Exam Questions - Block 5: Bayesian factor

[SebastianVeuskens]

Illustrate the Bayesian factor and its use

[DesR99]

There is also some comments in Block 1 for this answer.

[DesR99]

The formula for Bayes Factor is:
$r = \frac{P(B|A)}{P(B|\bar{A})}$

This factor can also be obtained by:
$Bayes factor = \frac{Posterior\ odds}{Prior\ odds} = \frac{P(A|B)/P(\bar{A}|B)} {P(A)/P(\bar{A})}$

The bayes factor depends on the sample data and can be used to indicate how much of the data is in favour of one model over another and in hypothesis testing to reject or fail to reject hypotheses. A small value for Bayes Factor will see us reject the null hypothesis in favour of the alternative hypothesis with the highest posterior probability.

The Bayes Factor is also the Likelihood ratio for the simple null and alternative hypotheses:
$\frac{Posterior\ odds}{Prior\ odds} = Likelihood\ ratio$

In the case of composite null and alternative hypotheses the Bayes Factor is not only dependent on the data but also the choice of priors as it is the likelihood ratio weighted with a prior:
$B = \frac{P(H_0|x)}{P(H_1|x)} * \frac{π_1}{π_0}$

[SebastianVeuskens]

Amazing answer, thanks Des!

Addition
From her notes:

The Bayes factor is a measure of the support to the hypotheses.

This means that it is a possible method to decide for one hypothesis or the other.

However, before in her slides she says that $H_0$ is accepted if the posterior odds ratio $>1$. Thus, maybe the Bayesian factor is an indicator what the data supports (and the prior for the compound hypotheses). But the final decision should be made based on the posterior odds.