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Information Theoretic Generative Modeling

In this article we use rate-distortion theory, a branch of information theory devoted to the problem of lossy compression introduced by Claude Shannon in 1959 [1], 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. The result we present here runs much deeper than the particular modeling problem being solved - in formally connecting the latent variable modeling problem to rate-distortion theory, we have established a bridge where decades of work on either field can now be considered for possible cross-pollination; notably in a subsequent article we intend to ask the question next of whether practical algorithms in data compression can be used to design latent variables. 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.