  Statistical models with latent structure have a history going back to the
1950s and have seen widespread use in the social sciences and, more recently,
in computational biology and in machine learning. Here we study the basic
latent class model proposed originally by the sociologist Paul F. Lazarfeld for
categorical variables, and we explain its geometric structure. We draw
parallels between the statistical and geometric properties of latent class
models and we illustrate geometrically the causes of many problems associated
with maximum likelihood estimation and related statistical inference. In
particular, we focus on issues of non-identifiability and determination of the
model dimension, of maximization of the likelihood function and on the effect
of symmetric data. We illustrate these phenomena with a variety of synthetic
and real-life tables, of different dimension and complexity. Much of the
motivation for this work stems from the "100 Swiss Francs" problem, which we
introduce and describe in detail.
