  Conditional independence in a multivariate normal (or Gaussian) distribution
is characterized by the vanishing of subdeterminants of the distribution's
covariance matrix. Gaussian conditional independence models thus correspond to
algebraic subsets of the cone of positive definite matrices. For statistical
inference in such models it is important to know whether or not the model
contains singularities. We study this issue in models involving up to four
random variables. In particular, we give examples of conditional independence
relations which, despite being probabilistically representable, yield models
that non-trivially decompose into a finite union of several smooth submodels.
