Approximate Bayesian Inference

A generalized likelihood-based Bayesian approach for scalable joint regression and covariance selection in high dimensions

The paper addresses joint sparsity selection in the regression coefficient matrix and the error precision (inverse covariance) matrix for high-dimensional multivariate regression models in the Bayesian paradigm. The selected sparsity patterns are crucial to help understand the network of relationships between the predictor and response variables, as well as the conditional relationships among the latter. While Bayesian methods have the advantage of providing natural uncertainty quantification through posterior inclusion probabilities and credible intervals, current Bayesian approaches either restrict to specific sub-classes of sparsity patterns and/or are not scalable to settings with hundreds of responses and predictors. Bayesian approaches which only focus on estimating the posterior mode are scalable, but do not generate samples from the posterior distribution for uncertainty quantification. Using a bi-convex regression based generalized likelihood and spike-and-slab priors, we develop an algorithm called Joint Regression Network Selector (JRNS) for joint regression and covariance selection which (a) can accommodate general sparsity patterns, (b) provides posterior samples for uncertainty quantification, and (c) is scalable and orders of magnitude faster than the state-of-the-art Bayesian approaches providing uncertainty quantification. We demonstrate the statistical and computational efficacy of the proposed approach on synthetic data and through the analysis of selected cancer data sets. We also establish high-dimensional posterior consistency for one of the developed algorithms.

Meta-Strategy for Learning Tuning Parameters with Guarantees

Online learning methods, similar to the online gradient algorithm (OGA) and exponentially weighted aggregation (EWA), often depend on tuning parameters that are difficult to set in practice. We consider an online meta-learning scenario, and we propose a meta-strategy to learn these parameters from past tasks. Our strategy is based on the minimization of a regret bound. It allows us to learn the initialization and the step size in OGA with guarantees. It also allows us to learn the prior or the learning rate in EWA. We provide a regret analysis of the strategy. It allows to identify settings where meta-learning indeed improves on learning each task in isolation.

Approximate Bayesian Computation for Discrete Spaces

Many real-life processes are black-box problems, i.e., the internal workings are inaccessible or a closed-form mathematical expression of the likelihood function cannot be defined. For continuous random variables, likelihood-free inference problems can be solved via Approximate Bayesian Computation (ABC). However, an optimal alternative for discrete random variables is yet to be formulated. Here, we aim to fill this research gap. We propose an adjusted population-based MCMC ABC method by re-defining the standard ABC parameters to discrete ones and by introducing a novel Markov kernel that is inspired by differential evolution. We first assess the proposed Markov kernel on a likelihood-based inference problem, namely discovering the underlying diseases based on a QMR-DTnetwork and, subsequently, the entire method on three likelihood-free inference problems: (i) the QMR-DT network with the unknown likelihood function, (ii) the learning binary neural network, and (iii) neural architecture search. The obtained results indicate the high potential of the proposed framework and the superiority of the new Markov kernel.

Variationally Inferred Sampling through a Refined Bound

In this work, a framework to boost the efficiency of Bayesian inference in probabilistic models is introduced by embedding a Markov chain sampler within a variational posterior approximation. We call this framework "refined variational approximation". Its strengths are its ease of implementation and the automatic tuning of sampler parameters, leading to a faster mixing time through automatic differentiation. Several strategies to approximate evidence lower bound (ELBO) computation are also introduced. Its efficient performance is showcased experimentally using state-space models for time-series data, a variational encoder for density estimation and a conditional variational autoencoder as a deep Bayes classifier.