  Insights into complex, high-dimensional data can be obtained by discovering
features of the data that match or do not match a model of interest. To
formalize this task, we introduce the "data selection" problem: finding a
lower-dimensional statistic - such as a subset of variables - that is well fit
by a given parametric model of interest. A fully Bayesian approach to data
selection would be to parametrically model the value of the statistic,
nonparametrically model the remaining "background" components of the data, and
perform standard Bayesian model selection for the choice of statistic. However,
fitting a nonparametric model to high-dimensional data tends to be highly
inefficient, statistically and computationally. We propose a novel score for
performing both data selection and model selection, the "Stein volume
criterion", that takes the form of a generalized marginal likelihood with a
kernelized Stein discrepancy in place of the Kullback-Leibler divergence. The
Stein volume criterion does not require one to fit or even specify a
nonparametric background model, making it straightforward to compute - in many
cases it is as simple as fitting the parametric model of interest with an
alternative objective function. We prove that the Stein volume criterion is
consistent for both data selection and model selection, and we establish
consistency and asymptotic normality (Bernstein-von Mises) of the corresponding
generalized posterior on parameters. We validate our method in simulation and
apply it to the analysis of single-cell RNA sequencing datasets using
probabilistic principal components analysis and a spin glass model of gene
regulation.
