# A Short Note on Almost Sure Convergence of Bayes Factors in the General Set-Up

## TL;DR

This paper establishes the almost sure convergence of Bayes factors in a very general setting, including dependent data and misspecified models, extending previous asymptotic results.

## Contribution

It provides the first comprehensive almost sure convergence theory for Bayes factors in a broad, non-i.i.d. framework, leveraging Shalizi's result.

## Key findings

- Almost sure convergence of Bayes factors in general set-ups
- Applicable to dependent data and misspecified models
- Simplifies proof using Shalizi's result

## Abstract

Although there is a significant literature on the asymptotic theory of Bayes factor, the set-ups considered are usually specialized and often involves independent and identically distributed data. Even in such specialized cases, mostly weak consistency results are available. In this article, for the first time ever, we derive the almost sure convergence theory of Bayes factor in the general set-up that includes even dependent data and misspecified models. Somewhat surprisingly, the key to the proof of such a general theory is a simple application of a result of Shalizi (2009) to a well-known identity satisfied by the Bayes factor.