56 INFERENTIAL MODELS
draw U ∼ P
U
, i.e., S = S(U) with U ∼ P
U
. The intuition is that if a draw U ∼ P
U
is a good prediction for the unobserved u
?
, then the random set S = S(U) should be
even better in the sense that there is high probability that S 3 u
?
.
Example 4.1 (cont). In this example we may predict the unobserved u
?
with a pre-
dictive random set S defined by the set-valued mapping
S(u) =
u
0
∈ (0,1) : |u
0
−0.5| < |u −0.5|
, u ∈ (0,1). (4.8)
As this predictive random set is designed to predict an unobserved uniform variate,
we may also employ (4.8) in other problems, including the Poisson example.
There are, of course, other choices of S(u), e.g., [0, u), (u,1], (0.5u, 0.5 + 0.5u)
and more. Although some other choice of S = S(U) might perform slightly ...