## Probabilistic graphical models

Let's start with a refresher course in basic statistics.

Given two events or observations *X* and *Y*, the joint probability of *X* and *Y* is defined as *p(X,Y) = p(X∩Y)*. If the observations *X* and *Y* are not related, an assumption known as conditional independence, then *p(X,Y) = p(X).p(Y)*. The conditional probability of an event *Y*, given *X*, is defined as *p(Y|X) = p(X,Y)/p(X)*.

These two definitions are quite simple. However, **probabilistic reasoning** can be difficult to read in the case of large numbers of variables and sequences of conditional probabilities. As a picture is worth a thousand words, researchers introduced **graphical models** to describe a probabilistic relation between random variables using graphs [5:1].

There are two categories of graphs, and therefore, graphical models, which are as follows:

Directed graphs such as Bayesian networks

Undirected graphs such as conditional random fields (refer to the

*Conditional random fields*section in Chapter 7,*Sequential Data...*