Targeting functional parameters with semiparametric Bayesian inference

TL;DR

This paper introduces a $ heta$-augmentation technique enabling flexible Bayesian semiparametric inference for functional parameters without relying on likelihood functions, enhancing analysis in complex models like causal inference.

Contribution

The paper presents a novel $ heta$-augmentation method that transforms nonparametric models into semiparametric ones, allowing direct Bayesian inference on functional parameters without likelihoods.

Findings

Enables Bayesian inference for any functional of the empirical distribution.

Provides a flexible approach for models lacking well-defined likelihoods.

Facilitates analysis in causal inference and censoring problems.

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.