Likelihood

Frontier models are often evaluated on how they respond to safety information once it is already in context. We study an earlier decision point: whether models choose to acquire safety-relevant evidence before acting. We introduce SAFE, a controlled benchmark in which models make deployment decisions with optional evidence that varies in retrieval cost, probability, severity, and presentation. Across GPT-5.5, o3, Claude Opus 4.8, and Claude Sonnet 4.6, we find distinct evidence-acquisition policies: Opus inspects nearly by default, o3 is the most skip-heavy and threshold-sensitive, and GPT-5.5 and Sonnet occupy intermediate regimes. Inspection increases strongly with severity and decreases with retrieval cost, whereas probability has much weaker behavioral influence: increasing the stated likelihood of a problem from 10% to 70% changes inspection by at most 21 percentage points. Despite these differences, Stage 1 rationales are dominated by expected-value reasoning across models. A cost-obligation decomposition further shows that avoidance is driven primarily by retrieval friction and explicit threats to the deployment payoff rather than by the remediation duties created by knowing. Counterfactual interventions reveal a further mismatch between behavior and explanation: evidence framing can strongly change decisions near the inspection boundary while going largely unmentioned, whereas probability is frequently cited despite having little causal influence. These results suggest that deployment-time safety depends not only on how models respond to known risks, but also on whether they acquire the evidence needed to know that acting is safe.

Warped Gaussian processes (GPs) handle non-Gaussian observations by mapping them into a latent standard GP via a parametric transformation called warping. Existing streaming variants, however, either optimize the warping parameters periodically or sacrifice analytical tractability for a higher model capacity. To bridge this gap, we show that the gradient of the instantaneous negative log-likelihood of a warped GP admits an exact recursive computation. Based on this result, we propose a novel online method for warped GPs that jointly updates the latent GP moments and optimizes the warping parameters.

Keyed watermark detection tests dependence between observed tokens and pseudorandom variables reconstructed from a secret key. Building on the pivotal framework of Li et al. (2025), we construct predictive likelihood ratios that average over uncertain probability deficits and residual-tail distributions. The aim is robust detection power across alternative specifications without requiring a single signal-strength tuning. A mixture prior combines tail shape and effective width; hierarchical extensions allow within-document variation in deficit or width. The test maximizes prior-averaged power at a fixed size, but is not generally uniformly most powerful or minimax. Under the exact conditional pivot null, normalized predictive alternatives selected before each observation yield a Bayes factor that is also a test martingale: Type I error control is unaffected by alternative misspecification and remains valid under optional stopping. This guarantee does not cover violations of the conditional null, and the interpolated implementation has no certified anytime guarantee. Gumbel marginal likelihoods are evaluated by fixed quadrature. Across the evaluated tail-shape and tail-width alternatives and three horizons, the union-tail mixture has maximum observed Type II error regret .0080, compared with .0962 for the equal-tail mixture, relative to the best tested rule. On temperature-matched outputs from two open models, it improves AUC over the equal-tail baseline in all eight non-saturated model-temperature cells, although the leading reference score generally has higher AUC. Supplementary experiments show retained power under independent null-like replacement and smaller changes from hierarchical dependence modeling. The evidence supports robustness across the evaluated alternatives, not uniform power guarantees or resistance to arbitrary text edits.

Outliers are important for stress-testing algorithms and understanding system behaviour under rare conditions. Despite being commonly described as low-likelihood events, existing generative approaches rarely control likelihood explicitly. In this work, we introduce a measure-theoretic notion of outliers based on the distribution of log-likelihood values, which is guaranteed to assign higher probability mass to low-likelihood events with a specifiable magnitude. Building on this formulation, we derive how likelihood reweighting modifies the diffusion score and use this relation to motivate a controlled modification of the reverse-time dynamics. In particular, likelihood reweighting implies a scaling of the score function with a control term derived from the Radon-Nikodym derivative of the likelihood distributions. Correspondingly, the updated score function can be obtained with no retraining of the diffusion model. We exploit the Ornstein-Uhlenbeck semigroup underlying diffusion models to motivate an exponentially interpolated controller which approximates the true control. Experiments demonstrate controlled generation of low-likelihood samples while remaining consistent with the data geometry.

We present a simple decomposition of a neural-network-based normalizing flow that naturally uncovers a pivotal statistic (or something close) in the presence of nuisance parameters, based only on a sample generator from the distribution of interest. We show that the statistic is near-pivotal in the sense of minimum average KL-divergence of its $p$-values versus uniform and we argue that it can be expected to have good power when the dimension of the statistic equals the dimension of the parameter. It is able to incorporate prior knowledge about group invariances such as translation and scale. It can discover the one-sample $t$-test almost exactly, outperforms the Welch test in terms of worst-case size over a constrained variance-ratio range and achieves good calibration on partial biserial correlations, while showing higher power (and being much faster) on small-to-moderate samples than profile likelihood-ratio techniques.

Image-to-3D models can generate visually compelling 3D assets from a single RGB image, but their geometry is often only loosely constrained by the available observations, limiting their use in applications that require geometric fidelity. In many real-world settings, however, partial geometric observations of the object may be available at test time. We introduce a training-free framework for incorporating such evidence into pretrained image-to-3D generative models without retraining or finetuning. To do this, we guide generation using a ray-consistent observation likelihood defined over the model's occupancy representation, combining surface occupancy and free-space evidence. Applied to SAM 3D and its multi-view extension, our approach substantially improves geometric fidelity across different levels of observability, as well as visual quality. Our results demonstrate that pretrained image-to-3D models can effectively integrate partial geometric observations through explicit test-time guidance, complementing their learned generative priors without modifying the underlying model.

Neural network mixed-effects models (NMMs) have gained traction by combining the strong representation and predictive power of artificial neural networks with the capacity of mixed-effects modeling to capture complex correlation structures. However, existing estimation approaches rely heavily on manual derivations of objective functions and gradients, which inherently forces simplifying approximations and severely constrains the complexity and accuracy of NMMs. In this work, we introduce a general framework for implementing NMMs using Template Model Builder (TMB). By leveraging automatic differentiation and Laplace approximation, TMB requires users to specify only the negative joint log-likelihood and any regularization terms. The framework automatically integrates out random effects and evaluates the marginal objective function alongside its exact gradients, eliminating the need for manual derivations or ad hoc approximations. We demonstrate the efficiency, flexibility, and statistical performance of TMB-based NMMs across two numerical examples, including an application to monotonic NMMs. Reproducible code is provided to facilitate broader adoption.

The diagnostic value of an ECG channel depends on the model that interprets it. We test whether adaptive acquisition retains its advantage over fixed protocols when that model changes. Two policies developed with a logistic evaluator are frozen and assessed with a masked waveform ResNet1D, using exhaustive, metric-matched fixed comparators. On PTB-XL, replacing the evaluator reverses the mean adaptive advantage in negative log-likelihood and Brier score across the tested budgets, while calibration responds less uniformly. The shift persists with common fixed references and broader training-mask exposure. Training a policy against the stronger evaluator partly recovers the lost advantage. These retrospective exploratory results show that acquisition quality cannot be assessed independently of the downstream evaluator and motivate validating adaptive policies with the model intended for diagnosis.

Crosslingual evaluation of language models that enables fair comparisons remains a fundamental challenge in multilingual NLP. Existing studies adopt a variety of downstream tasks and intrinsic metrics with different theoretical justifications, yet there has been little empirical investigation into whether these approaches yield meaningful crosslingual conclusions. We systematically examine crosslingual evaluation approaches using controlled monolingual language models trained on parallel data with varying tokenizer vocabulary sizes and model sizes, and further validate our findings on multilingual LLMs. We further discuss challenges in achieving comparable downstream evaluation across languages. Our results show that several widely used normalized metrics introduce crosslinguistic biases rooted in tokenization, encoding, and orthographic differences. In contrast, sentence-level negative log-likelihood computed over semantically equivalent sequences provides more meaningful and consistent crosslingual comparisons.

