On the identifiability of Dirichlet mixture models
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911538185502720 |
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| author | Nguyen, Hien Duy Gupta, Mayetri |
| author_facet | Nguyen, Hien Duy Gupta, Mayetri |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_21914 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the identifiability of Dirichlet mixture models Nguyen, Hien Duy Gupta, Mayetri Statistics Theory 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. |
| title | On the identifiability of Dirichlet mixture models |
| topic | Statistics Theory |
| url | https://arxiv.org/abs/2603.21914 |