Maximum Likelihood Estimation in Gaussian AMP Chain Graph Models and Gaussian Ancestral ...

101 Pages·2012·0.77 MB·English

Maximum Likelihood Estimation in Gaussian AMP Chain Graph
Models and Gaussian Ancestral Graph Models
Mathias Drton
A dissertation submitted in partial ful(cid:12)llment of
the requirements for the degree of
Doctor of Philosophy
University of Washington
2004
Program Authorized to O(cid:11)er Degree: Statistics

University of Washington
Graduate School
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Mathias Drton
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examining committee have been made.
Co-Chairs of Supervisory Committee:
Michael D. Perlman
Thomas S. Richardson
Reading Committee:
Michael D. Perlman
Thomas S. Richardson
Steen A. Andersson
Date:

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University of Washington
Abstract
Maximum Likelihood Estimation in Gaussian AMP Chain Graph Models and
Gaussian Ancestral Graph Models
by Mathias Drton
Co-Chairs of Supervisory Committee:
Professor Michael D. Perlman
Department of Statistics
Professor Thomas S. Richardson
Department of Statistics
Graphical Markov models use graphs to represent dependencies between stochastic vari-
ables. Via Markov properties, missing edges in the graph are translated into conditional
independence statements, which, in conjunction with a distributional assumption, de(cid:12)ne a
statistical model. This thesis considers maximum likelihood (ML) estimation of the param-
etersoftworecentlyintroducedclassesofgraphicalMarkovmodelsinthecaseofcontinuous
variables with a joint multivariate Gaussian distribution. The two new model classes are
theAMPchaingraphmodels, basedonchaingraphsequippedwithanewMarkovproperty,
and the ancestral graph models, based on a new class of graphs. Both classes generalize the
widely used models based on acyclic directed graphs (Bayesian networks) and undirected
graphs (Markov random (cid:12)elds).
In this thesis, we (cid:12)rst show that the likelihood of AMP chain graph and ancestral graph
models may be multimodal. Next, we combine existing techniques (iterative proportional
(cid:12)tting, generalized least squares) into an algorithm for ML estimation in AMP chain graph
models. For the ancestral graphs, we develop an ML estimation algorithm based on a new
iterative conditional (cid:12)tting (ICF) idea, which in the considered Gaussian case can be imple-

mented using least squares regression on synthetic variables. We derive the ICF algorithm
in the special case of bidirected graphs, also termed covariance graphs, and subsequently
generalize it to cover arbitrary ancestral graphs.