  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.