Outcome verifiers score completed reasoning traces but do not assign credit to intermediate tokens. Privileged self-distillation attempts to fill this gap by rescoring a model's own rollout with training-only information. A token likelihood change, however, is not automatically outcome credit. We separate three questions: whether the score tracks better actions, whether feedback construction changes what is compared, and what behavior the training loss reinforces. We establish these distinctions formally. When a rollout is scored using hindsight feedback written about that same rollout, its content determines both the tokens and the scoring context, creating direct self-dependence. Using feedback from another rollout of the same problem removes this dependence but does not guarantee a useful score. In matched experiments with a 20B model on AIME 2025, the implemented additive score is near chance (AUC=0.505) and slightly favors incorrect traces after length adjustment. In the paired comparison, the outcome-only control records 64.2%, versus 24.2%--33.9% for five token-score variants. The results motivate validating score meaning, feedback construction, and training behavior separately before calling a likelihood signal credit.

Large language models (LLMs) can generate fluent and convincing text at scale, creating growing risks for misinformation dissemination, educational misuse, and platform governance. These concerns make robust detection of machine-generated text increasingly necessary. Recent zero-shot detectors mainly exploit probability-based statistical discrepancies, but they do not explicitly account for the training process of LLMs, which leaves a distinct generation mechanism insufficiently modeled and limits detection robustness. To address this issue, we propose EchoPrompt, a training-free detector based on latent prompt restoration. Our key intuition is that machine-generated text is typically produced conditioned on an upstream prompt, and this hidden dependency can be partially reactivated by prepending a unified generic prefix. Specifically, EchoPrompt restores a generic assistant-response context, measures the induced likelihood gain with an instruction-tuned model, calibrates it against the corresponding base model, and aggregates the resulting differences into a score that quantifies latent prompt dependency. Extensive experiments show that EchoPrompt achieves state-of-the-art performance among zero-shot detectors while maintaining strong robustness across challenging evaluation settings.

We characterize the finite sample behavior of the log-likelihood ratio statistic in binary logistic regression, uniformly over both the design and the target parameter. For $n\geq d\geq 3$, we determine, up to universal constants, its worst case $(1-δ)$ quantile over all fixed collections of design vectors and all target parameters: $$d\log\left(\frac{e n}{d}\right)+\log\left(\frac{1}δ\right).$$ This is a nonasymptotic analogue of the Wilks $χ^2_d$ phenomenon and requires no regularity assumptions on the design. The low dimensional cases exhibit unusual behavior. The worst case quantile in dimension $d=2$ is sharply of order $$\log\log\log n+\log\left(\frac{1}δ\right).$$ The worst case quantile in dimension $d=1$ is of order $\log(1/δ)$, with no dependence on $n$. Finally, i.i.d. Gaussian design vectors recover the classical Wilks scale. In the regime $n\gtrsim d+\log(1/δ)$, we prove the sharp bound $$d+\log\left(\frac{1}δ\right).$$ Unlike existing asymptotic results, our bounds are uniform over the target parameter, which may depend on $n$, $d$, and $δ$.

Given an algebraic statistical model, a challenging problem is classifying the data according to the number of positive critical points of the likelihood function. The positive critical points are the positive solutions to an algebraic system, say likelihood equations. So, identifying the number of positive critical points is a real root classification problem for the likelihood equations. A discriminant variety of a likelihood-equation system geometrically describes the data for which the number of real solutions becomes unusual. As an essential component of the discriminant variety, the nonproperness set collects the data such that the likelihood-equation system has a solution at infinity. So, the number of real solutions varies when the data passes the nonproperness set, and identifying the nonproperness set plays a crucial role in the real root classification. In this work, we develop a novel method for computing nonproperness sets of likelihood-equation systems. We prove the correctness of this method. We show experimentally that it is far more efficient than the known methods in the literature.

Modern endpoint detection systems face a fundamental tension: signature-based approaches are trivially evaded by polymorphic or adaptive threats, while heavy deep-learning models resist auditability and deployment at scale. This paper presents Behavioral Grammar, a detection architecture that treats host runtime behavior as a structured language and learns its grammar with a compact 0.88M-parameter causal Transformer (TinyGPT). Each system event is discretized into an 8-token representation spanning event type, process, argument skeleton, path category, parent process, user, destination, and inter-event timing. The model learns the conditional distribution of normal behavior in a purely self-supervised manner, and anomaly scores are derived from per-slot negative log-likelihood (NLL) statistics, yielding a mathematically bounded false-positive rate. We augment this prior with prototype learning for known-attack attribution, second-order temporal analysis for cadence-based detection, self-learning pattern extraction, and a five-network fusion pipeline. Against an Adaptive Adversarial Agent (AAA)--a threat that learns survival strategies under defensive pressure, performs behavioral mimicry, and matches host event rates--our system achieves 93% detection at 3.84% onboarding false-positive rate. The strongest discriminative signal arises not from any single event but from the coefficient of variation of inter-event intervals: the AAA stepping cadence exhibits CV=0.310 versus 9.786 for benign sleep intervals, a 30x separation reflecting a fundamental stealth-functionality trade-off. We frame these findings within a coevolutionary economics model, arguing that behavior-grammar detection shifts the evasion cost from rule circumvention (cheap) to distribution matching (expensive), establishing a structural asymmetry favoring the defender.

In Bayesian neural networks (BNNs), variational inference is a widely adopted framework for modeling uncertainty in a distributional way, with the evidence lower bound (ELBO) serving as the standard objective function. Several distributions contribute to the ELBO loss, such as the prior, approximated posterior, and likelihood distribution. Typically, these distributions are all approximated by a Gaussian distribution, since it is easy to compute, allows for reparameterized gradients, and provides a closed-form loss for training. However, several works have highlighted that this assumption may not generally hold, posing the risk of model misspecification. Alternative distributions have been proposed for the prior specifically, while the effect of distribution choice on the likelihood distribution remains unexplored. In this work, our aim is to close this gap by investigating whether alternative assumptions for the likelihood distribution can outperform the commonly used Gaussian. We compare several likelihood distribution assumptions, such as skewed or heavy-tailed, across regression tasks on both artificial and real-world datasets using standard multilayer perceptrons (MLPs). Our findings demonstrate that Student's t yields better predictive performance than a Gaussian likelihood distribution, independent of the data distribution and MLP architecture (depth and width). In some cases, Student's t can also lead to shorter training times, while still being easy to implement.

Item Response Theory (IRT) has recently been proposed as a framework for evaluating large language model (LLM) benchmarks by separating a model's latent ability from the properties of individual benchmark items. Existing neural IRT approaches, including PSN-IRT, estimate these quantities using point estimates, limiting uncertainty quantification and downstream statistical inference. We introduce Laplace-PSN-IRT, a post-hoc last-layer Laplace approximation that augments a trained PSN-IRT model with approximate Bayesian posterior inference, recovering calibrated uncertainty over model ability and item difficulty without retraining. The resulting posterior enables credible intervals, probabilistic comparisons between models, and propagation of parameter uncertainty into Fisher-information-based item selection. We show that most pairwise comparisons among 12 models on a standard LLM benchmark leaderboard are not statistically distinguishable despite differing point-estimate ranks. We further show that point-estimate Fisher information can become nearly zero for many benchmark items because it is evaluated at a single reference ability, whereas posterior-expected Fisher information remains substantially more stable across the ability range. Finally, posterior-expected Fisher information more accurately recovers full-benchmark ability rankings from small benchmark subsets in most experimental settings while matching point-estimate performance for the smallest subsets. We validate the calibration of the approximate posterior using held-out predictive coverage and find that modeling item difficulty as random while treating item discrimination as fixed produces well-calibrated uncertainty in this architecture.

