  This work introduces a new method for selecting the number of components in
finite mixture models (FMMs) using variational Bayes, inspired by the
large-sample properties of the Evidence Lower Bound (ELBO) derived from
mean-field (MF) variational approximation. Specifically, we establish matching
upper and lower bounds for the ELBO without assuming conjugate priors,
suggesting the consistency of model selection for FMMs based on maximizing the
ELBO. As a by-product of our proof, we demonstrate that the MF approximation
inherits the stable behavior (benefited from model singularity) of the
posterior distribution, which tends to eliminate the extra components under
model misspecification where the number of mixture components is
over-specified. This stable behavior also leads to the $n^{-1/2}$ convergence
rate for parameter estimation, up to a logarithmic factor, under this model
overspecification. Empirical experiments are conducted to validate our
theoretical findings and compare with other state-of-the-art methods for
selecting the number of components in FMMs.
