  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. Linking these two ideas, recent work has
interpreted the geometric mixture path as an exponential family of
distributions to analyze the thermodynamic variational objective (TVO).
  We extend these likelihood ratio exponential families to include solutions to
rate-distortion (RD) optimization, the information bottleneck (IB) method, and
recent rate-distortion-classification 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.
