APPLICATION: LARGE-SCALE MULTINOMIAL INFERENCE 221
Indeed, if (y,u) 7→ Θ
y
(u) is suitably compatible with the sampling model Y ∼ P
Y |θ
,
then the generalized IM can be shown to be valid, provided that the predictive random
set for U is valid. The challenge is in defining a meaningful notion of “compatibil-
ity” in this context. More work is needed along these lines, but we expect that the
developments will shed light on various model-free inference, such as M-estimation,
but from an IM perspective.
Remark 11.3. Using Monte Carlo approximations (11.11) and (11.17) to construct
exact frequentist inferential procedures is, to our knowledge, new. Despite its novelty,
the method is surprisingly simple and general. On the other hand, there is a compu-
tational ...