Title: On the interpretation of path weights in undirected Markov random fields
Authors: Alberto Roverato - University of Bologna (Italy) [presenting]
Robert Castelo - Universitat Pompeu Fabra (Spain)
Abstract: In graphical Gaussian models an undirected graph is used to represent the association structure of variables as a network, and if a pair of variables is not joined by an edge in the graph, then the corresponding partial correlation is equal to zero. Although in graphical Gaussian models the structure of the network can be inferred from the zero pattern of the inverse covariance matrix, if the probability distribution of the variables is faithful to the network, then paths along the network connect random variables with non-zero entries in the covariance matrix. In the analysis of graphical Gaussian models, it has been associated a weight with every path in the network and showed that the covariance between two variables can be computed as the sum of the weights of all the paths joining the two variables. Path weights allow one to identify the relative contribution of a path to the value of the corresponding covariance. However, it is not clear either how to interpret the value of a single path or how to compare two paths with different endpoints. We provide an interpretation of the value taken by the weight of a path by decomposing it into a partial weight and an inflation factor. Furthermore, we identify a class of paths, called chordless paths, whose weights have a remarkably straightforward interpretation.