We provide the definitions of important terms used throughout the paper. Assumption 2.3 when the demand distribution is exponential. Note that Lemma B.1 implies that In the following result, we show that there exist appropriate constants such that prior distribution satisfies Assumption 2.3 when the demand distribution is a multivariate Gaussian with unknown The proof is a direct consequence of Theorem 3.2, Lemmas B.6, B.7, B.8, B.9, and Proposition 3.2. Theorem 6.19] the prior induced by Assumption 2.2 is a direct consequence of Assumption 2.4 and 2.5 are straightforward to satisfy since the model risk function Lemma B.13. F or a given Using the result above together with Proposition 3.2 implies that the RSVB posterior converges at C.1 Alternative derivation of LCVB We present the alternative derivation of LCVB. We prove our main result after a series of important lemmas.

To alleviate the risk of model-misspecification in MLE, we propose to generate samples from the generativemodel and monitor the quality of the samples in the process of training until the samples and the real data are indistinguishable.

We present the winning strategy for the EVA2025 Data Challenge, which aimed to estimate the probability of extreme precipitation events. These events occurred at most once in the dataset making the challenge fundamentally one of extrapolating extreme values. Given the scarcity of extreme events, we argue that a simple, robust modeling approach is essential. We adopt univariate models instead of multivariate ones and model Peaks Over Thresholds using Extreme Value Theory. Specifically, we fit an exponential distribution to model exceedances of the target variable above a high quantile (after seasonal adjustment). The novelty of our approach lies in using martingale testing to evaluate the extrapolation power of the procedure and to agnostically select the level of the high quantile. While this method has several limitations, we believe that framing extrapolation as a game opens the door to other agnostic approaches in Extreme Value Analysis.

Missing data problems have many manifestations across many scientific fields.

Large language model (LLM) generation often requires balancing output quality against inference cost, especially when using multiple generations. We introduce a new framework for inference-time optimization based on the classical Pandora's Box problem. Viewing each generation as opening a costly "box" with random reward, we develop algorithms that decide when to stop generating without knowing the underlying reward distribution. Our first contribution is a UCB-style Pandora's Box algorithm, which achieves performance that is provably close to Weitzman's algorithm, the optimal strategy when the distribution is known. We further adapt this method to practical LLM settings by addressing reward scaling across prompts via a Bradley-Terry inspired transformation. This leads to an adaptive inference-time optimization method that normalizes rewards and learns stopping thresholds on the fly. Experiments on the AlpacaFarm and HH-RLHF datasets, using multiple LLM-reward model pairs, show that our adaptive strategy can obtain the same performance as non-adaptive Best-of-N sampling while requiring 15-35 percent fewer generations on average. Our results establish a principled bridge between optimal stopping theory and inference-time scaling, providing both theoretical performance bounds and practical efficiency gains for LLM deployment.

Conditional random fields, which model the distribution of a multivariate response conditioned on a set of covariates using undirected graphs, are widely used in a variety of multivariate prediction applications. Popular instances of this class of models such as categorical-discrete CRFs, Ising CRFs, and conditional Gaussian based CRFs, are not however best suited to the varied types of response variables in many applications, including count-valued responses. We thus introduce a "novel subclass of CRFs", derived by imposing node-wise conditional distributions of response variables conditioned on the rest of the responses and the covariates as arising from univariate exponential families. This allows us to derive novel multivariate CRFs given any univariate exponential distribution, including the Poisson, negative binomial, and exponential distributions. Also in particular, it addresses the common CRF problem of specifying feature'' functions determining the interactions between response variables and covariates.

Evidence is presented that the accuracy of Nonextensive Statistical Mechanics framework is improved using the coupled entropy, which carefully establishes the physical measures of complex systems. While Nonextensive Statistical Mechanics (NSM) has developed into a powerful toolset, questions have persisted as to how to evaluate whether its proposed solutions properly characterize the uncertainty of heavy-tailed distributions. The entropy of the generalized Pareto distribution (GPD) is $1+κ+\lnσ$, where $κ$ is the shape or nonlinear coupling and $σ$ is the scale. A generalized entropy should retain the uncertainty due to the scale, while minimizing the dependence of the nonlinear coupling. The Tsallis entropy of the GPD instead subtracts a function of the inverse-scale and converges to one as $κ\rightarrow\infty$. Colloquially, the Tsallis entropy is too cold. The normalized Tsallis entropy (NTE) rectifies the positive dependence on the scale but introduces a nonlinear term multiplying the scale and the coupling, making it too hot. The coupled entropy measures the uncertainty of the GPD to be $1+\ln_\fracκ{1+κ}σ=1+\frac{1+κ}κ(σ^\fracκ{1+κ}-1)$, which converges to $σ$ as $κ\rightarrow\infty$. One could say, the coupled entropy allows scientists, engineers, and analysts to eat their porridge, confident that its measure of uncertainty reflects the mathematical physics of the scale of non-exponential distributions while minimizing the dependence on the shape or nonlinear coupling. The training of the coupled variational autoencoder is an example of the unique ability of the coupled entropy to improve the performance of complex systems.