## Learning by inference

In the introduction to this chapter, we saw that learning can be done in a frequentist way by counting data. In most cases, it will be sufficient, but it is also a narrow view of the notion of learning. More generally speaking, learning is the problem of integrating data into the domain knowledge in order to create a new model or improve an existing model. Therefore, learning can be seen as an inference problem, where one updates an existing model toward a better model.

Let's consider a simple problem: modeling the results of tossing a coin. We want to test if the coin is fair or not. Let's call θ the probability that the coin lands on its head. A fair throw would have a probability of 0.5. By tossing the coin several times we want to estimate this probability. Let's say the *i ^{-th}* toss outcome is

*v*

_{i}*= 1*if the coin shows a head and 0 otherwise. We also assume there is no dependence between each toss, which means observations are

*i.i.d*. And finally, we consider each toss...