  The inferential model (IM) framework provides valid prior-free probabilistic
inference by focusing on predicting unobserved auxiliary variables. But,
efficient IM-based inference can be challenging when the auxiliary variable is
of higher dimension than the parameter. Here we show that features of the
auxiliary variable are often fully observed and, in such cases, a simultaneous
dimension reduction and information aggregation can be achieved by
conditioning. This proposed conditioning strategy leads to efficient IM
inference, and casts new light on Fisher's notions of sufficiency,
conditioning, and also Bayesian inference. A differential equation-driven
selection of a conditional association is developed, and validity of the
conditional IM is proved under some conditions. For problems that do not admit
a valid conditional IM of the standard form, we propose a more flexible class
of conditional IMs based on localization. Examples of local conditional IMs in
a bivariate normal model and a normal variance components model are also given.
