Posterior odds on a model

$P(M_j|y)=\frac{p(y|M_j)P_j}{{\sum }_{i=1}^{J}p(y|M_i)P_i}$

A way to check among $J$ models which one has the best odds of being right. Note this is just a meta version of Bayes Theorem

• $P(M_j|y)$ - Probability of model j being “the best” among all the choices
• $p(y|M_j)$ - The likelihood of the data under model j
• $P_j$ - Prior beliefs about the probability of model j being the best
• ${\sum }_{i=1}^{J}p(y|M_i)P_i$ - This is a weighted average of the likelihoods of the data under each model with the prior probabilities as weights - basically the expected value of the model mixture

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References

@lancaster2004 - chapter 2