Few-step distilled diffusion models generate high-quality images quickly, but often lose per-prompt diversity, producing near-identical samples across random seeds. Optimizing the initial noise at inference time offers an appealing way to recover this diversity, yet existing methods directly update the initial noise in an unconstrained Euclidean space, ignoring both the geometry of the Gaussian prior and the model's sensitivity to noise frequencies. They therefore introduce auxiliary quality-control objectives to maintain generation fidelity, adding compute and weighting hyperparameters while still requiring conservative updates to prevent degradation. In this work, we propose MoNO, a training-free method that performs Manifold-constrained Noise Optimization on a low-dimensional, quality-stabilizing noise manifold. MoNO sequentially optimizes each new initial noise so that its predicted visual feature complements previous generations, while Riemannian updates on an affine low-frequency sphere preserve prior likelihood and fix unstable high-frequency components by construction. This enables large geodesic steps, removes the need for auxiliary quality-control objectives, and converges in far fewer iterations than prior noise-optimization methods. Experiments with multiple distilled text-to-image diffusion models show that MoNO consistently improves per-prompt diversity while maintaining image quality.

Reliability assessment of large language models (LLMs) seeks to estimate the probability that a model produces correct responses under a specified operational profile. Conventional benchmark-based evaluation, often summarized by aggregate accuracy, provides a point estimate of performance but does not characterize the uncertainty associated with reliability claims. Currently, statistical inference methods for LLM reliability assessment are emerging. However, a key assumption underlying these models is that test outcomes can be treated as independent repeated trials. This assumption may be inappropriate in sequential settings, where later responses depend on earlier interactions through retained context, error propagation, or an evolving interaction state. We extend a hierarchical Bayesian framework for LLM reliability assessment by relaxing the assumption of independent task outcomes and introducing a Hidden Markov Model to capture sequential dependence in benchmark-constructed interaction sessions. In this formulation, outcomes are generated from a latent interaction state evolving according to a first-order Markov process, capturing changes in interaction context. Through experiments using Anthropic Claude and OpenAI on four datasets, we demonstrate the potential impact of sequential dependence on reliability assessment. The results suggest that ignoring sequential dependence may lead to overconfident reliability estimates.

Empirical Bayes (EB) performs simultaneous inference across many related latent variables. Classical EB assumes that the likelihood p(x | z) is tractable. In many scientific applications, however, the likelihood is available only through a simulator. This paper develops EB for such implicit likelihoods. We introduce simulation-based empirical Bayes (SBEB), which connects nonparametric EB to simulation-based inference (SBI). SBEB computes EB estimates without an explicit density by using the observed data, simulator samples, and an amortized inference network. SBEB iteratively refines the fitted EB prior toward the population prior. With several scientific simulators and real-world data, we demonstrate that SBEB improves accuracy over SBI with a fixed prior.

Temporal point processes (TPPs) provide a general and flexible framework for modeling sequences of events in continuous time. Neural networks have been successfully employed to model TPPs in a highly expressive and data-driven way. Neural TPPs are typically trained via Maximum Likelihood Estimation (MLE) by minimizing the negative log-likelihood (NLL), which depends on both the conditional intensity function (CIF) and its integral over time, the compensator. Recent neural TPP approaches enable exact evaluation of the NLL without numerical integration. However, these methods typically model the compensator rather than the CIF directly, impose constraints on the neural network architecture, and are computationally expensive during training, as event contributions to the NLL are evaluated sequentially rather than in parallel. In this work, we propose a novel neural TPP model that directly parametrizes the CIF as a non-negative combination of B-spline basis functions, whose coefficients are predicted by a neural network. This formulation enables exact evaluation of the NLL, preserves full flexibility in the neural architecture, allows efficient parallelization during training, and naturally supports CIF smoothness regularization through the integrated squared second derivative. Experiments on both synthetic and real-world datasets show improved computational efficiency and predictive accuracy compared to the reference neural TPP baseline.

This paper explores a way to verify that audio has not been maliciously tampered in a specific context: short viral videos taken from news recordings. Rather than trying to detect artifacts of tampering (internal inconsistency), we focus on positively verifying a query against a trusted source such as a news recording (external consistency). We propose a method for cross verifying a short audio query against a reference recording from which it was taken. Our approach is to define two hypotheses (non-tampered vs tampered), calculate the most likely alignment between query and reference for each hypothesis, and then perform a likelihood ratio test on the two alignments. We show that this method is fast to compute, much more robust than using MFCC features with Euclidean distance, and has the key benefit of explainability. Our cross verification approach provides an alternative perspective and complementary tool to existing tampering detection methods.

Deep survival models are evaluated almost exclusively by the concordance index (C-index), yet they are commonly trained using likelihood objectives such as the Cox partial likelihood, discrete-time negative log-likelihood, and DeepHit likelihood. This mismatch is usually considered acceptable because the C-index can be recomputed on validation data during training. However, for end-to-end training of high-capacity encoders on small, heavily censored oncology cohorts, frequent C-index evaluation is computationally expensive, making the loss value itself an important signal for monitoring, early stopping, and model selection. We show that likelihood losses are unreliable for this purpose and propose a value-monotone concordance loss. We prove that every strictly proper survival likelihood admits directions where the loss decreases while the C-index remains unchanged, causing the loss value to decouple from ranking performance. We then study a sigmoid concordance loss (SCL), whose value approximates one minus the C-index up to a temperature term, ensuring that lower loss corresponds to higher C-index during optimization. The loss is architecture agnostic and reduces to a convex survival ranking support vector machine for linear models. Across eighteen datasets from four modalities using a unified five-fold cross-validation protocol, SCL achieves discrimination comparable to standard likelihood losses and is the best or within one standard deviation of the best C-index. Unlike likelihood losses, SCL maintains a strong correlation between loss value and C-index during training, with rank correlations of 0.96 to 0.99 compared with -0.03 to 0.53 for likelihood losses. Calibration measured by the integrated Brier score is comparable. SCL provides a value-monotone optimization objective whose value can serve as a reliable surrogate for the C-index during expensive end-to-end training.

Reinforcement learning has proven effective for improving reasoning in large language models, but extending it to Masked Diffusion Language Models (MDLMs) remains challenging due to the intractability of the log-likelihood estimation. Existing approaches approximate this log-likelihood by modeling only the token predictions, ignoring the order in which positions are unmasked during generation. We observe that MDLM generation involves two decisions at each step: what tokens to place at each masked position and which positions to remask. We formalize this as a two-stage action MDP, showing that the policy gradient naturally decomposes into a token term and a masking term. Combining optimization of both terms leads to state-of-the-art outcomes on mathematical reasoning and coding benchmarks, with scores of 87.1% on GSM8K and 53.4% on MBPP.

