  We present a new approach for inference in Bayesian networks, which is mainly
based on partial differentiation. According to this approach, one compiles a
Bayesian network into a multivariate polynomial and then computes the partial
derivatives of this polynomial with respect to each variable. We show that once
such derivatives are made available, one can compute in constant-time answers
to a large class of probabilistic queries, which are central to classical
inference, parameter estimation, model validation and sensitivity analysis. We
present a number of complexity results relating to the compilation of such
polynomials and to the computation of their partial derivatives. We argue that
the combined simplicity, comprehensiveness and computational complexity of the
presented framework is unique among existing frameworks for inference in
Bayesian networks.
