Phase Transitions for the Uniform Distribution in the PML Problem and its Bethe Approximation

TL;DR

This paper investigates phase transition phenomena in the PML estimate for probability distributions and explores whether similar transitions occur in its Bethe approximation, revealing insights into the optimization landscape.

Contribution

It identifies a phase transition in the PML estimate and analyzes its presence in the Bethe approximation, linking statistical physics concepts with distribution estimation.

Findings

Uniform distribution shifts from local maximum to minimum at a sharp threshold.

Evidence of a similar phase transition in the Bethe approximation.

Analysis involves properties of random matrices with fixed row and column sums.

Abstract

The pattern maximum likelihood (PML) estimate, introduced by Orlitsky et al., is an estimate of the multiset of probabilities in an unknown probability distribution $\mathbf{p}$, the estimate being obtained from $n$ i.i.d. samples drawn from $\mathbf{p}$. The PML estimate involves solving a difficult optimization problem over the set of all probability mass functions (pmfs) of finite support. In this paper, we describe an interesting phase transition phenomenon in the PML estimate: at a certain sharp threshold, the uniform distribution goes from being a local maximum to being a local minimum for the optimization problem in the estimate. We go on to consider the question of whether a similar phase transition phenomenon also exists in the Bethe approximation of the PML estimate, the latter being an approximation method with origins in statistical physics. We show that the answer to this question is a qualified "Yes". Our analysis involves the computation of the mean and variance of the $(i,j)$th entry, $a_{i,j}$, in a random $k \times k$ non-negative integer matrix $A$ with row and column sums all equal to $M$, drawn according to a distribution that assigns to $A$ a probability proportional to $\prod_{i,j} \frac{(M-a_{i,j})!}{a_{i,j}!}$.