  Variational auto-encoders (VAE) are popular deep latent variable models which
are trained by maximizing an Evidence Lower Bound (ELBO). To obtain tighter
ELBO and hence better variational approximations, it has been proposed to use
importance sampling to get a lower variance estimate of the evidence. However,
importance sampling is known to perform poorly in high dimensions. While it has
been suggested many times in the literature to use more sophisticated
algorithms such as Annealed Importance Sampling (AIS) and its Sequential
Importance Sampling (SIS) extensions, the potential benefits brought by these
advanced techniques have never been realized for VAE: the AIS estimate cannot
be easily differentiated, while SIS requires the specification of carefully
chosen backward Markov kernels. In this paper, we address both issues and
demonstrate the performance of the resulting Monte Carlo VAEs on a variety of
applications.
