# 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:

 $P(Intelligent\mid Grade{=}A).$

If it was also observed that $Works\_hard$ is false, the posterior distribution of $Intelligent$ is:

 $P(Intelligent\mid Grade{=}A\land Works\_hard{=}false).$

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.