Neural networks are the cornerstone of modern machine learning, yet can be difficult to interpret, give overconfident predictions and are vulnerable to adversarial attacks. Bayesian neural networks (BNNs) provide some alleviation of these limitations, but have problems of their own. The key step of specifying prior distributions in BNNs is no trivial task, yet is often skipped out of convenience. In this work, we propose a new class of prior distributions for BNNs, the Dirichlet scale mixture (DSM) prior, that addresses current limitations in Bayesian neural networks through structured, sparsity-inducing shrinkage. Theoretically, we derive general dependence structures and shrinkage results for DSM priors and show how they manifest under the geometry induced by neural networks. In experiments on simulated and real world data we find that the DSM priors encourages sparse networks through implicit feature selection, show robustness under adversarial attacks and deliver competitive predictive performance with substantially fewer effective parameters. In particular, their advantages appear most pronounced in correlated, moderately small data regimes, and are more amenable to weight pruning. Moreover, by adopting heavy-tailed shrinkage mechanisms, our approach aligns with recent findings that such priors can mitigate the cold posterior effect, offering a principled alternative to the commonly used Gaussian priors.

Sparse regression problems, where the goal is to identify a small set of relevant predictors, often require modeling not only main effects but also meaningful interactions through other variables. While the pliable lasso has emerged as a powerful frequentist tool for modeling such interactions under strong heredity constraints, it lacks a natural framework for uncertainty quantification and incorporation of prior knowledge. In this paper, we propose a Bayesian pliable lasso that extends this approach by placing sparsity-inducing priors, such as the horseshoe, on both main and interaction effects. The hierarchical prior structure enforces heredity constraints while adaptively shrinking irrelevant coefficients and allowing important effects to persist. We extend this framework to Generalized Linear Models (GLMs) and develop a tailored approach to handle missing responses. To facilitate posterior inference, we develop an efficient Gibbs sampling algorithm based on a reparameterization of the horseshoe prior. Our Bayesian framework yields sparse, interpretable interaction structures, and principled measures of uncertainty. Through simulations and real-data studies, we demonstrate its advantages over existing methods in recovering complex interaction patterns under both complete and incomplete data. Our method is implemented in the package \texttt{hspliable} available on Github.

The horseshoe prior, a widely used handy alternative to the spike-and-slab prior, has proven to be an exceptional default global-local shrinkage prior in Bayesian inference and machine learning. However, designing tests with frequentist false discovery rate (FDR) control using the horseshoe prior or the general class of global-local shrinkage priors remains an open problem. In this paper, we propose a frequentist-assisted horseshoe procedure that not only resolves this long-standing FDR control issue for the high dimensional normal means testing problem but also exhibits satisfactory finite-sample FDR control under any desired nominal level for both large-scale multiple independent and correlated tests. We carry out the frequentist-assisted horseshoe procedure in an easy and intuitive way by using the minimax estimator of the global parameter of the horseshoe prior while maintaining the remaining full Bayes vanilla horseshoe structure. The results of both intensive simulations under different sparsity levels, and real-world data demonstrate that the frequentist-assisted horseshoe procedure consistently achieves robust finite-sample FDR control. Existing frequentist or Bayesian FDR control procedures can lose finite-sample FDR control in a variety of common sparse cases. Based on the intimate relationship between the minimax estimation and the level of FDR control discovered in this work, we point out potential generalizations to achieve FDR control for both more complicated models and the general global-local shrinkage prior family.

Prediction-powered inference (PPI) enables valid statistical inference by combining experimental data with machine learning predictions. When a sufficient number of high-quality predictions is available, PPI results in more accurate estimates and tighter confidence intervals than traditional methods. In this paper, we propose to inform the PPI framework with prior knowledge on the quality of the predictions. The resulting method, which we call frequentist, assisted by Bayes, PPI (FAB-PPI), improves over PPI when the observed prediction quality is likely under the prior, while maintaining its frequentist guarantees. Furthermore, when using heavy-tailed priors, FAB-PPI adaptively reverts to standard PPI in low prior probability regions. We demonstrate the benefits of FAB-PPI in real and synthetic examples.

A probabilistic model based on the horseshoe prior is proposed for learning dependencies in the process of identifying relevant features for prediction. Exact inference is intractable in this model. However, expectation propagation offers an approximate alternative. Because the process of estimating feature selection dependencies may suffer from over-fitting in the model proposed, additional data from a multi-task learning scenario are considered for induction. The same model can be used in this setting with few modifications. Furthermore, the assumptions made are less restrictive than in other multi-task methods: The different tasks must share feature selection dependencies, but can have different relevant features and model coefficients. Experiments with real and synthetic data show that this model performs better than other multi-task alternatives from the literature. The experiments also show that the model is able to induce suitable feature selection dependencies for the problems considered, only from the training data.

