
166 PREDICTION OF FUTURE OBSERVATIONS
• Singleton. A singleton assertion is of the form A = {
˜
Y = ˜y} for a fixed ˜y. The
optimal predictive random set worked out in [180] for this assertion is compli-
cated, but a natural choice that is suitable in most cases (and optimal in some
cases) is the “default” predictive random set S = {w : |w −0.5| ≤ |W −0.5|},
for W ∼ Unif(0, 1). Then the C-step gives the plausibility function
pl
Y
(A) = P
S
{G
−1
Y
(S) ∩A 6= ∅}= 1 −|2G
Y
( ˜y) −1| (9.6)
The prediction region (9.8) based on the plausibility function in (9.6) will be a
two-sided prediction bound for
˜
Y .
It is important to note that, although the general P- and C-steps ...