Maximum likelihood estimation of the Latent Class Model through model boundary decomposition

TL;DR

This paper investigates the geometric structure of binary latent class models to improve understanding and computation of maximum likelihood estimators, especially addressing the limitations of the EM algorithm.

Contribution

It characterizes the boundary stratification of the model and provides exact MLE computation methods for small models, enhancing theoretical understanding and practical estimation.

Findings

Exact MLE computation for small models using boundary stratification

Characterization of boundary stratification in binary latent class models

Analysis of EM fixed point ideal and minimal primes

Abstract

The Expectation-Maximization (EM) algorithm is routinely used for the maximum likelihood estimation in the latent class analysis. However, the EM algorithm comes with no guarantees of reaching the global optimum. We study the geometry of the latent class model in order to understand the behavior of the maximum likelihood estimator. In particular, we characterize the boundary stratification of the binary latent class model with a binary hidden variable. For small models, such as for three binary observed variables, we show that this stratification allows exact computation of the maximum likelihood estimator. In this case we use simulations to study the maximum likelihood estimation attraction basins of the various strata. Our theoretical study is complemented with a careful analysis of the EM fixed point ideal which provides an alternative method of studying the boundary stratification and maximizing the likelihood function. In particular, we compute the minimal primes of this ideal in the case of a binary latent class model with a binary or ternary hidden random variable.