AIS with geometric paths is often considered the goldstandard for evaluating decoder-based generative models (Wu et al., 2017). In this section, we evaluate whether qpaths can improve marginal likelihood estimation for a vari-  First, we use AIS to evaluate the trained generative model on the true test set, with a Gaussian prior π 0 (z) = p(z) as the base distribution and true posterior π 1 (z) = p(z|x) ∝ p(x, z) as the target. Intermediate distributions then becomẽ π β (z) = p(z)p(x|z) β . We report stochastic lower bound estimates (Grosse et al., 2015) of E pdata(x) log p(x) in Fig. 6( When exact posterior samples are available, we can use a reverse AIS chain from the target density to the base to obtain a stochastic upper bound on the log marginal likelihood (Grosse et al., 2015). While such samples are not available on the real data, we can use simulated **data** drawn from the model using ancestral sampling x, z ∼ p(z)p(x|z) as the dataset, and interpret z as a posterior sample. We use the Bidirectional Monte Carlo (BDMC) gap, or difference between the stochastic lower and upper bounds obtained from forward and reverse chains on simulated data, to evaluate the quality of the AIS procedure.