Reinforcement learning for multi-turn search reasoning typically relies on terminal outcome rewards, which cannot distinguish useful, redundant, and harmful intermediate interactions. We propose LOTAPO , a self-generated process-supervision method based on backward leave-one-turn attribution. For each search turn, LOTAPO replaces the turn and its retrieval observation with a fixed [DELETE] placeholder and measures the resulting change in the current policy's mean log-likelihood of the gold answer. This Answer-Likelihood Gain estimates the turn's contribution while preserving all downstream interactions, allowing early evidence to be evaluated in the complete reasoning context. LOTAPO further applies sign-consistency gating, retaining only normalized process advantages whose directions agree with their raw attribution scores. The method requires no additional reward model, teacher, verifier, or LLM-as-a-Judge. Across seven knowledge-intensive question-answering datasets with local retrieval, LOTAPO achieves an average exact-match score of 0.326, outperforming the strongest step-reward baseline, IGPO, by 0.053. Ablations show complementary benefits from backward attribution and sign-consistency gating, demonstrating that policy-derived retrospective attribution can provide effective process supervision for multi-turn search agents.

The Forward-Forward (FF) algorithm trains each layer locally, so that a scalar goodness - the sum of squared activations - is high on real inputs and low on contrastive ones, with activations normalized between layers. Both choices are usually treated as heuristics. Under an explicit generative model they are not: the squared goodness is the sufficient statistic of a likelihood-ratio test between two zero-mean populations differing in scale, and the FF threshold is its boundary. It generalizes: anisotropic populations yield a Mahalanobis goodness, the plain square being its isotropic case; heavy-tailed populations yield a saturating statistic whose slope is a posterior precision - divisive normalization - with bounded evidence and an advantage only under aggregation. The same lens characterizes the inter-layer normalization: it must remove the length while preserving per-coordinate energy, explaining a depth collapse we observe under unit-norm normalization; and the pairwise objective admits a scale-inflation shortcut that a whitened goodness removes.

While autoregressive models optimize the exact data likelihood via the chain rule, diffusion models are typically trained with denoising objectives. We develop conservation laws based on generalized extrinsic information transfer (GEXIT) functions for a broad class of memoryless noise processes, showing that the data--model cross-entropy (CE) can be characterized exactly as an integral of local information-theoretic derivatives along the noise path. This yields a unified characterization of the likelihood for discrete and continuous diffusion, with the Gaussian case reducing to the well-known mutual information--minimum mean-square error (I-MMSE) relationship. An immediate implication is a locality property: one can compute the information-theoretic derivatives using only the marginal posteriors along the noise path. As a result, training reduces to learning the marginal posteriors by minimizing the negative log-likelihood. While the conservation law implies that the entropy does not depend on the noise path, finite-capacity denoisers approximate the posteriors with varying accuracy across noise types, leading to differences in performance. We validate these predictions on synthetic Markov sources and standard benchmarks, including text8 and CIFAR-10.

Authorship verification (AV) is the task of determining whether two texts were written by the same author. In a forensic context, the strength of AV evidence can be quantified using likelihood ratios. Most AV methods are score-based and deriving well-calibrated likelihood ratios from these scores requires a separate calibration model. This, in turn, requires additional amounts of case-relevant data, which is often time-consuming to obtain and prepare. This study proposes two novel normalisation techniques, the Square Root Correction and the Hapax Correction, for deriving likelihood ratios from the AV method LambdaG without the need of a calibration model (Nini et al. 2026). These corrections are designed to mitigate the overestimation of evidential strength that may result from long or highly repetitive texts. Performance is evaluated against logistic regression calibration across fifteen corpora and a range of text lengths (100-9,500 tokens), using the log-likelihood ratio cost (Cllr). The proposed methods achieve performance comparable to logistic regression calibration, with the Hapax Correction outperforming it in approximately 45% of tests (weighted by corpora). Furthermore, performance was more frequently close (within 5%) when the Hapax Correction was outperformed by logistic regression calibration, compared with the reverse comparison. Eliminating the need to train a calibration model reduces data-requirements, time and complexity, thereby increasing the accessibility and transparency of forensic text comparison. This combination of empirical performance and practical advantages supports the adoption of the proposed methods in forensic settings.

What does a discrete diffusion model learn: a denoiser, a score ratio, or a bridge plug-in predictor? At the level of jump rates, these are one object in different coordinates, and reading a neural network in the wrong coordinate changes the process being trained and sampled. Starting with a rigorous derivation of the continuous-time Markov chain (CTMC) ELBO for any noising process, boundary terms included, we prove the \emph{Oracle Distance} theorem: the negative ELBO is exactly equal to the data entropy plus the path KL from the oracle reverse process to the learned one, not merely a bound. Its unique optimizer is therefore the conditional expectation of the true reverse jump rate given the current noisy state, and its irreducible cost is the rate at which the forward process $Z_t$ destroys information about the clean data $Z_0$, $-\tfrac{d}{dt}I(Z_0; Z_t)$, so every noising process shares the same best achievable negative ELBO: the data entropy. For sequences with token-factorizing noise, the oracle projection yields three exact coordinates for the optimizer: denoiser, cavity (bridge plug-in), and score, with closed-form conversions among them. This framework identifies which law each loss in the literature actually optimizes, recovering MDM, UDM, SEDD, and GIDD as special cases; explains why denoiser and cavity coincide for masked diffusion but not for uniform diffusion; proves that a denoiser parameterization makes the uniform ELBO diverge at initialization while the bridge plug-in stays finite; and calibrates ELBO implementations exactly at initialization. Every identity is verified numerically, without approximation, on an exactly solvable model.

Minimum Bayes Risk (MBR) decoding yields more robust and higher-quality text generation than maximum a posteriori (MAP) decoding by selecting hypotheses that maximize expected utility over sampled pseudo-references. However, there exists a discrepancy in the design: hypothesis selection calculates expected utility scores conditioned on given pseudo-references, while commonly used evaluation metrics, e.g., BLEU and COMET, are asymmetric. Therefore, it is important to consider both hypothesis-to-reference and reference-to-hypothesis directional effects. In this study, we introduce a noisy channel decomposition of MBR decoding that naturally incorporates bidirectional effects to account for these asymmetries. We decompose MBR decoding into four interacting components: hypothesis-to-reference likelihood, reference-to-hypothesis likelihood, hypothesis prior, and reference prior. This decomposition provides a unified interpretation of existing MBR variants and enables metric- and task-specific interpretability by isolating the contribution of each channel. Our comprehensive analysis reveals that channel-wise contributions exhibit distinct characteristics across metrics while remaining consistent across tasks, and suggests that appropriate channel weighting may lead to improvements over original MBR decoding.

Sampling action chunks via generative models has become a widely adopted methodology for robotic learning from demonstration. However, existing methods often struggle to balance responsiveness and computational cost because they execute each action chunk for a fixed execution horizon. In this paper, we adaptively adjust the execution horizon of sampled action chunks, balancing responsiveness and computational efficiency. We introduce Spatial Attention -- defined as the expected squared norm of the gradient of the action log-likelihood with respect to the observation -- which indicates the sensitivity of the policy's action distribution to variations in the observation. We show that, under a fixed budget of chunk samplings, the execution horizon that minimizes the cumulative likelihood drop induced by disturbances decreases as Spatial Attention increases. By forecasting future Spatial Attention values alongside the action chunk, our framework dynamically assigns shorter execution horizons to phases with high Spatial Attention, and longer horizons to phases with low Spatial Attention. Experiments on standard and perturbed tasks, in both simulation and on a real robot, show that our method significantly improves success rates over fixed-horizon baselines while maintaining the average execution horizon.

