Strong relevance

Structural properties of Bayesian networks express various forms of structural relevance. An essential relevance measure called strong relevance is related to the concept of minimal Markov blanket (a.k.a. Markov Boundary Set)⁷⁴. Assuming that a given graph G properly represents all dependency relationships defined by the data, the Markov blanket set is represented by the neighborhood of a node (variable) Y which consists of those variables X_i that are in either direct structural relationship with Y or that are interaction terms. In other words, the Markov blanket set of Y ‘isolates’ Y from the effects of the rest of the network, such that if the values of the variables within the Markov blanket are known then no further information is required to infer the value of Y. Throughout the paper we refer to the Markov blanket set as strongly relevant set of variables which are represented by structural properties (i.e. nodes) of a Bayesian network. In the Bayesian framework a posterior probability can be induced for several types of structural properties. The posterior of strong relevance of a variable X_i can be estimated by model averaging, that is assessing the probability mass of those Markov blanket sets of which X_i is a member. The higher is the sum of posterior probabilities related to Markov blanket sets of which X_i is part of, the more relevant X_i is considered.

Technically, for each variable X_i an indicator function called Markov blanket membership can be defined as IMBM(Xi,Y,G), which takes the value of 1 if X_i is a member of the Markov blanket of Y in DAG structure G. Assuming that the data D admits multiple possible dependency structures (i.e. models) G, and that the indicator function can be evaluated for each such G, then the posterior probability of strong relevance P(MBM(X_i, Y, G|D)) can be defined by model averaging⁷⁵ based on the posteriors of possible structures^21,76. Therefore, strong relevance can be considered as an aggregate of possible multivariate dependency models in the form of a pairwise relevance measure between X_i and Y. The challenge of this approach is to properly assess the posterior probability of structural features such as Markov blanket sets.