Statistical physical view of statistical inference in Bayesian linear regression model

Abstract

This paper considers similarities between statistical physics and Bayes inference through the Bayesian linear regression model. Some similarities have been discussed previously, such as the analogy between the marginal likelihood in Bayes inference and the partition function in statistical mechanics. In particular, this paper considers the proposal to associate discrete sample size with inverse temperature [C. H. LaMont and P. A. Wiggins, Phys. Rev. E 99, 052140 (2019)2470-004510.1103/PhysRevE.99.052140]. The previous study suggested that incorporating this similarity motivates the derivation of analogs of thermodynamic functions such as energy and entropy. The study also anticipated that those analogous functions have potential to describe Bayes estimation from physical points of view and to provide physical insights into mechanisms of estimation. This paper incorporates a macroscopic perspective as an asymptotics similar to the thermodynamic limit into the previous suggestion. Its motivation stems from the statistical mechanical concept of deriving thermodynamic functions that characterize macroscopic properties of macroscopic systems. This incorporation not only allows analogs of macroscopic thermodynamic functions to be considered but also suggests a candidate for an analog of inverse temperature with continuity, which is partly consistent with the previous proposal to associate the discrete sample size with inverse temperature. On the basis of this suggestion, we analyze analogs of macroscopic thermodynamic functions for a Bayesian linear regression model which is the basis of various machine learning models. We further investigate, through the behavior of these functions, how Bayes estimation is described from the perspective of physics and what kind of physical insight is obtained. As a result, the estimation of regression coefficients, which is the primary task of regression, appears to be described by the physical picture of balance between decreasing energy and increasing entropy as in equilibrium states of thermodynamic systems. More specifically we observe the physical view of Bayes inference as follows: the estimation succeeds where the effect of decreasing energy is dominant at low temperature. On the other hand, the estimation fails where the effect of increasing entropy is dominant at high temperature.