Likelihood Ratio Exponential Families

Rob Brekelmans, Frank Nielsen, Alireza Makhzani, Aram Galstyan, Greg Ver Steeg

Keywords: rate-distortion, thermodynamic variational objective, free energy, hypothesis testing, legendre duality

TL;DR: We analyse likelihood ratio exponential families, which include TVO, RD, and IB as special cases, using Legendre duality and hypothesis testing.

Abstract:

The exponential family is well known in machine learning and statistical physics as the maximum entropy distribution subject to a set of observed constraints, while the geometric mixture path is common in MCMC methods such as annealed importance sampling (AIS). Linking these two ideas, recent work has interpreted the geometric mixture path as an exponential family of distributions to analyse the thermodynamic variational objective (TVO).

In this work, we extend \textit{likelihood ratio exponential families} to include solutions to RD optimization, the IB method, and recent `` RDC' approaches which combine RD and IB. This provides a common mathematical framework for understanding these methods via the conjugate duality of exponential families and hypothesis testing. Further, we collect existing results to provide a variational representation of intermediate RD or TVO distributions as a minimizing an expectation of KL divergences. This solution also corresponds to a size-power tradeoff using the likelihood ratio test and the Neyman Pearson lemma. In thermodynamic integration bounds such as the TVO, we identify the intermediate distribution whose expected sufficient statistics match the log partition function.