Linearity Properties of Bayes Nets with Binary Variables

TL;DR

This paper investigates the linearity properties of Bayesian networks with binary variables, demonstrating that certain properties hold universally or under specific conditions, especially when using noisy-or and noisy-and parameterizations.

Contribution

It proves that property (1) holds for all binary-variable Bayes nets and property (5) for singly trek-connected nets, also showing all five properties hold with noisy-or and noisy-and gates.

Findings

Property (1) holds for all binary-variable Bayes nets.

Property (5) holds for all singly trek-connected nets.

All five properties hold with noisy-or and noisy-and parameterizations.

Abstract

It is "well known" that in linear models: (1) testable constraints on the marginal distribution of observed variables distinguish certain cases in which an unobserved 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 confounded by unobserved common causes; (3) the association (or conditional probability distribution of one variable given another) of two variables connected by a path or trek can be computed directly from the parameter values associated with each edge in the path or 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 units conditional on the sums over all units of each of the original conditioning variables.These properties are exploited in search procedures. It is also known that properties (2)-(5) do not hold for all Bayes nets with binary variables. We show that (1) holds for all Bayes nets with binary variables and (5) holds for all singly trek-connected Bayes nets of that kind. We further show that all five properties hold for Bayes nets with any DAG and binary variables parameterized with noisy-or and noisy-and gates.