
110 CONDITIONAL INFERENTIAL MODELS
Proof. For the given predictive random set S for U in the baseline association, the
corresponding random set Θ
x
(S) in the C-step can be written as
Θ
x
(S) =
[
u∈S
{θ : T (x) = a
T
(τ(u),θ ), H(x) = η(u)}
=
[
u∈S
{θ : T (x) = a
T
(τ(u),θ )}∩{θ : H(x) = η(u)}
=
[
u∈R
H(x)
{θ : T (x) = a
T
(τ(u),θ )}
= Θ
T (x)
(τ(R
H(x)
)),
where R
H(x)
:= S ∩{u : η(u) = H(x)}. That is, given S, the random sets in the
baseline C-step match the random sets in the conditional C-step, but with a modified
predictive random set S
H(x)
:= τ(R
H(x)
). Since P
S
{Θ
x
(S) 6= ∅} > 0 for all x, the
two belief functions
bel
x
(A;S) = P
S
{Θ
x
(S) ⊆ A | Θ
x
(S) 6= ∅},
cbel
T (x)
(A;S
H(x)
)