Information Theoretic Lower Bounds on Negative Log Likelihood

TL;DR

This paper applies rate-distortion theory to derive fundamental lower bounds on negative log likelihood in latent variable models, linking prior and likelihood optimization through information theory.

Contribution

It formally connects prior optimization in latent models to rate-distortion theory and derives a lower bound on negative log likelihood, demonstrating practical relevance in image modeling.

Findings

Derived a lower bound on negative log likelihood using rate-distortion theory.

Showed equivalence between prior and likelihood optimization in latent models.

Experimentally validated the usefulness of information-theoretic quantities in image modeling.

Abstract

In this article we use rate-distortion theory, a branch of information theory devoted to the problem of lossy compression, to shed light on an important problem in latent variable modeling of data: is there room to improve the model? One way to address this question is to find an upper bound on the probability (equivalently a lower bound on the negative log likelihood) that the model can assign to some data as one varies the prior and/or the likelihood function in a latent variable model. The core of our contribution is to formally show that the problem of optimizing priors in latent variable models is exactly an instance of the variational optimization problem that information theorists solve when computing rate-distortion functions, and then to use this to derive a lower bound on negative log likelihood. Moreover, we will show that if changing the prior can improve the log likelihood, then there is a way to change the likelihood function instead and attain the same log likelihood, and thus rate-distortion theory is of relevance to both optimizing priors as well as optimizing likelihood functions. We will experimentally argue for the usefulness of quantities derived from rate-distortion theory in latent variable modeling by applying them to a problem in image modeling.