GEOMETRY OF MAXIMUM LIKELIHOOD ESTIMATION IN GAUSSIAN GRAPHICAL MODELS

被引：40

作者：

Uhler, Caroline ^{[1
]}

机构：

[1] IST Austria, A-3400 Klosterneuburg, Austria

来源：

ANNALS OF STATISTICS | 2012年 / 40卷 / 01期

关键词：

Gaussian graphical model; maximum likelihood estimation; matrix completion problems; duality; algebraic statistics; algebraic variety; number of observations; sufficient statistics; treewidth; elimination ideal; ML degree; bipartite graphs;

D O I：

10.1214/11-AOS957

中图分类号：

O21 [概率论与数理统计]; C8 [统计学];

学科分类号：

020208 ; 070103 ; 0714 ;

摘要：

We study maximum likelihood estimation in Gaussian graphical models from a geometric point of view. An algebraic elimination criterion allows us to find exact lower bounds on the number of observations needed to ensure that the maximum likelihood estimator (MLE) exists with probability one. This is applied to bipartite graphs, grids and colored graphs. We also study the ML degree, and we present the first instance of a graph for which the MLE exists with probability one, even when the number of observations equals the treewidth.

引用

页码：238 / 261

页数：24

共 22 条

[1] Basicness of semialgebraic sets [J].

Acquistapace, F ;

Broglia, F ;

Vélez, MP .

GEOMETRIAE DEDICATA, 1999, 78 (03) :229-240

[2]

[Anonymous], I MATH STAT LECT NOT

[3]

[Anonymous], 1978, WILEY SERIES PROBABI

[4]

[Anonymous], MACAULAY2 SOFTWARE S

[5]

[Anonymous], 1980, Multivariate Analysis

[6] THE REAL POSITIVE-DEFINITE COMPLETION PROBLEM FOR A SIMPLE CYCLE [J].

BARRETT, W ;

JOHNSON, CR ;

TARAZAGA, P .

LINEAR ALGEBRA AND ITS APPLICATIONS, 1993, 192 :3-31

[7]

BARRETT W, 1996, MEM AM MATH SOC, V122

[8]

BUHL SL, 1993, SCAND J STAT, V20, P263

[9]

Cox D., 1997, Ideals, Varieties, and Algorithms

[10] COVARIANCE SELECTION [J].

DEMPSTER, AP .

BIOMETRICS, 1972, 28 (01) :157-&

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