COMBINING LOCAL INFORMATION

 

Given a Bayesian network, there is a unique way to obtain a joint distribution [29]. This property does not generalize to Quasi-Bayesian models: given a Quasi-Bayesian network, there are several ways to combine the local conditional credal sets into a joint credal set. This paper focuses on two approaches to combination of local credal sets.

The first approach takes the joint credal set as the largest set of distributions that can generate the specified conditionals. This is in many ways the most natural way to represent the joint credal set as it incorporates all possible constraints in the model. The present paper is the first analysis of this method and its algorithmic implications for Bayesian networks.

The second approach multiplies element-wise all local credal sets and considers the convex hull of all resulting joint distributions. The joint credal set is constructed as follows. For each combination of vertices from the local credal sets, construct a joint distribution by multiplying the conditionals together. Now take the convex hull of all these joint distributions as the joint credal set. For example, consider a Quasi-Bayesian network with two credal sets associated with variables x₁ and x₂. To produce a joint credal set, take the convex hull of the distributions {(p_1,j p_2,k prod_i>2 p_i} for all j and k. This technique forms the largest joint credal set whose vertices respect the independence relations displayed in the network [37]. The axiomatic underpinnings and algorithmic properties of this method for Bayesian networks have been studied previously [5, 6].

The method of the first approach is referred to as a natural extension and the result of the second method is referred to as a type-1 combination, to use terms proposed by Walley in a similar setting [37, pp. 453, 455,]. Note that the natural extension is not identical to a type-1 combination; in most cases a type-1 joint credal set will be smaller than the largest possible credal set given local constraints.