Variational inference for approximate objective priors using neural networks

TL;DR

This paper introduces a neural network-based variational inference algorithm to approximate objective priors, especially Jeffreys priors, in Bayesian models, facilitating better prior selection and posterior recovery in complex statistical settings.

Contribution

The authors develop a flexible variational inference method using neural networks to approximate objective priors, including Jeffreys priors, and demonstrate its effectiveness through numerical experiments.

Findings

Successfully approximates Jeffreys and modified priors.

Recovers relevant posterior distributions using MCMC.

Performs well across increasing model complexities.

Abstract

In Bayesian statistics, the choice of the prior can have an important influence on the posterior and the parameter estimation, especially when few data samples are available. To limit the added subjectivity from a priori information, one can use the framework of objective priors, more particularly, we focus on reference priors in this work. However, computing such priors is a difficult task in general. Hence, we consider cases where the reference prior simplifies to the Jeffreys prior. We develop in this paper a flexible algorithm based on variational inference which computes approximations of priors from a set of parametric distributions using neural networks. We also show that our algorithm can retrieve modified Jeffreys priors when constraints are specified in the optimization problem to ensure the solution is proper. We propose a simple method to recover a relevant approximation of the parametric posterior distribution using Markov Chain Monte Carlo (MCMC) methods even if the density function of the parametric prior is not known in general. Numerical experiments on several statistical models of increasing complexity are presented. We show the usefulness of this approach by recovering the target distribution. The performance of the algorithm is evaluated on both prior and posterior distributions, jointly using variational inference and MCMC sampling.