Linearity Properties of Bayes Nets with Binary Variables

It is "well known" that in linear models: (1) testable constraints on the marginal distribution of observed variables distinguish certain cases in which an Wlobserved cause jointly influences several observed variables; (2) the technique of "instrumental variables" sometimes permits an estimation of the influence of one variable on another even when the association between the variables may be confoWided by Wlobserved common causes; (3) the association (or conditional probability distribution of one variable given another) of two variables connected by a path or pair of paths with a single common vertex (a trek) can be computed directly from the parameter values associated with each edge in the trek; (4) the association of two variables produced by multiple treks can be computed from the parameters associated with each trek; and (5) the independence of two variables conditional on a third implies the corresponding independence of the sums of the variables over all Wlits conditional on the sums over all Wlits of each of the original conditioning variables. Clark Glymour Carnegie-Mellon University; Institute for Human & Machine Cognition, University of West Florida; & University of California, San Diego These properties are exploited in search procedures. We show that (1) and (2) hold for all Bayes nets with binary variables. We further show that for Bayes nets parameterized as noisy­ OR and noisy-AND gates, all of these properties save ( 4) hold.