
186 SIMULTANEOUS INFERENCE ON MULTIPLE ASSERTIONS
The following result, a generalization of Theorem 10.2, shows that restricting
our consideration to predictive random sets supported on intersection focal elements
results in no loss of efficiency.
Theorem 10.3. Let {A
j
: j ∈ J} be a collection of assertions, where A
j
= A
j1
∪A
j2
partitions as a union of disjoint simple assertions. Let S
j
be the optimal predictive
random set for A
j
, j ∈ J. Then, for any predictive random set T , there exists an S,
whose focal elements are intersections of the focal elements of the S
j
s, such that S
is at least as efficient as T with respect to {A
j
: j ∈ J}.
Proof. The proof