Differential Equations for Gaussian Statistical Models with Rational Maximum Likelihood Estimator
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arXiv
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| Main Authors: | , , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866915475532808192 |
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| author | Améndola, Carlos Gustafsson, Lukas Kohn, Kathlén Marigliano, Orlando Seigal, Anna |
| author_facet | Améndola, Carlos Gustafsson, Lukas Kohn, Kathlén Marigliano, Orlando Seigal, Anna |
| contents | We study multivariate Gaussian statistical models whose maximum likelihood estimator (MLE) is a rational function of the observed data. We establish a one-to-one correspondence between such models and the solutions to a nonlinear first-order partial differential equation (PDE). Using our correspondence, we reinterpret familiar classes of models with rational MLE, such as directed (and decomposable undirected) Gaussian graphical models. We also find new models with rational MLE. For linear concentration models with rational MLE, we show that homaloidal polynomials from birational geometry lead to solutions to the PDE. We thus shed light on the problem of classifying Gaussian models with rational MLE by relating it to the open problem in birational geometry of classifying homaloidal polynomials. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2304_12054 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Differential Equations for Gaussian Statistical Models with Rational Maximum Likelihood Estimator Améndola, Carlos Gustafsson, Lukas Kohn, Kathlén Marigliano, Orlando Seigal, Anna Algebraic Geometry Statistics Theory 62R01, 62F10, 62H22, 14E05, 35C11, 35F20 We study multivariate Gaussian statistical models whose maximum likelihood estimator (MLE) is a rational function of the observed data. We establish a one-to-one correspondence between such models and the solutions to a nonlinear first-order partial differential equation (PDE). Using our correspondence, we reinterpret familiar classes of models with rational MLE, such as directed (and decomposable undirected) Gaussian graphical models. We also find new models with rational MLE. For linear concentration models with rational MLE, we show that homaloidal polynomials from birational geometry lead to solutions to the PDE. We thus shed light on the problem of classifying Gaussian models with rational MLE by relating it to the open problem in birational geometry of classifying homaloidal polynomials. |
| title | Differential Equations for Gaussian Statistical Models with Rational Maximum Likelihood Estimator |
| topic | Algebraic Geometry Statistics Theory 62R01, 62F10, 62H22, 14E05, 35C11, 35F20 |
| url | https://arxiv.org/abs/2304.12054 |