Analytic Mutual Information in Bayesian Neural Networks

TL;DR

This paper derives an analytical formula for mutual information in Bayesian neural networks, enhancing understanding of epistemic uncertainty and improving active learning strategies.

Contribution

It provides the first explicit analytical expression for mutual information in Bayesian neural networks, linking it to point process entropy and applying it to active learning.

Findings

Analytical formula for mutual information derived

Improved active learning performance demonstrated

Application to Dirichlet distribution parameter estimation

Abstract

Bayesian neural networks have successfully designed and optimized a robust neural network model in many application problems, including uncertainty quantification. However, with its recent success, information-theoretic understanding about the Bayesian neural network is still at an early stage. Mutual information is an example of an uncertainty measure in a Bayesian neural network to quantify epistemic uncertainty. Still, no analytic formula is known to describe it, one of the fundamental information measures to understand the Bayesian deep learning framework. In this paper, we derive the analytical formula of the mutual information between model parameters and the predictive output by leveraging the notion of the point process entropy. Then, as an application, we discuss the parameter estimation of the Dirichlet distribution and show its practical application in the active learning uncertainty measures by demonstrating that our analytical formula can improve the performance of active learning further in practice.