Tutorials
Belief and Decision Networks

## Tutorial Five (Supplementary): Conditional Independence

This tutorial gives a short introduction to the concepts of conditional independence and d-separation. For more in-depth information about these subjects, try some of the links listed at the end of this tutorial.

A Bayesian network is a graphical representation of conditional independence and conditional probabilities. Informally, a variable is conditionally independent of another, if your belief in the value of the latter wouldn't influence your belief in the value of the former. By the same token, a variable is conditionally independent of another, given some information, if your belief in the value of the second variable wouldn't influence your belief in the value of the first, given that you have the information. For example, my belief in whether or not my cat has knocked over a plant on the balcony is independent of my belief in whether or not it is windy today. However, my belief in whether or not a plant on the balcony has been knocked over is not conditionally independent of either of these two factors, since either could be a cause of a plant being knocked over. Additionally, if I happen to know that a plant on my balcony has been knocked over, my belief in whether or not my cat has done it is no longer independent of my belief in whether or not it is windy. If it is windy, that gives me a reason to believe that the plant's being knocked over is a result of the wind. In other words, knowing that it is windy can explain away a reason for my plant's being knocked over.

More formally, random variable X is independent of random variable Y given random variable Z, if P( X | Y & Z ) = P( X | Z ). One of the most important independence assumptions in a belief network is that each random variable is independent of its nondescendents given its parents. A broader formalisation is given by d-separation. The definition below is from [Pearl, 2000]:

A path p is said to be d-separated (or blocked) by a set of nodes Z if and only if
1. p contains a chain i->m->j or a fork i<-m->j such that the middle node m is in Z, or
2. p contains an inverted fork (or collider) i->m<-j such that the middle node m is not in Z and such that no descendent of m is in Z.

A set Z is said to d-separate X from Y if and only if Z blocks every path from a node in X to a node in Y.

The algorithm used to determine irrelevant variables in Verbose Query Mode is the same algorithm used by the Quiz to determine the answers to conditional independence questions. To start quizzing yourself, click the 'Independence Quiz' button on the toolbar in Solve mode. The Quiz window will appear with a copy of the graph currently visible in the main applet.

When the quiz window opens, the 'Quiz Yourself' tab will be selected. You will see a button labelled 'Answer a Question'. Below them will be some disabled options. There is also a text area at the bottom of the screen, where the questions you ask and answer will appear.