Maximum likelihood estimation in Gaussian models under total positivity
© 2019 Institute of Mathematical Statistics. We analyze the problem of maximum likelihood estimation for Gaussian distributions that are multivariate totally positive of order two (MTP2). By exploiting connections to phylogenetics and single-linkage clustering, we give a simple proof that the maximu...
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| Format: | Article |
| Language: | English |
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Institute of Mathematical Statistics
2021
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| Online Access: | https://hdl.handle.net/1721.1/134422 |
| _version_ | 1811096004506681344 |
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| author | Lauritzen, Steffen Uhler, Caroline Zwiernik, Piotr |
| author_facet | Lauritzen, Steffen Uhler, Caroline Zwiernik, Piotr |
| author_sort | Lauritzen, Steffen |
| collection | MIT |
| description | © 2019 Institute of Mathematical Statistics. We analyze the problem of maximum likelihood estimation for Gaussian distributions that are multivariate totally positive of order two (MTP2). By exploiting connections to phylogenetics and single-linkage clustering, we give a simple proof that the maximum likelihood estimator (MLE) for such distributions exists based on n = 2 observations, irrespective of the underlying dimension. Slawski and Hein [Linear Algebra Appl. 473 (2015) 145-179], who first proved this result, also provided empirical evidence showing that the MTP2 constraint serves as an implicit regularizer and leads to sparsity in the estimated inverse covariance matrix, determining what we name the ML graph. We show that we can find an upper bound for the ML graph by adding edges corresponding to correlations in excess of those explained by the maximum weight spanning forest of the correlation matrix. Moreover, we provide globally convergent coordinate descent algorithms for calculating the MLE under the MTP2 constraint which are structurally similar to iterative proportional scaling. We conclude the paper with a discussion of signed MTP2 distributions. |
| first_indexed | 2024-09-23T16:36:49Z |
| format | Article |
| id | mit-1721.1/134422 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T16:36:49Z |
| publishDate | 2021 |
| publisher | Institute of Mathematical Statistics |
| record_format | dspace |
| spelling | mit-1721.1/1344222022-03-31T14:31:26Z Maximum likelihood estimation in Gaussian models under total positivity Lauritzen, Steffen Uhler, Caroline Zwiernik, Piotr © 2019 Institute of Mathematical Statistics. We analyze the problem of maximum likelihood estimation for Gaussian distributions that are multivariate totally positive of order two (MTP2). By exploiting connections to phylogenetics and single-linkage clustering, we give a simple proof that the maximum likelihood estimator (MLE) for such distributions exists based on n = 2 observations, irrespective of the underlying dimension. Slawski and Hein [Linear Algebra Appl. 473 (2015) 145-179], who first proved this result, also provided empirical evidence showing that the MTP2 constraint serves as an implicit regularizer and leads to sparsity in the estimated inverse covariance matrix, determining what we name the ML graph. We show that we can find an upper bound for the ML graph by adding edges corresponding to correlations in excess of those explained by the maximum weight spanning forest of the correlation matrix. Moreover, we provide globally convergent coordinate descent algorithms for calculating the MLE under the MTP2 constraint which are structurally similar to iterative proportional scaling. We conclude the paper with a discussion of signed MTP2 distributions. 2021-10-27T20:04:56Z 2021-10-27T20:04:56Z 2019 2019-07-09T18:10:20Z Article http://purl.org/eprint/type/JournalArticle https://hdl.handle.net/1721.1/134422 en 10.1214/17-AOS1668 The Annals of Statistics Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/ application/pdf Institute of Mathematical Statistics arXiv |
| spellingShingle | Lauritzen, Steffen Uhler, Caroline Zwiernik, Piotr Maximum likelihood estimation in Gaussian models under total positivity |
| title | Maximum likelihood estimation in Gaussian models under total positivity |
| title_full | Maximum likelihood estimation in Gaussian models under total positivity |
| title_fullStr | Maximum likelihood estimation in Gaussian models under total positivity |
| title_full_unstemmed | Maximum likelihood estimation in Gaussian models under total positivity |
| title_short | Maximum likelihood estimation in Gaussian models under total positivity |
| title_sort | maximum likelihood estimation in gaussian models under total positivity |
| url | https://hdl.handle.net/1721.1/134422 |
| work_keys_str_mv | AT lauritzensteffen maximumlikelihoodestimationingaussianmodelsundertotalpositivity AT uhlercaroline maximumlikelihoodestimationingaussianmodelsundertotalpositivity AT zwiernikpiotr maximumlikelihoodestimationingaussianmodelsundertotalpositivity |