The $θ$-augmented model for Bayesian semiparametric inference on functional parameters

Semiparametric Bayesian inference has so far relied on models for the observable that partition into two parts, one being parametric and the other nonparametric, with the target parameter being dependent on the parametric component. While a partitioned structure makes specification of the marginal prior on the target parameter simple to perform, it often arises from conditional modelling which is subject to misspecification and ultimately a lack of consistency. We introduce a new type of semiparametric model to allow easy prior specification for a parameter that is defined as a functional of the distribution for the observable. Our semiparametric model is obtained as an extension of nonparametric models that are consistent under very general conditions. This type of Bayesian semiparametric model can be used to obtain Bayesian versions of Frequentist estimators that are defined as functionals of the empirical distribution. This gives us new opportunities to conduct Bayesian analysis in problems where Frequentist estimators exist but not well-accepted likelihoods.