MLE and MAP learning

Let's suppose we have a data generating process p_data, used to draw a dataset X:

In many statistical learning tasks, our goal is to find the optimal parameter set θ according to a maximization criterion. The most common approach is based on the likelihood and is called MLE. In this case, the optimal set θ is found as follows:

This approach has the advantage of being unbiased by wrong preconditions, but, at the same time, it excludes any possibility of incorporating prior knowledge into the model. It simply looks for the best θ in a wider subspace, so that p(X|θ) is maximized. Even if this approach is almost unbiased, there's a higher probability of finding a sub-optimal solution that can also be quite different from a reasonable (even if not sure) prior. After all, several models are too complex to allow us to define a suitable prior probability (think, for example, of reinforcement learning strategies where there's a huge number of complex states). Therefore, MLE offers...