Comparing two probability distributions is a basic building block of statistics and machine learning, and the right family is well understood: the Rényi divergences of order $α\in[0,\infty]$ are the unique family monotone under data processing and additive on independent products. Many problems instead compare more than two distributions at once -- multi-population fairness, multi-prior PAC-Bayes bounds, multi-hypothesis testing -- and the right multi-distribution generalization of the Rényi family has been an open question. We characterize it. Every functional of $W$-tuples of distributions that is monotone under data processing and additive on independent products is a positive integral of multi-way coincidence divergences $C_α(π_1,\dots,π_W) := -\log\int π_1^{α_1}\cdotsπ_W^{α_W}$ (with $\sum_k α_k = 1$) over a parameter space with four strata: the simplex interior; mixed-sign exponent cones (the analogue of Rényi orders $>1$); a tropical boundary at infinity carrying max-divergences; and pairwise Kullback-Leibler edges at the simplex vertices. Each stratum is necessary -- the destination of an explicit data-processing-monotone, product-additive divergence the others cannot reproduce -- and each is a clean limit of simplex-interior atoms. The same family arises from several independent routes -- the structural axioms, Kolmogorov-Nagumo means with Rényi's entropy axiomatics, classical entropy characterizations, multi-hypothesis testing error exponents, and a multi-lottery betting interpretation -- structural evidence that this is the canonical multi-distribution Rényi calculus rather than an artefact of any one axiomatic input. The two-prior case recovers the standard Rényi result; a worked $W=3$ instance, numerical verification, and a conditional extension round out the treatment.

We prove a single algebraic mixed coincidence identity that unifies a broad swath of information-theoretic variational results. For any family of priors $\{π_i\}$ and real exponents $\{ α_i \}$, the log of the mixed count $E_{x\simν}\!\left[\prod_{i=1}^W π_i^{α_i}(x)\right]$ is simultaneously a Boltzmann coincidence weight, an exponential-family normalizer, a maximum-entropy value, and a KL-barycenter optimum. The identity yields a unified derivation of classical cornerstones of information theory: concentration of empirical distributions (Sanov-type decompositions and Gibbs conditioning), hypothesis-testing error exponents (Chernoff information and its multi-way analogue), change-of-measure inequalities (Donsker-Varadhan and PAC-Bayes), and laws governing rare-pattern coincidences (Erdos-Renyi run-length, iterative guesswork, rate-distortion, and birthday thresholds). Each is recovered as a specialization of the same algebraic equality. It strictly generalizes the classical Renyi entropy and divergence variational formulas (one and two priors respectively) to a $W$-prior simplex, and holds for unnormalized and continuum-indexed priors. Among its consequences are an exact multi-prior PAC-Bayes penalty that subtracts an explicit "coincidence bonus" from the usual single-prior posterior penalty, and the asymptotic MAP error exponent for $W$-ary hypothesis testing as an edge-restricted simplex optimum. We demonstrate the calculus at scale on two large alphabets encoding richly modeled sequential languages: on language-model next-token predictives where we recover contrastive decoding, and on human genomic regulatory sequence where it separates correlated from diverse prior families along a sliding-window trace.

Mean Field Variational Inference (MFVI) is widely understood to underestimate posterior variance. By analysing conjugate Bayesian Linear Regression (BLR), we show that this characterization is incomplete: while MFVI underestimates the variance in parameter space, it can overestimate the predictive variance compared to the exact posterior. We show that if the MFVI posterior underestimates predictive variances in some directions, it necessarily overestimates them in others. Crucially, this overestimation occurs in directions where the training data concentrates. This leads to the surprising result that, for a test point drawn from the training distribution, MFVI's expected predictive variance exceeds that of the exact posterior. We demonstrate a pathological case of this effect, where the MFVI posterior fails to reduce predictive variance compared to the prior on in distribution data. We connect these results to the Cold Posterior Effect, arguing that varying the temperature can correct this overestimation, yielding predictions closer to those of the exact posterior. We validate our theory on synthetic and real-world regression tasks.

We present a theoretical framework showing that popular LLM alignment methods--including RLHF and its variants--can be understood as divergence estimators between aligned (safe or preferred) and unaligned (harmful or less-preferred) distributions. This perspective explains the emergence of separation in the latent space between safe and harmful prompts after alignment. As an application of our general divergence framework, we propose KLDO, a novel KL divergence-based alignment method, and empirically validate its effectiveness. We further show that using compliance-refusal datasets, rather than standard preference-based datasets, leads to stronger separation and improved safety alignment. Finally, to quantify the separation effect, we propose a distance-based metric in the prompt representation space, which also acts as a statistically significant indicator for model safety.

