Targeting functional parameters with semiparametric Bayesian inference

Abstract

Typical Bayesian inference requires parameter identification via likelihood parameterization, which has invited criticism for being less flexible than the Frequentist framework and subject to misspecification. Though misspecification may be avoided by functional parameter inference under a nonparametric model space, there does not exist a flexible Bayesian semiparametric model that would allow full control over the marginal prior over any general functional parameter. We present the technique of $\theta$-augmentation which helps us manipulate nonparametric models into semiparametric ones that directly target any functional parameter. The method allows Bayesian probabilistic statements to be drawn for any estimator that is defined as a functional of the empirical distribution without requiring a likelihood function, thus providing a path to Bayesian analysis in problems like causal inference and censoring where there do not exist well-accepted likelihood functions.