The paper presents the algorithm for clustering a dataset by grouping the optimal, from the point of view of the BIC criterion, number of Gaussian clusters into the optimal, from the point of view of their statistical separability, superclusters. The algorithm consists of three stages: representation of the dataset as a mixture of Gaussian distributions - clusters, which number is determined based on the minimum of the BIC criterion; using the Mahalanobis distance, to estimate the distances between the clusters and cluster sizes; combining the resulting clusters into superclusters using the DBSCAN method by finding its hyperparameter (maximum distance) providing maximum value of introduced matrix quality criterion at maximum number of superclusters. The matrix quality criterion corresponds to the proportion of statistically significant separated superclusters among all found superclusters. The algorithm has only one hyperparameter - statistical significance level, and automatically detects optimal number and shape of superclusters based of statistical hypothesis testing approach. The algorithm demonstrates a good results on test datasets in noise and noiseless situations. An essential advantage of the algorithm is its ability to predict correct supercluster for new data based on already trained clusterer and perform soft (fuzzy) clustering. The disadvantages of the algorithm are: its low speed and stochastic nature of the final clustering. It requires a sufficiently large dataset for clustering, which is typical for many statistical methods.

Network complexity and computational efficiency have become increasingly significant aspects of deep learning. Sparse deep learning addresses these challenges by recovering a sparse representation of the underlying target function by reducing heavily over-parameterized deep neural networks. Specifically, deep neural architectures compressed via structured sparsity (e.g. node sparsity) provide low latency inference, higher data throughput, and reduced energy consumption. In this paper, we explore two well-established shrinkage techniques, Lasso and Horseshoe, for model compression in Bayesian neural networks. To this end, we propose structurally sparse Bayesian neural networks which systematically prune excessive nodes with (i) Spike-and-Slab Group Lasso (SS-GL), and (ii) Spike-and-Slab Group Horseshoe (SS-GHS) priors, and develop computationally tractable variational inference including continuous relaxation of Bernoulli variables. We establish the contraction rates of the variational posterior of our proposed models as a function of the network topology, layer-wise node cardinalities, and bounds on the network weights. We empirically demonstrate the competitive performance of our models compared to the baseline models in prediction accuracy, model compression, and inference latency.

Modern genomic studies are increasingly focused on identifying more and more genes clinically associated with a health response. Commonly used Bayesian shrinkage priors are designed primarily to detect only a handful of signals when the dimension of the predictors is very high. In this article, we investigate the performance of a popular continuous shrinkage prior in the presence of relatively large number of true signals. We draw attention to an undesirable phenomenon; the posterior mean is rendered very close to a null vector, caused by a sharp underestimation of the global-scale parameter. The phenomenon is triggered by the absence of a tail-index controlling mechanism in the Bayesian shrinkage priors. We provide a remedy by developing a global-local-tail shrinkage prior which can automatically learn the tail-index and can provide accurate inference even in the presence of moderately large number of signals. The collapsing behavior of the Horseshoe with its remedy is exemplified in numerical examples and in two gene expression datasets.

Regularization is a fascinating topic, that puzzles me for a long time. First introduced in a machine learning course as a given, it always raised a question why it works. Then I started uncover a connection of regularization to the statistical properties of the underlying model. Indeed, if we consider linear regression model, it is easy to show, that L2 regularization is equivalent to adding Gaussian noise to the input. In fact, the latter is preferred if we consider feature interactions (or we have to use a non-trivial Tikhonov Matrix, e.g.

Feature subset selection arises in many high-dimensional applications of statistics, such as compressed sensing and genomics. The $\ell_0$ penalty is ideal for this task, the caveat being it requires the NP-hard combinatorial evaluation of all models. A recent area of considerable interest is to develop efficient algorithms to fit models with a non-convex $\ell_\gamma$ penalty for $\gamma\in (0,1)$, which results in sparser models than the convex $\ell_1$ or lasso penalty, but is harder to fit. We propose an alternative, termed the horseshoe regularization penalty for feature subset selection, and demonstrate its theoretical and computational advantages. The distinguishing feature from existing non-convex optimization approaches is a full probabilistic representation of the penalty as the negative of the logarithm of a suitable prior, which in turn enables efficient expectation-maximization and local linear approximation algorithms for optimization and MCMC for uncertainty quantification. In synthetic and real data, the resulting algorithms provide better statistical performance, and the computation requires a fraction of time of state-of-the-art non-convex solvers.