Bayes Factor

$B_{12}=\frac{P(y|M_1)}{P(y|M_2)}=\frac{\int l_1({\theta }_1|\theta )p({\theta }_1)d{\theta }_1}{\int l_2({\theta }_2|\theta )p({\theta }_2)d{\theta }_2}$

This is the ratio of the likelihoods under model 1 and 2

If $B_{12}>1$, then the data makes model 1 seem more probable than before.

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References

@lancaster2004 - Chapter 2 2 model posterior odds