Maximum Likelihood Estimation of the Latent Class Model through Model Boundary Decomposition

Elizabeth S. Allman, Hector Banos, Robin Evans, Serkan Hosten*, Kaie Kubjas, Daniel Lemke, John A. Rhodes, Piotr Zwiernik

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

127 Downloads (Pure)


The Expectation-Maximization (EM) algorithm is routinely used for maximum likelihood estimation in latent class analysis. However, the EM algorithm comes with no global 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 and performance of the EM algorithm. 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.

Original languageEnglish
Pages (from-to)51-84
Number of pages34
JournalJournal of Algebraic Statistics
Issue number1
Publication statusPublished - 2019
MoE publication typeA1 Journal article-refereed


  • Maximum likelihood estimation
  • Expectation Maximization
  • latent class models
  • fixed point ideals
  • boundary stratification
  • RANK

Cite this