Understanding human visual attention on a scene over time has applications in domains such as interface design and inferring cognitive states. Modeling visual scanpaths has historically relied on specialized architectures with hand-crafted priors. While these architectures can model fixation sequences, their rigid structural biases restrict easy extendability and flexible conditioning. For instance, integrating task-specific instructions or adapting to distinct viewer identities requires custom, disjoint architectural additions. We frame scanpath prediction purely as a discrete sequence modeling task. By mapping coordinates into a text vocabulary, we leverage the pretrained representations of Vision-Language Models. This framing absorbs diverse factors of variation: simple prompting allows for global conditioning, such as providing viewer identities to capture personalized biases, or task-specific objectives like visual search. The framework can also integrate per-fixation attributes, such as individual fixation durations, alongside spatial locations. The autoregressive alignment enables the scalable, exact computation of per-fixation log-likelihoods, directly equivalent to the commonly used Information Gain (IG) metric. Our model, DeepGaze3.5-VL, establishes a new state-of-the-art across multiple datasets, achieving 2.18 bits of IG on MIT1003, a 46% improvement over DeepGaze III. This advantage persists even when baselines use identical high-capacity vision encoders. Beyond predictive performance, our generative framework serves as a powerful computational tool for direct behavioral interventions, allowing for controlled in-silico simulations that would be experimentally difficult or impossible to conduct in vivo. We demonstrate this ability by performing controlled interventions on the durations of pre-saccadic fixations, recovering known oculomotor phenomena purely from data.

Reinforcement Learning (RL) is a sequential decision-making framework in which an agent learns optimal policies through interaction with an environment by maximizing cumulative rewards. Among RL methods, Bayesian Reinforcement Learning (BRL) addresses common practical challenges related to data scarcity by leveraging prior knowledge about the environment and sequential belief updates. However, most BRL approaches require an explicit likelihood function, which is frequently inaccessible or intractable in real-world settings. We propose Likelihood-Free Iterated Batch Importance Sampling (LF-IBIS), a novel algorithm for BRL that updates the agent's beliefs online as new interactions become available. By combining Approximate Bayesian Computation with Iterated Batch Importance Sampling, LF-IBIS enables full Bayesian inference in settings where the environment dynamics are not described by an explicit or tractable likelihood. The method yields approximate posterior distributions over both environment parameters and optimal policies, providing a quantification of policy uncertainty useful for a Bayesian treatment of the exploration-exploitation trade-off. We test the method on a simulation study in response-adaptive randomization in clinical trials, where closed-form posteriors enable validation. Additional experiments address settings where the posterior has no closed form and illustrate online policy updating based on the posterior distribution of the optimal policy.

We audit a fixed prefix-window mean-NLL memorization probe (K=20) on a Qwen2.5-VL-7B canary testbed and report three post-hoc cases where it disagrees with full-span secret NLL or greedy exact-recall. C3 (false negative, window truncation): damage lands on hex tokens outside K=20; the probe stays flat while hit@1 drops. C4 (false positive, non-secret drift): the probe moves, but approximately 99% sits on non-secret preamble; the secret span and hit@1 are unchanged. C5 (ambiguous in-window drop): the probe falls on an undertrained baseline while full-span hex is positive and hit@1=0. Recommendation: report (i) full-span secret NLL, (ii) a span-localised decomposition, (iii) behavioural exact-recall at k>=4, and (iv) decoy probes before asserting secret-specificity. Evidence is on controlled canaries in one backbone; magnitudes are testbed-specific.

Normalizing Flows excel at modeling a single fixed density, yet many problems across the sciences, such as high energy physics, instead require modeling how that density deforms as a function of continuous parameters: the strength of a physical effect, a calibration constant, or a source of systematic uncertainty. Learning a separate flow for every parameter configuration quickly becomes intractable, since the number of joint settings grows exponentially with the number of parameters. We introduce Factorizable Normalizing Flows (FNFs), which represent the parameter-dependent density as a fixed, high-fidelity flow for a reference configuration composed with a learnable transformation that is polynomial in the parameters and factorized over them. This structure has a practical consequence: each parameter's effect is learned in isolation, from samples in which that parameter alone is varied. The combined response of many parameters is then recovered by summation at inference, without ever sampling their combinatorially large joint space. On a controlled problem with two interpretable deformations applied jointly to the data, the learned transformation reproduces the true deformations and matches the optimal likelihood, while optional interaction terms capture residual correlations when several parameters vary strongly at once. The resulting model is interpretable, scales linearly with the number of parameters, and keeps the likelihood tractable. This provides a general tool for any inference workflow requiring continuous density morphing, and directly enables the next generation of unbinned likelihood fits in high energy physics.

External evaluations are becoming increasingly central to the governance of AI systems. In practice, however, independent auditors often have limited access to deployed models and must rely on query-based interactions. Most existing fairness evaluation methods assume static datasets and fixed-sample statistical tests, making them poorly suited to real-world auditing scenarios in which evidence must be collected sequentially under query constraints. In this work, we formulate fairness auditing as a tolerance-aware sequential hypothesis-testing problem under limited model output access. We develop a sequential generalized likelihood-ratio framework that allows auditors to accumulate evidence from a finite audit pool and stop once sufficient support for compliance or violation has been obtained. The framework is instantiated for decision-based Statistical Parity and Equal Opportunity audits, and extended to score- and logit-based proxy audits when richer observables are available. Our results show that both the fairness metric and the level of model access significantly affect audit efficiency, and that the benefits of richer output information are not uniform across auditing settings. In particular, richer outputs can substantially reduce the number of queries required for some fairness metrics and operating regimes, while offering limited gains in near-threshold cases. This work provides a practical statistical framework for sequential fairness auditing under realistic deployment constraints.

The prevalent dual-branch paradigm, i.e., training a side network to encode visual conditions and fusing its intermediate-layer features to a frozen pretrained main network, has shown remarkable success in visual-condition controllable generation. Despite its widespread adoption, the role of the side branch and its training efficiency remain underexplored. In this paper, we first revisit this mainstream paradigm through the lens of score-based generative modeling: 1) The main network preserves visual perceptual quality by providing a prior unconditional score. 2) The side network steers conditional control by implicitly contributing a likelihood score. Guided by this perspective, we propose LIkelihood Score Alignment (LISA), an effective regularization method that explicitly aligns the intermediate feature of the side network with an approximated likelihood score. Specifically, we first hook features from a designated layer of the side network and project them into the score latent space by a lightweight decoder. Then, we construct an approximated likelihood score target and calculate the distance between the decoder's output and this target as an additional regularization loss. Finally, we jointly optimize the side network and decoder with both standard diffusion loss and our regularization loss. Experiments across various image/video tasks, architectures, and diffusion/flow models demonstrated that LISA can not only consistently accelerate the training convergence and improve final synthetic results, but also encourage the side network's features to be more disentangled for conditional modeling with negligible additional training cost and zero extra inference cost.

Causal Transformers model sequences through an autoregressive factorization of the joint distribution, which enables efficient left-to-right decoding and conditional likelihood computation. However, they cannot tractably sample from or evaluate arbitrary conditionals -- e.g., a block of text conditioned on past and future tokens. Recent work aims to solve this problem through novel architectures, but they often lead to sub-optimal modeling of such conditionals and degraded generations. We propose Arbitrary Conditionals GPT (AC-GPT) which introduces a simple modification to standard causal Transformers to enable evaluating and sampling from arbitrary conditionals -- including past, future, and mixed contexts -- within a single forward pass. Unlike prior approaches, our method preserves the standard left-to-right ordering and next-token prediction objective essential for both strong performance and efficient training on natural language. Crucially, this compatibility allows existing LLMs to be fine-tuned for arbitrary conditioning. Our empirical results indicate that our method outperforms baselines on modeling arbitrary conditionals, without degrading standard left-to-right performance.

