Thermodynamic Bayesian model comparison

Thermodynamics have been shown to have direct applications in Bayesian model evaluation. Within a tempered transitions scheme, the Boltzmann-Gibbs distribution pertaining to different Hamiltonians is implemented to create a path which links the distributions of interest at the endpoints. As illustrated here, an optimal temperature exists along the path which directly provides the free energy, which in this context corresponds to the marginal likelihood and/or Bayes factor. Estimators which have been developed under this framework are organised here using a unifying approach, in parallel with their stepping-stone sampling counterparts. New estimators are presented and the use of compound paths is introduced. As a byproduct, it is shown how the thermodynamic integral allows for the estimation of probability distribution divergences and measures of statistical entropy. A geometric approach is employed here to illustrate the importance of the choice of the path in terms of the corresponding estimator's error (path-related variance), which provides a more intuitive approach in tuning the error sources.