  Maximizing the log-likelihood is a crucial aspect of learning latent variable
models, and variational inference (VI) stands as the commonly adopted method.
However, VI can encounter challenges in achieving a high log-likelihood when
dealing with complicated posterior distributions. In response to this
limitation, we introduce a novel variational importance sampling (VIS) approach
that directly estimates and maximizes the log-likelihood. VIS leverages the
optimal proposal distribution, achieved by minimizing the forward $\chi^2$
divergence, to enhance log-likelihood estimation. We apply VIS to various
popular latent variable models, including mixture models, variational
auto-encoders, and partially observable generalized linear models. Results
demonstrate that our approach consistently outperforms state-of-the-art
baselines, both in terms of log-likelihood and model parameter estimation.
