# Empirical Bayes Method for Boltzmann Machines

## TL;DR

This paper introduces a fast, non-iterative empirical Bayes algorithm for Boltzmann machines that uses the replica method and Plefka expansion to estimate hyperparameters efficiently, despite some bias issues.

## Contribution

It proposes a novel, simple, and fast empirical Bayes method for Boltzmann machines that bypasses computational intractability using advanced approximation techniques.

## Key findings

- The method is computationally efficient and does not require iterative procedures.
- It introduces a bias in estimates due to the Plefka expansion.
- The peculiar bias behavior is linked to the approximation method used.

## Abstract

In this study, we consider an empirical Bayes method for Boltzmann machines and propose an algorithm for it. The empirical Bayes method allows estimation of the values of the hyperparameters of the Boltzmann machine by maximizing a specific likelihood function referred to as the empirical Bayes likelihood function in this study. However, the maximization is computationally hard because the empirical Bayes likelihood function involves intractable integrations of the partition function. The proposed algorithm avoids this computational problem by using the replica method and the Plefka expansion. Our method does not require any iterative procedures and is quite simple and fast, though it introduces a bias to the estimate, which exhibits an unnatural behavior with respect to the size of the dataset. This peculiar behavior is supposed to be due to the approximate treatment by the Plefka expansion. A possible extension to overcome this behavior is also discussed.