Bayes factors

Bayes factors can be considered as a Bayesian (Bayesian statistics) analogue of the likelihood-ratio test. They are often used in hypothesis testing and model selection. Along with p-values and e-values, they are considered measures of evidence against the null.

For parameters ${\theta }_1,{\theta }_2$ (perhaps representing different hypotheses or models) and given data $X$, the Bayes factor is

$$B:=\frac{{\mathbb{P}}_{{\theta }_1}(X)}{{\mathbb{P}}_{{\theta }_2}(X)}=\frac{\mathbb{P}({\theta }_1|X)\mathbb{P}({\theta }_1)}{\mathbb{P}({\theta }_2|X)\mathbb{P}({\theta }_2)},$$

where the second equality follows from Bayes’ theorem if we place prior $\mathbb{P}(\theta )$ over the parameter space. Thus, it is really the second equality where Bayesianism enters the picture—the first equality is simply a likelihood ratio and need not be Bayesian at all.

Jeffreys (a Bayesian) gave a table summarizing how much evidence is provided by different values of $B$. So did Kass and Raftery in 1995. This is pretty silly, as it depends on the application and what actions we’re considering on the basis of $B$.