The most popular type of extension discussed in the literature is the type-1 extension [4,27]. A type-1 extension is the convex hull of all the joint distributions formed by cross-multiplication of extreme points of local credal sets; consequently, a type-1 extension is the largest joint credal set where all extreme points satisfy Expression (1).
The appeal of type-1 extensions comes from their intuitive similarity with standard Bayesian networks. The following theorem formalizes this intuition using Walley's definition of independence:
This result demonstrates that type-1 extensions mimic the properties of standard Bayesian networks as independence-maps [22, page 119]. The theorem also complements results by Cano et al. [4]. They give conditions on independence concepts that satisfy d-separation in type-1 extensions, but they do not provide any definition of independence to illustrate their result. The theorem demonstrates that Walley's independence relations exhibit the desired correspondence with d-separation.
D-separation has important algorithmic consequences. Graphical operations that are guaranteed by d-separation can be performed in a type-1 extension. In particular, consider a query involving a variable Xq and evidence E. All variables that do not affect computation of p(Xq|E) can be detected through d-separation computations [13]. This greatly reduces the computational effort in Quasi-Bayesian inferences with type-1 extensions both for exact (enumeration) and approximate (sampling, iterative) algorithms [9]. The theorem in this section completes that investigation with a formal proof that d-separation can (and should) be used to handle type-1 extensions.