  Bayesian mixture models are widely used for clustering of high-dimensional
data with appropriate uncertainty quantification. However, as the dimension of
the observations increases, posterior inference often tends to favor too many
or too few clusters. This article explains this behavior by studying the random
partition posterior in a non-standard setting with a fixed sample size and
increasing data dimensionality. We provide conditions under which the finite
sample posterior tends to either assign every observation to a different
cluster or all observations to the same cluster as the dimension grows.
Interestingly, the conditions do not depend on the choice of clustering prior,
as long as all possible partitions of observations into clusters have positive
prior probabilities, and hold irrespective of the true data-generating model.
We then propose a class of latent mixtures for Bayesian clustering (Lamb) on a
set of low-dimensional latent variables inducing a partition on the observed
data. The model is amenable to scalable posterior inference and we show that it
can avoid the pitfalls of high-dimensionality under mild assumptions. The
proposed approach is shown to have good performance in simulation studies and
an application to inferring cell types based on scRNAseq.
