Posterior Predictive Distribution

Likelihood of the new data given the parameter

is Posterior distribution of the parameter given the data

How to derive this form

Lets start with the

Joint Probability

Posterior Predictive Distribution can be represented as

Marginalization

We can use above Joint distribution

Assuming conditional independence of and given

And then we got the first form

Bayes model averaging (BMA)

This can be viewed as a form of Bayes model averaging (BMA) since we are making predictions using an infinite set of models with parameters values, each one weighted by how likely it is. The use of BMA reduces the chance of overfitting since we are not just using the single best model.

Posterior Predictive Distribution

How to derive this form

Bayes model averaging (BMA)

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