  Scientists and engineers are often interested in learning the number of
subpopulations (or components) present in a data set. A common suggestion is to
use a finite mixture model (FMM) with a prior on the number of components. Past
work has shown the resulting FMM component-count posterior is consistent; that
is, the posterior concentrates on the true, generating number of components.
But consistency requires the assumption that the component likelihoods are
perfectly specified, which is unrealistic in practice. In this paper, we add
rigor to data-analysis folk wisdom by proving that under even the slightest
model misspecification, the FMM component-count posterior diverges: the
posterior probability of any particular finite number of components converges
to 0 in the limit of infinite data. Contrary to intuition, posterior-density
consistency is not sufficient to establish this result. We develop novel
sufficient conditions that are more realistic and easily checkable than those
common in the asymptotics literature. We illustrate practical consequences of
our theory on simulated and real data.
