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On the identifiability of Dirichlet mixture models

Hien Duy Nguyen, Mayetri Gupta

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.

On the identifiability of Dirichlet mixture models

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 shifted Dirichlet densities, where 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 coefficients sharing a common sign, and deduce that mixtures with fewer than 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.
Paper Structure (7 sections, 21 theorems, 175 equations)

This paper contains 7 sections, 21 theorems, 175 equations.

Table of Contents

  1. Introduction
  2. Non-identifiability of $\mathcal{F}$
  3. Identifiable restrictions of $\mathcal{F}$
  4. The class $\mathcal{F}\left(\Theta_{A}\right)$
  5. The class $\mathcal{F}\left(\Theta_{\mathrm{FD}}\right)$
  6. Identifiability of $\mathcal{F}_{J-1}$
  7. Implications

Key Result

Theorem 2

For every $\alpha\in(0,\infty)^{J}$ and every $x\in\Delta_{J-1}^{\circ}$,

Theorems & Definitions (46)

  • Definition 1
  • Theorem 2
  • proof
  • Corollary 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Lemma 6
  • ...and 36 more