A General Framework for Updating Belief Distributions

TL;DR

This paper introduces a flexible Bayesian inference framework that updates beliefs using loss functions instead of likelihoods, enabling coherent inference even when traditional models are difficult or impossible to specify.

Contribution

It generalizes Bayesian updating by incorporating loss functions, allowing belief updates without requiring a full data-generating model or likelihood, applicable to a broader range of problems.

Findings

Framework coincides with Bayesian updating when likelihood is known

Enables belief updating for parameters not directly linked to density functions

Provides coherent inference in complex or model-misspecified settings

Abstract

We propose a framework for general Bayesian inference. We argue that a valid update of a prior belief distribution to a posterior can be made for parameters which are connected to observations through a loss function rather than the traditional likelihood function, which is recovered under the special case of using self information loss. Modern application areas make it is increasingly challenging for Bayesians to attempt to model the true data generating mechanism. Moreover, when the object of interest is low dimensional, such as a mean or median, it is cumbersome to have to achieve this via a complete model for the whole data distribution. More importantly, there are settings where the parameter of interest does not directly index a family of density functions and thus the Bayesian approach to learning about such parameters is currently regarded as problematic. Our proposed framework uses loss-functions to connect information in the data to functionals of interest. The updating of beliefs then follows from a decision theoretic approach involving cumulative loss functions. Importantly, the procedure coincides with Bayesian updating when a true likelihood is known, yet provides coherent subjective inference in much more general settings. Connections to other inference frameworks are highlighted.