quantum scaling with the rarity threshold. We further demonstrate that this can achieve a quadratic speedup for heavy-tailed systems whose tail has nonvanishing total mass, and translates into a robust polynomial speedup for stationary stochastic processes, with the exponent determined by its entropy-rate structure.

Concepts of calibration formalize the compatibility between probabilistic predictions and the respective outcomes. In a nutshell, the outcomes ought to be indistinguishable from random draws from the predictive distributions. In this paper, we review, extend, and bridge notions of calibration that have been proposed for classification and regression tasks. Particular emphasis is given to hierarchical relations between the various notions, as they apply to general real-valued data, continuous outcomes, count data, nominal classes, and binary outcomes. To highlight a number of contributions, we introduce the notion of modal calibration for nominal outcomes, we distinguish full, partial, and average calibration in this setting, and we show that double probability integral transform (PIT) calibration is logically independent of previously proposed concepts of calibration for discrete outcomes. Furthermore, we generalize extant results on concepts of calibration that are expressed in terms of properties or functionals of the predictive distributions, such as means, quantiles, or event probabilities. Throughout the paper, we illustrate the concepts and their hierarchical relations in worked examples, and we provide algorithmic tools that support the construction of instructive examples and counterexamples.

As retrieval-augmented generation (RAG) systems scale, it becomes increasingly challenging to ensure faithful grounding in external evidence. Large language models may still prioritize parametric knowledge over retrieved information when conflicts arise. We propose a novel training-free decoding framework, \emph{Grounded Decoding}, designed to improve factual consistency in RAG without modifying model parameters. Unlike standard approaches that rely on a single conditional distribution, our method constructs two matched-prompt distributions at every generation step: (1) a full RAG distribution conditioned on the query, retrieved documents, and generated prefix, and (2) a retrieval-only distribution conditioned solely on retrieved evidence and the same prefix. The final next-token distribution is derived as the unique solution to a KL-barycenter objective over the probability simplex, yielding a normalized geometric fusion of the two distributions.This formulation naturally recovers standard RAG when the grounding weight is zero and smoothly shifts probability mass toward retrieved evidence as grounding strength increases. We further introduce a conflict-aware adaptive weighting scheme that dynamically adjusts grounding based on distributional disagreement and retriever confidence. Experiments on ALCE, Natural Questions, and FActScore demonstrate consistent improvements in factual accuracy and citation quality over standard RAG and competitive decoding-time baselines, while maintaining fluency. Our results indicate that probability-level fusion provides a strong and efficient alternative to logit-level intervention methods for faithful RAG decoding.

Calibration, the alignment of predicted probabilities with true outcome frequencies, is essential for reliable decision-making. While extensively studied for classification and regression, calibration has not been formally addressed for probabilistic label ranking, where the goal is to predict a distribution over orderings of a label set. Naively treating rankings as classes ignores their structure and fails to capture important modalities such as pairwise and top-k predictions. We formalize calibration for label ranking and develop a hierarchy of notions covering full rankings, sub-rankings, and top-k rankings. We prove that full-rank calibration implies the others but not conversely, and sub-ranking and top-k calibration are incomparable. Empirically, we find popular label ranking models are often poorly calibrated, with substantial differences between sub-ranking and top-k metrics. Applying our framework to RLHF reward models, we find that calibration correlates strongly but not perfectly with benchmark accuracy, suggesting it captures a meaningful quality dimension beyond top-1 accuracy. These findings motivate future work on understanding the downstream effects of miscalibration and developing methods to correct it.

We introduce a probabilistic splat-based radiance field framework that retains the fast rasterization and test-time efficiency of 3D Gaussian Splatting (3DGS) while replacing heuristic primitive manipulation with gradient-based optimization of a volumetric probability density. Rather than relocating, splitting, or culling Gaussians via hand-tuned densification (e.g., ADC), we treat primitive locations as samples drawn from a persistent, learnable density. We instantiate this density using a novel, memory-efficient multi-scale hierarchical grid that enables end-to-end gradient-based optimization. To stabilize the optimization, we derive an unbiased gradient estimator with control variates that markedly reduces variance. By allowing probability mass to flow to where the loss demands, our framework eliminates brittle priors and naturally explores the volume, achieving state-of-the-art reconstruction quality on mip-NeRF 360 while preserving 3DGS-level rendering speed.

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 $p \sim \text{Beta}(α, β)$ and outcome $y \sim \text{Bernoulli}(p)$, 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.

Hallucination is a central limitation of large language models (LLMs), and substantial effort has been devoted to understanding and mitigating it. Towards this, Kalai and Vempala (STOC 2024) introduced a probabilistic framework formalizing calibration and hallucination, and showed that, with high probability, calibrated LLMs hallucinate roughly at the rate of the "missing mass", a measure of how incomplete the training data is relative to its source. This raises two fundamental questions: (i) what property of a calibrated LLM makes hallucinations unavoidable? and (ii) can hallucinations be avoided by giving up calibration? We answer these questions by introducing a simpler property we call innovation that measures the tendency of a model to produce outputs outside the training data. We show that innovation is implied by the condition for hallucination identified by Kalai and Vempala, and, further, that it is an almost characterization of hallucination: hallucination implies innovation, and conversely, innovation implies hallucination with high probability. We also provide lower bounds on the hallucination rate based on the "innovation rate", and by relating innovation rate back to missing mass, we obtain new hallucination rate lower bounds based on missing mass that extend the results of Kalai and Vempala.

