Bayes’s theorem for improper mixtures
While looking for references for a Master summer project at Warwick on Bayesian inference on the Cauchy location parameter, I came across a 2011 Annals of Statistics paper by Peter McCullagh and Han Han. Which expands the Bayesian framework to the improper case by considering a Poisson process over the parameter set with mean measure ν the improper prior. Instead of a single random parameter, this construct returns a countable collection of pairs (θ,y), while the observations induce a subset of that collection constrained by y∈A, a “sampling region” both capital to the derivation of the joint distribution and obscure in that A remains unspecified (but such that 0<ν(A)<∞ and conveniently returning the observed sample of y’s).
“Provided that the key finiteness condition is satisfied, this probabilistic analysis of the extended model may be interpreted as a vindication of improper Bayes procedures derived from the original model.”
“Thus, the existence of a joint probability model associated with an improper prior does not imply optimality in the form of coherence, consistency or admissibility.”
This is definitely fascinating!, even though I have troubles linking this infinite sequence of θ‘s with regular Bayesian inference, since the examples in the paper seem to revert to a single parameter value, as in §4.1, for the Normal model and §5 for the Cauchy model. The authors also revisit the marginalisation paradoxes of Dawid, Stone and Zidek (1973), with the argument that the improper measure leading to the paradox is not compatible with ν(A)<∞, hence does not define a natural conditional, while the “other” improper measure avoids the paradox.
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