Maximum A Posteriori Learning

Instead of Full Bayesian Learning we try to choos the MAP hypothesis $h_{\text{MAP}}$ that maximizes $P(h_i\mid d).$ This can be rewritten using Bayes Rule to $P(d\mid h_i)\cdot P(h_i)$ (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 ${\log }_2\left(P\left(\mathrm{d}\mid h_i\right)\right)+{\log }_2\left(P\left(h_i\right)\right).$ So we get $h_{\text{MAP}}$ with $\underset{h_i}{\mathrm{argmax}}{\log }_2\left(P\left(\mathrm{d}\mid h_i\right)\right)+{\log }_2\left(P\left(h_i\right)\right)$ which requires solving an optimization problem instead.

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

$$\mathrm{P}(X\mid d)\approx \mathrm{P}(X\mid h_{\mathrm{MAP}})$$

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