In the first post, we discussed the importance of model updating, now we want to focus on the mathematics that comes into play. To do so, we will make use of Bayes’ theorem. In particular, we will need an alternative formulation of the (hopefully) well-known result.

First of all, we will briefly revise conditional probabilities.

**Definition.** Given two events and in a probability space, with , the *conditional probability of given * is the quantity

The idea is the following: we know that some event ( in the definition above) has happened, and we want to renormalise our notion of probability to the new reality we are living in, the one where is certain, as it already happened. One can notice that the conditional probability with respect to is itself a probability measure, and, in particular, satisfies all the properties of a probability. At the same time, it is worth noticing, with an example, that , in general, is not equal to . Among inmates in Italy, roughly 90% are male, so we can write. However, it is definitely false that the probability that an Italian male is an inmate, that is , is 90%. At the same time, the two probabilities are connected with one another, as the following result shows.

**Theorem (Bayes, probability form).** Given two events with nonzero probability and , the following identity holds:

Continue reading “Bayes’ Theorem – Part 1”