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The Wasserstein distance between mixing measures has come to occupy a centralplace in the statistical analysis of mixture models. This work proposes a newcanonical interpretation of this distance and provides tools to performinference on the Wasserstein distance between mixing measures in topic models.  We consider the general setting of an identifiable mixture model consistingof mixtures of distributions from a set $\mathcal{A}$ equipped with anarbitrary metric $d$, and show that the Wasserstein distance between mixingmeasures is uniquely characterized as the most discriminative convex extensionof the metric $d$ to the set of mixtures of elements of $\mathcal{A}$. TheWasserstein distance between mixing measures has been widely used in the studyof such models, but without axiomatic justification. Our results establish thismetric to be a canonical choice.  Specializing our results to topic models, we consider estimation andinference of this distance. Though upper bounds for its estimation have beenrecently established elsewhere, we prove the first minimax lower bounds for theestimation of the Wasserstein distance in topic models. We also establish fullydata-driven inferential tools for the Wasserstein distance in the topic modelcontext. Our results apply to potentially sparse mixtures of high-dimensionaldiscrete probability distributions. These results allow us to obtain the firstasymptotically valid confidence intervals for the Wasserstein distance in topicmodels.

Estimation and inference for the Wasserstein distance between mixing measures in topic models