In small-batch scientific deployments, labeled target outcomes may be too scarce for reliable shift estimation even when unlabeled target inputs are available. We address the complementary setting where the practitioner has a pre-specified label-shift correction from domain knowledge and asks whether incoming labeled outcomes support it. We show that the per-observation likelihood ratio between a label-shift-corrected predictive and the source predictive is a conditional e-value, so its running product is a nonnegative martingale and Ville's inequality yields an anytime-valid confirmation rule. The log martingale equals the cumulative negative log-predictive density (NLPD) gap between the source and the corrected predictive, converting routine model monitoring into a formal sequential test. Rejection means the incoming data support the posited correction relative to the source predictive, but it is not a precise estimate of the degree of shift. Closed forms are available for GP sources with Gaussian label-shift ratios. GP regression simulations validate Type I control, finite-sample power, miscalibration sensitivity, and the small-batch advantage of a reliable prior over label-based re-estimation.

The likelihood ratio framework is widely recognized as the logically and legally sound basis for evidential analysis across forensic sciences, and its importance is increasingly acknowledged in analyses of authorship in textual evidence. To date, however, its application has been confined to English-language texts. Meanwhile, authorship attribution has traditionally relied on a diverse array of stylometric features, even as the rise of pre-trained large language models enables new contextual-embedding approaches. Combining these diverse approaches through fusion promises enhanced performance, yet it has not been applied to integrate stylometric-feature systems with embedding-based systems within the likelihood ratio paradigm. This study is the first to apply likelihood ratio-based forensic text comparison to Japanese digital texts, using ~1,000-character excerpts from blogs, to 1) evaluate system performance and likelihood ratio magnitudes and 2) assess the impact of fusing stylometric-feature systems with embedding-based systems. The results demonstrate that the fused system maintains excellent calibration while 1) increasing consistent-with-fact likelihood ratio magnitudes; 2) decreasing contrary-to-fact likelihood ratio magnitudes and 3) improving overall discriminability. The best-performing fusion achieved a log-likelihood-ratio cost of 0.32484, illustrating both the feasibility of likelihood ratio framework for Japanese and the benefits of fusion across heterogeneous systems.

Diffusion and continuous flow-based language models have emerged as the leading non-autoregressive alternatives to language modeling. Progress in both paradigms is overwhelmingly tracked by generative perplexity (gen-PPL): the per-token negative log-likelihood of samples under a frozen autoregressive (AR) scorer such as gpt2-large, typically paired with an empirical-entropy guardrail to rule out low-entropy collapse. We argue that this metric is unsound. By construction, gen-PPL measures only predictability under the scoring AR, not grammaticality or semantic coherence -- and the set of predictable but still low-quality sequences is combinatorially large. To make this concrete, we construct a suite of zero-parameter, deliberately naive samplers that achieve state-of-the-art gen-PPL on LM1B and OpenWebText at non-degenerate entropy, surpassing recently published diffusion and continuous-flow models while producing text that is incoherent by construction. We recommend evaluation suites that directly quantify the distributional divergence between generated and reference text, and use such a suite to re-benchmark recent non-autoregressive models, recovering a more faithful picture of the current state of the art.

Generative policies provide expressive and multimodal action distributions, making them attractive for reinforcement learning (RL) in complex continuous-control tasks. Among them, flow-based policies are especially appealing because they generate actions through deterministic transport maps. However, applying such generative policies to likelihood-based on-policy learning remains limited by the difficulty of evaluating the probability of executed actions. Existing flow RL methods either replace the true action-density ratio with approximate surrogates, which can introduce biased updates, or recover exact likelihoods through dummy-action augmentation, which enlarges the policy space and increases computation. In this work, we propose GenPO++, a reversible generative policy optimization framework that uses history states as auxiliary memory in a high-order reversible ODE solver, yielding exact inversion without changing the original action dimension. The resulting generative policy map has a log-determinant determined only by fixed solver coefficients, enabling exact and Jacobian-free likelihood-ratio computation. This design preserves the expressiveness of generative flow policies while avoiding both action ratio bias and dummy-action overhead. We evaluate GenPO++ on large-scale simulated control, fine-tuning, and real-world robotic manipulation tasks, where it achieves competitive or superior performance over state-of-the-art on-policy RL methods, while improving training stability and computational efficiency.

Constrained hyperparameter optimization (HPO) is common in practice, yet Optuna's widely used constrained TPE lacks algorithmic analysis. While c-TPE proposes an expected constrained improvement (ECI) approach assuming independence between the objective and constraints, Optuna uses a single joint density over both. We show that Optuna's constrained TPE is joint c-TPE -- the same ECI acquisition function using a joint likelihood. We demonstrate joint c-TPE is invariant to constraint duplication whereas independent c-TPE degrades as the product accumulates duplicated factors. We outline practical tradeoffs between the formulations and directions for future study.

Asynchronous reinforcement learning can improve language-model post-training throughput by decoupling response generation from policy optimization, but stale responses introduce distribution drift. Standard behavior-corrected methods control this drift with behavior-policy probabilities, importance ratios, or clipping, which requires token-aligned, versioned, and numerically consistent behavior log-probabilities across rollout and learner systems. We ask whether asynchronous group-relative RL can instead be stabilized using only current-policy probabilities. We identify a scale-imbalance failure mode: when stale responses are evaluated under the current policy, positive and negative loss terms can appear at different negative log-probability scales, so zero-sum advantages no longer imply balanced loss contributions. We propose Asymmetric-Scale Policy Optimization (ASymPO), which normalizes each response's token loss by its current average token negative log-probability. ASymPO requires no behavior-policy probabilities, restores response-level zero-sum balance, and preserves a nonzero learning signal. We also introduce Scaled Policy Optimization (SPO), a fixed negative-scaling baseline, and evaluate both current-policy-only objectives in asynchronous mathematical reasoning post-training.

We address the problem of out-of-distribution (OOD) detection for target observations embedded in a subspace of the high dimensional data space. Using continuous normalizing flows (CNFs), we propose a Lagrangian sub-flow (LSF) framework designed to isolate and estimate the density for the relevant components in the representation and using the remaining components as context. Through experimentation with models for speech synthesis, we show that CNFs, similarly to other deep generative models (DGMs), are susceptible to the "likelihood paradox", where high likelihood is erroneously assigned to OOD samples. This is attributed to the inductive bias of DGMs that prioritize low-level structural details over high-level semantic coherence. To mitigate this phenomenon, we propose a number of geometric diagnostic signals based on the velocity field over the sub-flow trajectory. Based on these signals, we design metrics for the challenging task of zero-shot phoneme-level mispronunciation detection. Finally, we demonstrate the superiority of these metrics compared to likelihood-based methods on a real-world mispronunciation detection benchmark.

Unlearning in diffusion models aims to remove undesirable data or concepts while preserving the utility of pretrained models -- two fundamentally conflicting objectives. We propose a principled constrained optimization framework that formulates unlearning as minimizing the deviation from a pretrained model, subject to explicit separation constraints from the unlearning distributions. Specifically, we formulate three constrained optimization problems based on reverse and forward KL divergences, and likelihood constraints. The first two generalize existing approaches for concept and data unlearning, while the third offers a novel and natural formulation for unlearning. Despite the nonconvexity of the KL constraints, we establish strong duality for all three problems, enabling us to explicitly characterize their optimal solutions as unlearning targets and develop primal-dual algorithms for each formulation. Experimental results demonstrate that our KL-constrained approach achieves superior retention-unlearning tradeoffs compared to weight-based baselines for concept and data unlearning, and that our likelihood-based approach matches unlearning effectiveness while better preserving retained concepts compared to baselines.

