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 cannot exploit the latent structure.Comment: update test statistic (MCMC version

Stochastic Stein Discrepancies

Stein discrepancies (SDs) monitor convergence and non-convergence in approximate inference when exact integration and sampling are intractable. However, the computation of a Stein discrepancy can be prohibitive if the Stein operator - often a sum over likelihood terms or potentials - is expensive to evaluate. To address this deficiency, we show that stochastic Stein discrepancies (SSDs) based on subsampled approximations of the Stein operator inherit the convergence control properties of standard SDs with probability 1. In our experiments with biased Markov chain Monte Carlo (MCMC) hyperparameter tuning, approximate MCMC sampler selection, and stochastic Stein variational gradient descent, SSDs deliver comparable inferences to standard SDs with orders of magnitude fewer likelihood evaluations

Testing Goodness of Fit of Conditional Density Models with Kernels

We propose two nonparametric statistical tests of goodness of fit for conditional distributions: given a conditional probability density function $p(y|x)$ and a joint sample, decide whether the sample is drawn from $p(y|x)r_x(x)$ for some density $r_x$. Our tests, formulated with a Stein operator, can be applied to any differentiable conditional density model, and require no knowledge of the normalizing constant. We show that 1) our tests are consistent against any fixed alternative conditional model; 2) the statistics can be estimated easily, requiring no density estimation as an intermediate step; and 3) our second test offers an interpretable test result providing insight on where the conditional model does not fit well in the domain of the covariate. We demonstrate the interpretability of our test on a task of modeling the distribution of New York City's taxi drop-off location given a pick-up point. To our knowledge, our work is the first to propose such conditional goodness-of-fit tests that simultaneously have all these desirable properties.Comment: In UAI 2020. http://auai.org/uai2020/accepted.ph

Characterizations of non-normalized discrete probability distributions and their application in statistics

From the distributional characterizations that lie at the heart of Stein's method we derive explicit formulae for the mass functions of discrete probability laws that identify those distributions. These identities are applied to develop tools for the solution of statistical problems. Our characterizations, and hence the applications built on them, do not require any knowledge about normalization constants of the probability laws. To demonstrate that our statistical methods are sound, we provide comparative simulation studies for the testing of fit to the Poisson distribution and for parameter estimation of the negative binomial family when both parameters are unknown. We also consider the problem of parameter estimation for discrete exponential-polynomial models which generally are non-normalized.Comment: 24 pages, 3 figure