Conditional and Marginal Independence

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

  1. \(A \leftarrow B \rightarrow C\)
  2. \(A \rightarrow B \rightarrow C\)
  3. \(A \rightarrow B \leftarrow C\)

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.

  1. \(P(A, B, C) = P(B)P(A|B)P(C|B)\)
  2. \(P(A, B, C) = P(A)P(B|A)P(C|B)\)
  3. \(P(A, B, C) = P(A)P(C)P(B|A, C)\)