Many domain experts do not have the time or expertise to write formal Bayesian models. This paper takes an informal problem description as input, and combines a large language model and a probabilistic programming language to define a joint distribution over formal models, latent variables, and data. A posterior over latent variables follows by conditioning on observed data and integrating over formal models. This presents a challenging inference problem. We suggest an inference recipe that amounts to generating many formal models from the large language model, performing approximate inference on each, and then doing a weighted average. This is justified and analyzed as a combination of self-normalized importance sampling, MCMC, and importance-weighted variational inference. Experimentally, this produces sensible predictions from only data and an informal problem description, without the need to specify a formal model.

As machine learning increasingly relies on large, opaque foundation models powering generative and agentic AI, deploying these systems in safety-critical contexts demands rigorous generalization guarantees beyond training data. PAC-Bayes theory provides principled certificates linking training performance to generalization risk, yet existing approaches remain impractical: simple theoretical priors yield vacuous bounds, while data-dependent priors require costly second-stage training or introduce bias. To bridge this critical gap, we propose a localized PAC-Bayes prior--a structured, computationally efficient prior softly concentrated around parameters favored during standard training. By integrating this localized prior directly into the standard training objective, we deliver practically tight generalization certificates with minimal workflow disruption. Under standard neural tangent kernel assumptions, our bound shrinks as networks widen and datasets grow, becoming negligible in realistic regimes. Empirically, we demonstrate tight generalization certificates on tasks ranging from image classification (MNIST, CIFAR, ImageNet) and NLP fine-tuning (GLUE) to semantic segmentation (Cityscapes), typically within three percentage points of test error at ImageNet scale. Additionally, our approach provides rigorous guarantees for individual predictions, selective rejection of uncertain predictions, adversarial robustness, and accurate calibration--directly addressing key requirements for trustworthy AI deployment.

Pseudo label based semi-supervised learning (SSL) for single-label and multilabel classification tasks has been extensively studied; however, semi-supervised label distribution learning (SSLDL) remains a largely unexplored area. Existing SSL methods fail in SSLDL because the pseudo-labels they generate only ensure overall similarity to the ground truth but do not preserve the ranking relationships between true labels, as they rely solely on KL divergence as the loss function during training. These skewed pseudo-labels lead the model to learn incorrect semantic relationships, resulting in reduced performance accuracy. To address these issues, we propose a novel SSLDL method called RankMatch. RankMatch fully considers the ranking relationships between different labels during the training phase with labeled data to generate higher-quality pseudo-labels. Furthermore, our key observation is that a flexible utilization of pseudo-labels can enhance SSLDL performance. Specifically, focusing solely on the ranking relationships between labels while disregarding their margins helps prevent model overfitting. Theoretically, we prove that incorporating ranking correlations enhances SSLDL performance and establish generalization error bounds for RankMatch.

Figure outperforms original 1: 3DGS.

We show that replacing the standard MSE denoising loss in diffusion models with a nonlinear transformation induced by an f-divergence yields a simple robust training surrogate that empirically improves performance under data contamination, with small additional computational overhead. The theoretical foundation rests on a local divergence construction: under the Gaussian reverse-kernel structure of DDPM, each per-step likelihood ratio follows a lognormal distribution parameterized by a scalar mismatch, so the conditional f-divergence at each step reduces to a one-dimensional function of the denoising error. Summing these local divergences yields a training objective that unifies diffusion training as divergence induced weighted denoising, where the derivative of the induced divergence acts as a residual-space influence weight that controls the contribution of each sample. Bounded-influence divergences (Hellinger, negative exponential) suppress large error samples, with Hellinger yielding an explicit exponential weight, connecting the framework to robust M-estimation. Empirically, on CIFAR-10 under 30% contamination, NED reduces FID from 93.0 (KL) to 77.5, while also outperforming standard robust losses such as Huber and clipped MSE.

We analyze the generalization gap (gap between the training and test errors) when training a potentially over-parametrized model using a Markovian stochastic training algorithm, initialized from some distribution θ0 p0. We focus on Langevin dynamics with a positive temperature β 1, i.e. gradient descent on a training loss Lwith infinitesimal step size, perturbed with β 1-variances Gaussian noise, and lightly regularized or bounded. There, we bound the generalization gap, at any time during training, by p (βEL(θ0)+ln(1/δ))/N with probability 1 δ over the dataset, where N is the sample size, and EL(θ0) = O(1)with standard initialization scaling. In contrast to previous guarantees, we have no dependence on either training time or reliance on mixing, nor a dependence on dimensionality, gradient norms, or any other properties of the loss or model. This guarantee follows from a general analysis of any Markov process-based training that has a Gibbs-style stationary distribution. The proof is surprisingly simple, once we observe that the marginal distribution divergence from initialization remains bounded, as implied by a generalized second law of thermodynamics.