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.
Use rate-distortion theory to bound how much a latent variable model can be improved Addresses problems of optimization of the prior in the latent variable model and the selection of the likelihood function by proposing criteria based on a lower-bound on the negative log-likelihood. Presents a theorem which gives a lower bound on negative log likelihood of rate-distortion for latent-variable modeling The authors argue that the rate-distortion theory for lossy compression provides a natural toolkit for studying latent variable models proposes a lower bound.