  This paper focuses on variational inference with intractable likelihood
functions that can be unbiasedly estimated. A flexible variational
approximation based on Gaussian mixtures is developed, by adopting the mixture
population Monte Carlo (MPMC) algorithm in \cite{cappe2008adaptive}. MPMC
updates iteratively the parameters of mixture distributions with importance
sampling computations, instead of the complicated gradient estimation of the
optimization objective in usual variational Bayes. Noticing that MPMC uses a
fixed number of mixture components, which is difficult to predict for real
applications, we further propose an automatic component--updating procedure to
derive an appropriate number of components. The derived adaptive MPMC algorithm
is capable of finding good approximations of the multi-modal posterior
distributions even with a standard Gaussian as the initial distribution, as
demonstrated in our numerical experiments.
