Asymptotic Normality of the Maximum Pseudolikelihood Estimator for Fully Visible Boltzmann Machines

TL;DR

This paper proves that the maximum pseudolikelihood estimator for fully visible Boltzmann machines is asymptotically normal, enabling statistical inference, and confirms this through simulation results.

Contribution

It establishes the asymptotic normality of MPLE for fully visible Boltzmann machines, extending previous consistency results.

Findings

MPLE is asymptotically normal for fully visible BMs

Theoretical results are supported by simulation studies

Enables confidence interval construction and hypothesis testing

Abstract

Boltzmann machines (BMs) are a class of binary neural networks for which there have been numerous proposed methods of estimation. Recently, it has been shown that in the fully visible case of the BM, the method of maximum pseudolikelihood estimation (MPLE) results in parameter estimates which are consistent in the probabilistic sense. In this article, we investigate the properties of MPLE for the fully visible BMs further, and prove that MPLE also yields an asymptotically normal parameter estimator. These results can be used to construct confidence intervals and to test statistical hypotheses. We support our theoretical results by showing that the estimator behaves as expected in a simulation study.