# Conditional and Marginal Independence

Let \(A, B, C\) be three random variables. Consider the following dependency structures modeled with Bayesian networks.

- \(A \leftarrow B \rightarrow C\)
- \(A \rightarrow B \rightarrow C\)
- \(A \rightarrow B \leftarrow C\)

The first two cases both say that \(A\) and \(C\) are *conditionally independent* given \(B\). The last case says that \(A\) and \(C\) are *marginally independent* (when \(B\) is marginalized out); however, they are dependent conditioning on \(B\).

The following are the precise descriptions using the probability notion.

- \(P(A, B, C) = P(B)P(A|B)P(C|B)\)
- \(P(A, B, C) = P(A)P(B|A)P(C|B)\)
- \(P(A, B, C) = P(A)P(C)P(B|A, C)\)