
130 MARGINAL INFERENTIAL MODELS
sense that, for any A ⊂ Ψ and any α ∈ (0,1), the marginal belief function satisfies
sup
(ψ,ξ )∈A
c
×Ξ
P
X|(ψ,ξ )
mbel
X
(A;S) ≥ 1 −α
≤ α.
Since this holds for all A, we also have
sup
(ψ,ξ )∈A×Ξ
P
X|(ψ,ξ )
mpl
X
(A;S) ≤ α
≤ α.
Proof. Similar to the validity theorem proofs in Chapters 4 and 6. This result is also
covered by the proof of Theorem 7.3 below.
Therefore, if the baseline association is regular and the predictive random set is
valid, then the marginal IM constructed has the desirable frequency calibration prop-
erty. In particular, this means that marginal plausibility intervals based on mpl
x
will
achieve the nominal frequentist ...