The Wasserstein distance between mixing measures has come to occupy a central place in the statistical analysis of mixture models. This work proposes a new canonical interpretation of this distance and provides tools to perform inference on the Wasserstein distance between mixing measures in topic models. We consider the general setting of an identifiable mixture model consisting of mixtures of distributions from a set A equipped with an arbitrary metric d, and show that the Wasserstein distance between mixing measures is uniquely characterized as the most discriminative convex extension of the metric d to the set of mixtures of elements of A. The Wasserstein distance between mixing measures has been widely used in the study of such models, but without axiomatic justification. Our results establish this metric to be a canonical choice. Specializing our results to topic models, we consider estimation and inference of this distance. Although upper bounds for its estimation have been recently established elsewhere, we prove the first minimax lower bounds for the estimation of the Wasserstein distance between mixing measures, in topic models, when both the mixing weights and the mixture components need to be estimated. Our second main contribution is the provision of fully data-driven inferential tools for estimators of the Wasserstein distance between potentially sparse mixing measures of high-dimensional discrete probability distributions on p points, in the topic model context. These results allow us to obtain the first asymptotically valid, ready to use, confidence intervals for the Wasserstein distance in (sparse) topic models with potentially growing ambient dimension p.
Bing et al. (Fri,) studied this question.