The diachronic Bayesian

TL;DR

This paper explores how Bayesian probability measures evolve over time, proposing methods to test their adequacy and coherence through betting models, with applications to multi-step forecasting and decision-making.

Contribution

It introduces a framework for testing the coherence of evolving Bayesian beliefs using betting models, extending to multi-step forecasting and decision-making.

Findings

A betting-based method for testing Bayesian belief coherence.

Extension of the framework to multi-step ahead forecasting.

Application to making nearly optimal decisions over time.

Abstract

It is well known that a Bayesian probability forecast for all future observations should be a probability measure in order to satisfy a natural condition of coherence. The main topics of this paper are the evolution of the Bayesian probability measure and ways of testing its adequacy as it evolves over time. The process of testing evolving Bayesian beliefs is modelled in terms of betting, similarly to the standard Dutch book treatment of coherence. The resulting picture is adapted to forecasting several steps ahead and making almost optimal decisions.