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 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.
