Maximum A Posteriori Learning

Instead of Full Bayesian Learning we try to choos the MAP hypothesis $h_{\text{MAP}}$ that maximizes

This can be rewritten using Bayes Rule to

(we can leave out $P(d)$ as it is constant for all hypothesis. We could in theory even leave out $P(h_i)$ when the dataset is large → ML Learning).

or even better by utilizing the properties of the logarithm

So we get $h_{\text{MAP}}$ with

which requires solving an optimization problem instead.

Predictions
The predictions made this way are approximately Bayesian to the extent that

This means, for predictions we have to only compute this one probability and do not have to sum over all possible hypothesis.