  We characterize probabilities in Bayesian networks in terms of algebraic
expressions called quasi-probabilities. These are arrived at by casting
Bayesian networks as noisy AND-OR-NOT networks, and viewing the subnetworks
that lead to a node as arguments for or against a node. Quasi-probabilities are
in a sense the "natural" algebra of Bayesian networks: we can easily compute
the marginal quasi-probability of any node recursively, in a compact form; and
we can obtain the joint quasi-probability of any set of nodes by multiplying
their marginals (using an idempotent product operator). Quasi-probabilities are
easily manipulated to improve the efficiency of probabilistic inference. They
also turn out to be representable as square-wave pulse trains, and joint and
marginal distributions can be computed by multiplication and complementation of
pulse trains.
