foundations of computational agents
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.
Before there are any observations, the distribution over intelligence is , which is provided as part of the network. To determine the distribution over grades, , requires inference.
If a grade of is observed, the posterior distribution of is given by:
If it was also observed that is false, the posterior distribution of is:
Although and 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.