Maximum likelihood estimation in Gaussian models under total positivity
| Autor: | Piotr Zwiernik, Caroline Uhler, Steffen L. Lauritzen |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2019 |
| Předmět: |
Statistics and Probability
FOS: Computer and information sciences 15B48 inverse M-matrix Gaussian Dimension (graph theory) Mathematics - Statistics Theory Statistics Theory (math.ST) 01 natural sciences Upper and lower bounds nonfrustrated GRMF Combinatorics Methodology (stat.ME) 010104 statistics & probability symbols.namesake 60E15 62H99 15B48 FOS: Mathematics attractive Gaussian Markov random field (GMRF) 0101 mathematics Coordinate descent Ultrametric space Statistics - Methodology Mathematics Covariance matrix Order (ring theory) $\mathrm{MTP}_{2}$ distribution ultrametric Gaussian graphical model Linear algebra symbols Statistics Probability and Uncertainty 60E15 62H99 |
| Zdroj: | Lauritzen, S L, Uhler, C & Zwiernik, P 2019, ' Maximum likelihood estimation in Gaussian models under total positivity ', Annals of Statistics, vol. 47, no. 4, pp. 1835-1863 . https://doi.org/10.1214/17-AOS1668 Ann. Statist. 47, no. 4 (2019), 1835-1863 arXiv |
| DOI: | 10.1214/17-AOS1668 |
| Popis: | We analyze the problem of maximum likelihood estimation for Gaussian distributions that are multivariate totally positive of order two ($\mathrm{MTP}_{2}$). 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 $n\geq2$ 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 $\mathrm{MTP}_{2}$ 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 $\mathrm{MTP}_{2}$ constraint which are structurally similar to iterative proportional scaling. We conclude the paper with a discussion of signed $\mathrm{MTP}_{2}$ distributions. |
| Databáze: | OpenAIRE |
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