A Large-Deviation Analysis of the Maximum-Likelihood Learning of Markov Tree Structures

TL;DR

This paper analyzes the probability decay rate of errors in maximum-likelihood learning of Markov tree structures, revealing how the error probability diminishes exponentially and how noise impacts learning accuracy.

Contribution

It provides a large-deviation framework for understanding ML-estimation errors in Markov trees and introduces an SNR-based approximation for the error exponent in noisy conditions.

Findings

Error probability decays exponentially with sample size.

The dominant error event involves a non-neighbor node pair replacing a true edge.

SNR approximation accurately predicts error decay in noisy regimes.

Abstract

The problem of maximum-likelihood (ML) estimation of discrete tree-structured distributions is considered. Chow and Liu established that ML-estimation reduces to the construction of a maximum-weight spanning tree using the empirical mutual information quantities as the edge weights. Using the theory of large-deviations, we analyze the exponent associated with the error probability of the event that the ML-estimate of the Markov tree structure differs from the true tree structure, given a set of independently drawn samples. By exploiting the fact that the output of ML-estimation is a tree, we establish that the error exponent is equal to the exponential rate of decay of a single dominant crossover event. We prove that in this dominant crossover event, a non-neighbor node pair replaces a true edge of the distribution that is along the path of edges in the true tree graph connecting the nodes in the non-neighbor pair. Using ideas from Euclidean information theory, we then analyze the scenario of ML-estimation in the very noisy learning regime and show that the error exponent can be approximated as a ratio, which is interpreted as the signal-to-noise ratio (SNR) for learning tree distributions. We show via numerical experiments that in this regime, our SNR approximation is accurate.