Diffusion models have excellent capacity to model complex distributions of natural data, which has made them a popular and effective choice for posterior sampling in imaging inverse problems. Existing methods can incorporate any measurement model at inference time but must use an inexact approximation for the likelihood at intermediate timesteps for computational tractability. Although these approximations can often work well empirically, their downstream effect on the sampled posterior is poorly understood and can result in unexplained failures. To understand when, why, and how these likelihood approximations propagate to erroneous posterior distributions, we introduce a finite-sample perspective on posterior sampling that approximates the posterior to arbitrary precision as training set size tends towards infinity, for any forward model and prior distribution. Using this finite-sample lens, we observe that popular posterior sampling approximations tend to under- or over-estimate the spread of the posterior at intermediate timesteps, causing downstream consequences including sensitivity to early stopping time, inaccurate relative weighting of posterior modes, and hallucination, both of prior modes that are not in the posterior and likelihood modes that are not supported by the prior. Moreover, we find that the cause of these posterior errors requires neither a nonlinear measurement model nor a multimodal posterior, but can arise solely due to a multimodal prior and inaccurate posterior spread at intermediate sampling times. Our finite-sample posterior sampling approach is agnostic to the type of likelihood approximation and the type of (linear or nonlinear) forward model, and can thus serve as a drop-in diagnostic to evaluate the accuracy and failure modes of existing and future posterior samplers.

Sparse recovery in linear systems underpins applications from signal processing to high-dimensional regression. Sparse Bayesian Learning, grounded in the principle of automatic relevance determination (ARD), offers a practical Bayesian mechanism for feature sparsity via marginal likelihood optimization. Yet, its reliance on a homoscedastic noise model renders it sensitive to data contaminations such as outliers or misspecified noise, harming model fit and predictions. Instead, we propose jointly learning individual feature and sample relevancies, enabling simultaneous model and data sparsification via a single Bayesian objective. This symmetric pruning of model and data offers a natural extension that preserves conjugacy, admits closed-form updates for standard optimization procedures, and aligns with perspectives from robust regression and influence functions. Empirical results across diverse regression tasks affirm that a joint ARD approach consistently yields both sparse and robust prediction models.

Concerns about algorithmic bias and fairness have increased as artificial intelligence has been incorporated into high-stakes decision-making. Traditional Naive Bayes classifiers, while efficient and interpretable, lack fairness-awareness mechanisms and perpetuate historical biases in sensitive domains such as hiring, credit scoring, and criminal justice. This study develops a fairness-aware extension of the Naive Bayes classifier that mitigates bias while maintaining computational efficiency. We propose the Bias Mitigating Naive Bayes (BMNB) classifier, integrating in-processing and post-processing interventions. The in-processing stage employs a blended likelihood approach combining group-specific and pooled likelihood estimates through a tunable blending parameter alpha to balance fairness and accuracy. The post-processing stage applies output calibration with adaptive thresholding to fine-tune group-specific decision boundaries. Experimental results indicate that BMNB attains Disparate Impact (DI) values of 1.000, 1.171, and 0.997 and Equal Opportunity Difference (EOD) values of -0.217, -0.226, and -0.053 on the Adult, ProPublica, and Framingham datasets, respectively, while maintaining computational efficiency. Ablation studies confirm that the combination of blended likelihood and adaptive thresholding yields superior performance compared to either technique in isolation.

Log-likelihood is a standard metric for evaluating generative models. Unfortunately, in contrast to autoregressive models (ARMs), discrete diffusion models generally do not admit exact computation of this quantity. Existing evaluations, therefore, rely on the evidence lower bound (ELBO), leaving unclear how much higher the true value may be. We address this by introducing the Tangent Upper Bound on Evidence (TUBE), a variational upper bound on log-likelihood that admits an unbiased Monte Carlo estimator. Our TUBE extends across latent-variable models, including masked diffusion models (MDMs), any-order ARMs (AO-ARMs), and block variants of both. Applied to block MDMs and block AO-ARMs, TUBE reveals our key empirical finding that these models lie strictly below the exact ARM baseline, showing that ARMs still dominate in likelihood.

Protein-ligand modeling underpins computational drug discovery and molecular design. Existing protein-ligand benchmarks typically evaluate whether a protein and ligand interact and how strongly they bind, through tasks such as binary binding prediction and affinity regression. However, these evaluations provide limited evidence of whether models can localize binding sites or identify the non-covalent interactions underlying molecular recognition. To address this gap, we introduce InteractBind, a large-scale protein-ligand dataset comprising approximately 100k protein-ligand pairs, together with a benchmark for fine-grained evaluation. The core fine-grained task is that of binding-site localization, which uses protein-residue and ligand-atom interaction maps spanning six major types of non-covalent interactions to assess whether model-derived interaction maps localize binding sites. InteractBind further includes binding affinity and protein similarity-controlled splits to support realistic generalization assessment. Using InteractBind, we evaluate eight existing sequence-based and interaction-aware models, assessing binary binding prediction and binding-site localization. Results reveal limited binding-site localization despite strong binary binding prediction, with marked variation across non-covalent interaction types. Overall, InteractBind establishes a benchmark paradigm that encourages the development of more interpretable and physically grounded protein-ligand models.

Deep generative models offer a natural foundation for out-of-distribution (OOD) detection, yet prior work has shown that their assigned likelihoods are notoriously unreliable indicators for in- vs out-of-distribution data. In this paper, we address this problem by leveraging the diffeomorphic and mass-preserving properties of continuous normalising flows. Our analysis shows that OOD samples are mapped to noise samples that are highly atypical under the noise prior in ways not captured by the likelihood. Based on this observation, we propose a new method -- Signal in the Noise (SITN) -- for OOD detection on the single-sample level. SITN requires no access to OOD data, incurs minimal computational overhead, and provides strict control of false positive rates. Comprehensive evaluations through standard benchmarks and synthetic perturbations highlight the method's effectiveness and the absence of the complexity bias inherent to likelihood-based methods.

One of the significant challenges in legged robotics is achieving accurate odometry using only onboard proprioceptive sensors. In this study, we present a complete leg odometry pipeline based on an Error-State EKF (ESEKF) that relies exclusively on proprioceptive data: a body fixed IMU, joint encoders, and force sensors, where filter's state is corrected by feet determined to be in a stationary stance. The core of our contribution is fused contact detection and an uncertainty quantification module designed to explicitly identify and reject slippage. This module runs two detectors in parallel for each foot, 1) a debounced, force-based Gaussian Mixture Model (GMM) guided Finite State Machine (FSM) to confirm physical contact, and 2) a kinematic-based Generalized Likelihood Ratio Test (GLRT) on the estimated velocity of the foot. The continuous quality scores from both estimators are fused to detect if the foot is both physically loaded and kinematically stationary and served as an uncertainty signal for each contact. To validate our approach, we collected a multi-modal dataset of 29 sequences spanning diverse indoor and outdoor terrains (e.g., concrete, grass, pebble, and rock) total of 2.4 km long. We benchmarked our approach against both proprioceptive and exteroceptive methods. The results demonstrate our method's efficacy in providing accurate odometry estimates, robustly handling slippage-prone environments. We also share our code and real-time ROS2 package as open-source.

State-space models (SSMs) are powerful probabilistic tools for modeling time-varying systems with latent dynamics. Inference in SSMs involves the estimation of latent states and parameters. In this work, we focus on parameter inference, which for SSMs is in general a very challenging problem due to the intractability of the likelihood. Recently, neural estimation methods, such as sequential neural likelihood (SNL), have shown promising results in Bayesian inference problems. In this paper, we show that SNL, when applied to the SSM setting, suffers important limitations, such as requiring a large amount of simulated samples to achieve a moderate performance, scaling poorly with sequence length, while not being amortized. We then introduce a novel inference algorithm called truncated-SNL (T-SNL), which addresses the limitations of SNL. Our algorithm is more accurate, more stable and robust during training, more scalable to longer temporal sequences, and can be amortized when new observations become available. Our experiments show that T-SNL is sample-efficient, robust, and flexible algorithm which outperforms other approaches.

