A Kernel Stein Test for Comparing Latent Variable Models

We propose a kernel-based nonparametric test of relative goodness of fit, where the goal is to compare two models, both of which may have unobserved latent variables, such that the marginal distribution of the observed variables is intractable. The proposed test generalises the recently proposed kernel Stein discrepancy (KSD) tests (Liu et al., 2016, Chwialkowski et al., 2016, Yang et al., 2018) to the case of latent variable models, a much more general class than the fully observed models treated previously. As our main theoretical contribution, we prove that the new test, with a properly calibrated threshold, has a well-controlled type-I error. In the case of models with low-dimensional latent structure and high-dimensional observations, our test significantly outperforms the relative Maximum Mean Discrepancy test, which is based on samples from the models and does not exploit the latent structure.