Geometry, moments and conditional independence trees with hidden variables

Raffaella Settimi and Jim Q. Smith

Source: Ann. Statist. Volume 28, Number 4 (2000), 1179-1205.

Abstract

We study the geometry of the parameter space for Bayesian directed graphical models with hidden variables that have a tree structure and where all the nodes are binary.We show that the conditional independence statements implicit in such models can be expressed in terms of polynomial relationships among the central moments.This algebraic structure will enable us to identify the inequality constraints on the space of the manifest variables that are induced by the conditional independence assumptions as well as determine the degree of unidentifiability of the parameters associated with the hidden variables. By understanding the geometry of the sample space under this class of models we shall propose and discuss simple diagnostic methods.

Primary Subjects: 62F15

Secondary Subjects: 62H17, 68R10

Keywords: Conditional independence; Bayesian networks; Bayesian multinomial models; model identifiability

Full-text: Open access

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Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.aos/1015956712
Mathematical Reviews number (MathSciNet): MR1811324
Digital Object Identifier: doi:10.1214/aos/1015956712
Zentralblatt MATH identifier: 01828979

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