Graphical methods for efficient likelihood inference in Gaussian covariance models

TL;DR

This paper introduces a method to transform bi-directed graphs into maximal ancestral graphs with fewer arrowheads, improving likelihood inference efficiency in Gaussian covariance models.

Contribution

It presents a novel transformation technique that preserves independence structures while minimizing arrowheads, aiding more efficient likelihood maximization in Gaussian models.

Findings

Transformation preserves independence structure

Reduces complexity of ancestral graphs

Enhances efficiency of likelihood inference

Abstract

In graphical modelling, a bi-directed graph encodes marginal independences among random variables that are identified with the vertices of the graph. We show how to transform a bi-directed graph into a maximal ancestral graph that (i) represents the same independence structure as the original bi-directed graph, and (ii) minimizes the number of arrowheads among all ancestral graphs satisfying (i). Here the number of arrowheads of an ancestral graph is the number of directed edges plus twice the number of bi-directed edges. In Gaussian models, this construction can be used for more efficient iterative maximization of the likelihood function and to determine when maximum likelihood estimates are equal to empirical counterparts.