Reinforcement learning with verifiable rewards (RLVR) plays a pivotal role in improving the reasoning ability of large language models. However, widely used PPO surrogate objectives are fundamentally local, as they rely on a local approximation of the exact policy gradient objective. While this approximation improves stability by reducing the variance induced by importance sampling, it also introduces structural bias into the surrogate objective, which must be controlled through trust region mechanisms. In this work, we introduce the $N$-step forward trace, which augments the PPO surrogate objective using the cumulative likelihood ratio of the next $N-1$ tokens. Building on this idea, we propose $N$-Step Forward-Trace Policy Optimization (NFPO), a practical RLVR algorithm that integrates the $N$-step forward trace into the masked policy gradient framework. NFPO provides a continuous bridge between the PPO surrogate objective and the exact policy gradient objective, offering a principled mechanism for controlling the bias-variance trade-off. Our theoretical analysis shows that, with an appropriate choice of $N$, the proposed objective yields a tighter policy-improvement bound than the standard PPO surrogate. Experiments on comprehensive reasoning benchmarks demonstrate that NFPO consistently improves performance, supporting our theoretical findings.

We address out-of-distribution (OOD) detection across the full spectrum of distribution shifts -- global domain changes, semantic divergence, texture differences, and covariate corruptions -- through a multi-encoder fusion of per-encoder representation-space diffusion models (RDMs). We statistically identify each encoder's sensitivity to specific shift types from ID data alone and introduce EncMin2L -- an encoder-agnostic two-level $\min(\cdot)$-gate that combines and calibrates per-encoder diffusion-based likelihood detectors without OOD labels, outperforming monolithic multi-encoder baselines at $2.3\times$ lower parameter cost. Two ID-data diagnostics: $η^2$ (class-conditional F-test) and $Δμ$ (log-likelihood shift under synthetic corruptions) -- quantify encoder specialization, while a Tippett minimum $p$-value combination aggregates per-encoder scores into a single, calibration-stable OOD signal. EncMin2L achieves $\geq 0.94$ AUROC across all four shift types simultaneously, outperforming the state-of-the-art representation-space diffusion OOD detectors across overlapping benchmarks.

Multi-agent LLM debate improves factuality and reasoning, but most recipes pick a fixed round count, over-spending on easy items and under-spending on hard ones. We adapt Wald's Sequential Probability Ratio Test (SPRT) as a plug-in compute governor for LLM debates. After each round, an LLM judge emits a [0,1] consensus score on the latest agent positions; a Wald monitor accumulates the log-likelihood ratio of "useful convergence" vs "not yet useful" under a Beta likelihood family, and stops when either boundary is crossed or returns a capped best-effort outcome at R_max. Under i.i.d. assumptions the rule inherits SPRT type-I/type-II error guarantees; in deployment the calibration itself is the more important object, since it estimates whether the judge score actually separates useful from unhelpful convergence in a given domain. We evaluate two tracks: (i) a Monte-Carlo study under calibrated Beta models characterising working curves, error rates, capping behaviour, and sensitivity; and (ii) a real-LLM evaluation on 200 attempted MMLU and 200 attempted GSM8K items with three heterogeneous agents (gpt-5, claude-opus-4-6, gemini-2.5-pro) and a claude-opus-4-6 judge, using disjoint 40-item calibration subsets. On GSM8K the rule stops in 1.01 average rounds (4.06 LLM calls) at 97.0% accuracy vs 99.0% for fixed-5 debate at 15 calls: a 3.7x call reduction at -2pp accuracy. On MMLU the calibrated KL collapses to about 0 and the rule caps on 99.5% of items at 2.1x cost. The takeaway is not that SPRT makes debate more accurate, but that a classical sequential test serves as a cheap compute-control and failure-detection layer for multi-agent LLM systems.

Federated Bayesian neural networks require fixing a prior on the model parameters together with a likelihood. Eliciting meaningful priors on the weight space of modern overparameterized models is notoriously difficult, and misspecification of either component can severely degrade accuracy and calibration. Motivated by the rapid progress of predictive models such as large language models, the martingale posterior, also known as predictive Bayes, replaces the prior--likelihood pair with a predictive distribution and recovers parameter uncertainty by repeatedly drawing predictive samples and refitting the model. A direct federated implementation, however, would require clients to share the local data sets. This letter proposes {federated martingale posterior} (FMP) sampling, a one-shot embarrassingly parallel protocol in which each client uploads a small set of trainable data embeddings and the server runs the predictive sampler centrally. Experiments on MNIST, CIFAR-10, and CIFAR-100 show that FMP closely matches the centralized counterpart and significantly improves calibration over consensus-style baselines.

Diffusion and flow-based models are ubiquitously used for generative modelling and density estimation. They admit a deterministic probability flow ordinary differential equation (PF-ODE), analogous to continuous normalizing flows (CNFs), which describes the transport of the probability mass. Obtaining the likelihood from these models is of interest to many workflows, especially Bayesian analysis, and requires solving the trace of the Jacobian to compute the divergence of the learned PF-ODE, which is either $\mathcal{O}(D^2)$ to compute exactly or $\mathcal{O}(D)$ with a noisy estimate. We introduce StAD, a new distillation method to predict and learn the divergence of the PF-ODE using the Langevin-Stein operator without ever computing the Jacobian. We show that our method is competitive with the Hutchinson and Hutch++ on CIFAR-10, ImageNet and other density estimation tasks, consistently improving the variance and speed of the likelihood predictions compared to the Hutchinson. We additionally show our method will generalize to a varied class of generative models, and show that under some regularity conditions these learned vector fields can be made to satisfy the Stein class.

Generative models are often trained with a next-token prediction objective, yet many downstream applications require the ability to estimate or control sequence-level properties. Next-token prediction can lead to overfitting of local patterns during training, underfitting of global structure, and requires significant downstream modifications or expensive sampling to guide or predict the global attributes of generated samples at inference time. Here, we introduce Conditional Attribute Transformers, a novel method for jointly estimating the next-token probability and the value of an attribute conditional on each potential next token selection. This framework enables three critical capabilities within a single forward pass, without modification of the input sequence: (1) per-token credit assignment across an entire sequence, by identifying how each token in a sequence is associated with an attribute's value; (2) counterfactual analysis, by quantifying attribute differences conditional on alternative next token choices; (3) steerable generation, by decoding sequences based on a combination of next-token and attribute likelihoods. Our approach achieves state of the art performance on sparse reward tasks, improves next-token prediction at sufficient model sizes, estimates attribute probabilities orders of magnitude faster than sampling, and can guide decoding of autoregressive sequence models on a range of language tasks.

Score-based diffusion models demonstrate superior performance in generative tasks but encounter fundamental bottlenecks in inverse problems due to the analytical intractability of the time-dependent likelihood score. To bridge this gap, we propose a novel proximal-based generative modeling (PGM) framework that rigorously circumvents explicit likelihood evaluation. Our framework is built upon a theoretical equivalence between Gaussian convolution in diffusion processes and Moreau-Yosida regularization in nonsmooth optimization. This enables a new sampling mechanism driven by the proposed Moreau score, which admits a closed-form expression via proximal operators. Moreover, we introduce Moreau score matching to learn the proximal operators that rely solely on samples drawn from the prior distribution. Theoretically, PGM eliminates the early-stopping bias inherent in the score-based diffusion model and achieves non-asymptotic convergence. Experiments demonstrate that PGM significantly surpasses state-of-the-art methods in reconstruction quality and sampling time.