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| Main Authors: | , |
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| Format: | Preprint |
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2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2310.11560 |
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| _version_ | 1866914177764818944 |
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| author | Johnson, Joseph Sullivant, Seth |
| author_facet | Johnson, Joseph Sullivant, Seth |
| contents | The codegree of a lattice polytope is the smallest integer dilate that contains a lattice point in the relative interior. The weak maximum likelihood threshold of a statistical model is the smallest number of data points for which there is a non-zero probability that the maximum likelihood estimate exists. The codegree of a marginal polytope is a lower bound on the maximum likelihood threshold of the associated log-linear model, and they are equal when the marginal polytope is normal. We prove a lower bound on the codegree in the case of hierarchical log-linear models and provide a conjectural formula for the codegree in general. As an application, we study when the marginal polytopes of hierarchical models are Gorenstein, including a classification of Gorenstein decomposable models, and a conjectural classification of Gorenstein binary hierarchical models. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_11560 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The Codegree, Weak Maximum Likelihood Threshold, and the Gorenstein Property of Hierarchical Models Johnson, Joseph Sullivant, Seth Combinatorics The codegree of a lattice polytope is the smallest integer dilate that contains a lattice point in the relative interior. The weak maximum likelihood threshold of a statistical model is the smallest number of data points for which there is a non-zero probability that the maximum likelihood estimate exists. The codegree of a marginal polytope is a lower bound on the maximum likelihood threshold of the associated log-linear model, and they are equal when the marginal polytope is normal. We prove a lower bound on the codegree in the case of hierarchical log-linear models and provide a conjectural formula for the codegree in general. As an application, we study when the marginal polytopes of hierarchical models are Gorenstein, including a classification of Gorenstein decomposable models, and a conjectural classification of Gorenstein binary hierarchical models. |
| title | The Codegree, Weak Maximum Likelihood Threshold, and the Gorenstein Property of Hierarchical Models |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2310.11560 |