A correspondence between thermodynamics and inference

We explore a natural analogy between Bayesian statistics and thermal physics in which sample size corresponds to inverse temperature. This analogy motivates the definition of two novel statistical quantities: a learning capacity and a Gibbs entropy. The analysis of the learning capacity, corresponding to the heat capacity in thermal physics, leads to a critical insight into why some models have anomalously good learning performance. The mechanism is a statistical analogue of the failure of the equipartition theorem formula for the heat capacity. We explore the properties of the learning capacity in a number of examples, including a sloppy model. We also propose that the Gibbs entropy provides a natural device for counting distinguishable distributions in the context of Bayesian inference. This insight results in a new solution to a long-standing problem in Bayesian inference: the definition of an objective or uninformative prior. We use the Gibbs entropy to define a generalized principle of indifference (GPI) in which every distinguishable model is assigned equal a priori probability. This approach both resolves a number of long standing inconsistencies in objective Bayesian inference and unifies several seemingly unrelated Bayesian methods with the information-based paradigm of inference.