8.3 Belief Networks

8.3.1 Observations and Queries

A belief network specifies a joint probability distribution from which arbitrary conditional probabilities can be derived. The most common probabilistic inference task is to compute the posterior distribution of a query variable, or variables, given some evidence, where the evidence is a conjunction of assignment of values to some of the variables.

Example 8.14.

Before there are any observations, the distribution over intelligence is P(Intelligent), which is provided as part of the network. To determine the distribution over grades, P(Grade), requires inference.

If a grade of A is observed, the posterior distribution of Intelligent is given by:


If it was also observed that Works_hard is false, the posterior distribution of Intelligent is:


Although Intelligent and Works_hard are independent given no observations, they are dependent given the grade. This might explain why some people claim they did not work hard to get a good grade; it increases the probability they are intelligent.