Submodular and supermodular functions have found wide applicability in machine learning, capturing notions such as diversity and regularity, respectively. These notions have deep consequences for optimization, and the problem of (approximately) optimizing submodular functions has received much attention. However, beyond optimization, these notions allow specifying expressive probabilistic models that can be used to quantify predictive uncertainty via marginal inference. Prominent, well-studied special cases include Ising models and determinan-tal point processes, but the general class of log-submodular and log-supermodular models is much richer and little studied. In this paper, we investigate the use of Markov chain Monte Carlo sampling to perform approximate inference in general log-submodular and log-supermodular models. In particular, we consider a simple Gibbs sampling procedure, and establish two sufficient conditions, the first guaranteeing polynomial-time, and the second fast ( O ( n log n)) mixing. We also evaluate the efficiency of the Gibbs sampler on three examples of such models, and compare against a recently proposed variational approach.

Neural Information Processing Systems http://nips.cc/

We propose and investigate new complementary methodologies for estimating predictive variance networks in regression neural networks.

Neural Information Processing Systems http://nips.cc/

Neural Information Processing Systems http://nips.cc/

The emergence of in-context learning (ICL) in large language models (LLMs) remains poorly understood despite its consistent effectiveness, enabling models to adapt to new tasks from only a handful of examples. To clarify and improve these capabilities, we characterize how the statistical properties of the pretraining distribution (e.g., tail behavior, coverage) shape ICL on numerical tasks. We develop a theoretical framework that unifies task selection and generalization, extending and sharpening earlier results, and show how distributional properties govern sample efficiency, task retrieval, and robustness. To this end, we generalize Bayesian posterior consistency and concentration results to heavy-tailed priors and dependent sequences, better reflecting the structure of LLM pretraining data. We then empirically study how ICL performance varies with the pretraining distribution on challenging tasks such as stochastic differential equations and stochastic processes with memory. Together, these findings suggest that controlling key statistical properties of the pretraining distribution is essential for building ICL-capable and reliable LLMs.

Simultaneously performing variable selection and inference in high-dimensional models is an open challenge in statistics and machine learning. The increasing availability of vast amounts of variables requires the adoption of specific statistical procedures to accurately select the most important predictors in a high-dimensional space, while being able to control some form of selection error. In this work we adapt the Mirror Statistic approach to False Discovery Rate (FDR) control into a Bayesian modelling framework. The Mirror Statistic, developed in the classic frequentist statistical framework, is a flexible method to control FDR, which only requires mild model assumptions, but requires two sets of independent regression coefficient estimates, usually obtained after splitting the original dataset. Here we propose to rely on a Bayesian formulation of the model and use the posterior distributions of the coefficients of interest to build the Mirror Statistic and effectively control the FDR without the need to split the data. Moreover, the method is very flexible since it can be used with continuous and discrete outcomes and more complex predictors, such as with mixed models. We keep the approach scalable to high-dimensions by relying on Automatic Differentiation Variational Inference and fully continuous prior choices.

Bayesian optimal experimental design provides a principled framework for selecting experimental settings that maximize obtained information. In this work, we focus on estimating the expected information gain in the setting where the differential entropy of the likelihood is either independent of the design or can be evaluated explicitly. This reduces the problem to maximum entropy estimation, alleviating several challenges inherent in expected information gain computation. Our study is motivated by large-scale inference problems, such as inverse problems, where the computational cost is dominated by expensive likelihood evaluations. We propose a computational approach in which the evidence density is approximated by a Monte Carlo or quasi-Monte Carlo surrogate, while the differential entropy is evaluated using standard methods without additional likelihood evaluations. We prove that this strategy achieves convergence rates that are comparable to, or better than, state-of-the-art methods for full expected information gain estimation, particularly when the cost of entropy evaluation is negligible. Moreover, our approach relies only on mild smoothness of the forward map and avoids stronger technical assumptions required in earlier work. We also present numerical experiments, which confirm our theoretical findings.

With the increasing demand for interpretability in machine learning, functional ANOVA decomposition has gained renewed attention as a principled tool for breaking down high-dimensional function into low-dimensional components that reveal the contributions of different variable groups. Recently, Tensor Product Neural Network (TPNN) has been developed and applied as basis functions in the functional ANOVA model, referred to as ANOVA-TPNN. A disadvantage of ANOVA-TPNN, however, is that the components to be estimated must be specified in advance, which makes it difficult to incorporate higher-order TPNNs into the functional ANOVA model due to computational and memory constraints. In this work, we propose Bayesian-TPNN, a Bayesian inference procedure for the functional ANOVA model with TPNN basis functions, enabling the detection of higher-order components with reduced computational cost compared to ANOVA-TPNN. We develop an efficient MCMC algorithm and demonstrate that Bayesian-TPNN performs well by analyzing multiple benchmark datasets. Theoretically, we prove that the posterior of Bayesian-TPNN is consistent.

Ordinal regression (OR, also called ordinal classification) is classification of ordinal data, in which the underlying target variable is categorical and considered to have a natural ordinal relation for the underlying explanatory variable. A key to successful OR models is to find a data structure `natural ordinal relation' common to many ordinal data and reflect that structure into the design of those models. A recent OR study found that many real-world ordinal data show a tendency that the conditional probability distribution (CPD) of the target variable given a value of the explanatory variable will often be unimodal. Several previous studies thus developed unimodal likelihood models, in which a predicted CPD is guaranteed to become unimodal. However, it was also observed experimentally that many real-world ordinal data partly have values of the explanatory variable where the underlying CPD will be non-unimodal, and hence unimodal likelihood models may suffer from a bias for such a CPD. Therefore, motivated to mitigate such a bias, we propose approximately unimodal likelihood models, which can represent up to a unimodal CPD and a CPD that is close to be unimodal. We also verify experimentally that a proposed model can be effective for statistical modeling of ordinal data and OR tasks.