  For data assumed to come from a finite mixture with an unknown number of
components, it has become common to use Dirichlet process mixtures (DPMs) not
only for density estimation, but also for inferences about the number of
components. The typical approach is to use the posterior distribution on the
number of components occurring so far --- that is, the posterior on the number
of clusters in the observed data. However, it turns out that this posterior is
not consistent --- it does not converge to the true number of components. In
this note, we give an elementary demonstration of this inconsistency in what is
perhaps the simplest possible setting: a DPM with normal components of unit
variance, applied to data from a "mixture" with one standard normal component.
Further, we find that this example exhibits severe inconsistency: instead of
going to 1, the posterior probability that there is one cluster goes to 0.
