# Independence¶

As discussed in the previous section, **conditional probabilities** quantify the extent to which the knowledge of the occurrence of a certain event affects the probability of another event 1.
In some cases, it makes no difference: the events are independent. More formally, events \(A\) and \(B\) are **independent** if and only if

This definition is not valid if \(P (B) = 0\). The following definition covers this case and is otherwise equivalent.

Definition (Independence).

Let \((\Omega,\mathcal{F},P)\) be a probability space. Two events \(A,B \in \mathcal{F}\) are independent if and only if

Notation

This is often denoted \( A \indep B \)

Similarly, we can define **conditional independence** between two events given a third event.
\(A\) and \(B\) are conditionally independent given \(C\) if and only if

where \(P (A|B, C) := P (A|B \cap C)\). Intuitively, this means that the probability of \(A\) is not affected by whether \(B\) occurs or not, as long as \(C\) occurs.

Notation

This is often denoted \( A \indep B \mid C\)

## Graphical Models¶

There is a graphical model representation for joint distributions \(P(A,B,C)\) that encodes their conditional (in)dependence known as a **probabilistic graphical model**. For this situation \( A \indep B \mid C\), the graphical model looks like this:

The lack of an edge directly between \(A\) and \(B\) indicates that the two varaibles are conditionally independent. This image was produced with `daft`

, and there are more examples in Visualizing Graphical Models.

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This text is based on excerpts from Section 1.3 of NYU CDS lecture notes on Probability and Statistics