Definition:Bayes' Factor

Definition

Let $D$ be a set of data which may be assumed to have arisen from one of only $2$ possible models: $M_1$ or $M_2$.

Let the prior probabilities of $M_1$ and $M_2$ be denoted $\map \Pr {M_1}$ and $\map \Pr {M_2}$ such that $\map \Pr {M_2} = 1 - \map \Pr {M_1}$.

Let the posterior probabilities be denoted $\condprob {M_1} D$ and $\condprob {M_2} D$ such that $\condprob {M_2} D = 1 - \condprob {M_1} D$.

From Bayes' Theorem we have: $\dfrac {\condprob {M_1} D} {\condprob {M_2} D} = B_{1 2} \dfrac {\map \Pr {M_1} } {\map \Pr {M_2} }$ such that:

$B_{1 2} = \dfrac {\condprob D {M_1} } {\condprob D {M_2} }$

The coefficient $B_{1 2}$ is called the Bayes' factor.

Hence the posterior probability of $M_1$ is obtained by multiplying the prior probability of $M_1$ by the Bayes' factor, which is independent of the prior probability.

This entry was named for Thomas Bayes.