Event Diary
CRiSM Seminar
Alastair Young, Imperial College London
Objective Bayes and Conditional Inference
In Bayesian parametric inference, in the absence of
subjective prior information about the parameter of interest, it is natural
to consider use of an objective prior which leads to posterior probability
quantiles which have, at least to some higher order approximation in terms of
the sample size, the correct frequentist interpretation. Such priors are
termed probability matching priors. In many circumstances, however,
the appropriate frequentist inference is a conditional one. The key
contexts involve inference in multi-parameter exponential families,
where conditioning eliminates the nuisance parameter, and models which
admit ancillary statistics, where conditioning on the ancillary is indicated
by the conditionality principle of inference. In this talk, we
consider conditions on the prior under which posterior quantiles have, to
high order, the correct conditional frequentist interpretation. The
key motivation for the work is that the conceptually simple objective
Bayes route may provide accurate approximation to more complicated
frequentist procedures. We focus on the exponential family context, where it
turns out that the condition for higher order conditional frequentist
accuracy reduces to a condition on the model, not the prior: when the
condition is satisfied, as it is in many key situations, any first order
probability matching prior (in the unconditional sense) automatically yields
higher order conditional probability matching. We provide
numerical illustrations, discuss the relationship between the objective
Bayes inference and the parametric bootstrap, as well as giving a brief
account appropriate to the ancillary statistic context, where
conditional frequentist probability matching is more difficult. [This
is joint work with Tom DiCiccio, Cornell].