On the identifiability of Dirichlet mixture models

TL;DR

This paper investigates the conditions under which finite mixtures of Dirichlet distributions are identifiable, revealing limitations on identifiability on the full parameter space and conditions for recovery on restricted regions.

Contribution

It establishes new theoretical results on the identifiability of Dirichlet mixture models, including conditions for when mixtures are identifiable or not.

Findings

Mixtures with fewer than J atoms are identifiable.

Non-identifiability on the full parameter space due to shift identities.

Identifiability can be recovered on fixed slices and restricted regions.

Abstract

We study identifiability of finite mixtures of Dirichlet distributions on the interior of the simplex. We first prove a shift identity showing that every Dirichlet density can be written as a mixture of $J$ shifted Dirichlet densities, where $J-1$ is the dimension of the simplex support, which yields non-identifiability on the full parameter space. We then show that identifiability is recovered on a fixed-total parameter slice and on restricted box-type regions. On the full parameter space, we prove that any nontrivial linear relation among Dirichlet kernels must involve at least $J$ coefficients sharing a common sign, and deduce that mixtures with fewer than $J$ atoms are identifiable. We further report direct non-identifiability implications for unrestricted finite mixtures of generalized Dirichlet, Dirichlet-multinomial, fixed-topic-matrix latent Dirichlet allocation, Beta-Liouville, and inverted Beta-Liouville models.