Robustness analysis employs sets of distributions to model perturbations in the parameters of a probability distribution [Berger1985, Berger1990, Huber1980, Kadane1984, Wasserman1992b]. Robust Bayesian inference is the calculation of bounds on posterior values given such perturbations. This paper focuses on perturbations that are imposed on the global structure of a Bayesian network. The goal is to find exact solutions for calculation of posterior bounds when global perturbations are present. We concentrate on bounds for marginal probabilities, expectations and variances.
No efficient exact solutions are available for calculation of posterior bounds in arbitrary Bayesian networks with perturbations modeled by sets of probability distributions (the algorithmic challenge presented by robust Bayesian inference is discussed in a companion technical report [Cozman1996]). However, this does not preclude the existence of models that are amenable to efficient analysis (for example, Markov-like lower probabilities [Chrisman1996b]). This paper proposes global models that lead to such efficient exact analysis.
Section 2 introduces the theory of inferences which validates our procedures. Section 3 introduces the definitions needed to develop global neighborhoods of Bayesian networks, and Sections 4, 5, 6 and 7 present solutions for the epsi-contaminated, the constant density ratio, the constant density bounded and the total variation classes respectively. Section 8 extends the analysis to variance bounds. Section 9 discusses the relationship between global neighborhoods and independence relations in a Bayesian network.
Thu Jan 23 15:54:13 EST 1997