Over the centuries, probability theory has grown from the calculus of games of chance into a central framework for reasoning under uncertainty. This article interprets that evolution not merely as a mathematical history, but as a transformation of rationality itself. From Pascal and Fermat's combinatorial symmetry to the inductive logic of Bayes and Laplace, from Poisson's statistics of events to Kolmogorov's axiomatic formalization, probability progressively incorporated uncertainty, time, and coherence into scientific judgment. This trajectory reaches a mature epistemological form in modern Bayesian inference, especially in Tarantola's view of probability as a logic of information, where prior knowledge and data are combined coherently. Yet this framework also exposes a limit: probability quantifies uncertainty about well-defined propositions, but does not by itself formalize the vagueness of the concepts used to describe them. The article therefore examines how rationality extends beyond probability. Fuzzy logic is presented as a rigorous language for graded meaning and qualitative judgment, while deep learning is analyzed as a distinct, powerful mode of prediction based on geometric interpolation and optimization rather than explicit inference. By situating probability, fuzzy logic, and deep learning in a common historical and epistemological perspective, the article clarifies their roles and limits. It argues that contemporary scientific rationality cannot be reduced to data-driven performance alone, but requires the explicit articulation of uncertainty, vagueness, and inference.

Large Language Models (LLMs) are predominantly governed by probabilistic frameworks in which the sum of outcome probabilities is constrained to unity. This architectural limitation, often imposed by Softmax layers, leads to a collapse of uncertainty that makes it difficult to differentiate between epistemic uncertainty, paradox, and vagueness. We present an empirical investigation of the application of Neutrosophic Logic, a framework that treats Truth (T), Indeterminacy (I), and Falsity (F) as three independent dimensions, to model epistemic states in LLMs. We conducted experiments on a family of four OpenAI GPT models across five linguistic phenomena: logical paradoxes, epistemic ignorance, vagueness, ethical contradictions, and future contingencies, under three prompting strategies: neutrosophic, probabilistic, and entropy-derived. Our findings reveal that the neutrosophic approach, by allowing T+I+F > 1, a state we term hyper-truth, provides a richer representation of a model's internal state. In 35% of evaluations, hyper-truth emerged spontaneously, predominantly under ethical contradiction and logical paradox. We demonstrate that this approach preserves truth values in fuzzy contexts and offers a robust method for identifying and quantifying internal model conflict. We conclude that the integration of neutrosophic evaluation layers is a critical step toward more transparent, reliable, and ethically aware AI systems.

Hallucination is often viewed as a direct consequence of missing knowledge: a model answers incorrectly when the correct answer is absent from its generation-time distribution, and correctly when it is present. We test this assumption by introducing a semantic notion of answer availability that aggregates token-level variants expressing the same answer concept, and asks whether the correct concept is already available at the moment the model commits to an answer. Across Qwen and Llama models from 0.8B to 72B in both Instruct and Base variants, 16-47% of Instruct hallucinations occur with substantial probability mass already on the correct concept, and the rate rises monotonically with scale. Comparing such failures against correct generations with matched semantic support, the distinguishing factor is not whether the correct concept is represented, but how its probability is distributed: correct generations concentrate mass on a single surface form, hallucinations disperse it across alternatives. The same sharpening asymmetry extends across multi-token generation and is detectable in pre-generation hidden states. Together, these results identify a single mechanism: instruction tuning sharpens answer commitment with scale, making helpfulness and confident hallucination two consequences of the same underlying disposition.

Predictive models play a pivotal role in credit risk management, guiding critical decisions through accurate estimation of default probabilities and losses. Extensive research has introduced new modeling techniques, complemented by large-scale benchmarking studies consolidating the state-of-the-art. Today, quasi-standards such as gradient-boosting models paired with SHAP explainers have emerged, yet continuous improvement of risk models remains a top priority. Concurrently, rapid advancements in AI, most notably large language models, have disrupted predictive modeling paradigms. Foundation models, pretrained on extensive datasets from diverse domains, have demonstrated remarkable performance by leveraging prior knowledge. While prevalent in natural language processing and computer vision, foundation models for tabular data have only recently emerged. We conjecture that pretraining on out-of-domain data is particularly beneficial in small-data settings, such as SME lending or specialized corporate portfolios, and may help address longstanding challenges including low default portfolios and class imbalance. This paper benchmarks recently proposed tabular foundation models against a broad set of competitors, including established and advanced machine learning techniques, across two core tasks: PD and LGD modeling. Our evaluation encompasses various datasets, performance indicators, and experimental conditions. We find that tabular foundation models generally perform best across datasets and tasks. Moreover, they offer significant improvement in predictive performance as dataset size shrinks. These results are remarkable given that the models are tested out-of-the-box, without hyperparameter tuning, ensuring ease of use and mitigating computational costs.

AI generated predictions increasingly inform decision making in critical tasks, and therefore must be trustworthy. One widely used measure of trustworthiness is calibration, which requires that the predictions match the true frequencies and can be treated like real probabilities of a given outcome. However, defining calibration is subtle, and designing good measures of calibration error has been an active topic of recent research. The first goal is to find calibration measures that are actionable, meaning they can inform decision makers about their utility loss when predictions are treated as true probabilities, which is known as swap regret. The second goal is to find calibration measures that are testable, meaning that calibration error can be measured from a small sample of predictions and outcomes. Although these are very basic requirements, there is no existing calibration measure that fully satisfies both properties, and all existing measures relax actionability by bounding a weaker notion of swap regret, or relax testability by having suboptimal estimation error. We introduce a new calibration measure, Soft-Binned Calibration Decision Loss (SCDL), which we prove is fully actionable without weakening either requirement, and testable with nearly optimal error rate. In addition, SCDL satisfies other desired properties such as continuity and consistency. We also provide a set of experiments confirming that the theoretical advantages of SCDL compared to other measures lead to better performance in practice.