  Mixtures of product distributions are a powerful device for learning about
heterogeneity within data populations. In this class of latent structure
models, de Finetti's mixing measure plays the central role for describing the
uncertainty about the latent parameters representing heterogeneity. In this
paper posterior contraction theorems for de Finetti's mixing measure arising
from finite mixtures of product distributions will be established, under the
setting the number of exchangeable sequences of observed variables increases
while sequence length(s) may be either fixed or varied. The role of both the
number of sequences and the sequence lengths will be carefully examined. In
order to obtain concrete rates of convergence, a first-order identifiability
theory for finite mixture models and a family of sharp inverse bounds for
mixtures of product distributions will be developed via a harmonic analysis of
such latent structure models. This theory is applicable to broad classes of
probability kernels composing the mixture model of product distributions for
both continuous and discrete domain $\mathfrak{X}$. Examples of interest
include the case the probability kernel is only weakly identifiable in the
sense of Ho and Nguyen (2016), the case where the kernel is itself a mixture
distribution as in hierarchical models, and the case the kernel may not have a
density with respect to a dominating measure on an abstract domain
$\mathfrak{X}$ such as Dirichlet processes.
