Editing Conditional independence (section)

{{Short description|Probability theory concept}}
{{see also|Conditional dependence}}

{{Probability fundamentals}}

In [[probability theory]], '''conditional independence''' describes situations wherein an observation is irrelevant or redundant when evaluating the certainty of a hypothesis. Conditional independence is usually formulated in terms of [[conditional probability]], as a special case where the probability of the hypothesis given the uninformative observation is equal to the probability without. If <math>A</math> is the hypothesis, and <math>B</math> and <math>C</math> are observations, conditional independence can be stated as an equality:

:<math>P(A\mid B,C) = P(A \mid C)</math>

where <math>P(A \mid B, C)</math> is the probability of <math>A</math> given both <math>B</math> and <math>C</math>. Since the probability of <math>A</math> given <math>C</math> is the same as the probability of <math>A</math> given both <math>B</math> and <math>C</math>, this equality expresses that <math>B</math> contributes nothing to the certainty of <math>A</math>. In this case, <math>A</math> and <math>B</math> are said to be '''conditionally independent''' given <math>C</math>, written symbolically as: <math>(A \perp\!\!\!\perp B \mid  C)</math>.

The concept of conditional independence is essential to graph-based theories of statistical inference, as it establishes a mathematical relation between a collection of conditional statements and a [[graphoid]].