Conditional independence

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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 ${\displaystyle A}$ is the hypothesis, and ${\displaystyle B}$ and ${\displaystyle C}$ are observations, conditional independence can be stated as an equality:

${\displaystyle P(A\mid B,C)=P(A\mid C)}$

where ${\displaystyle P(A\mid B,C)}$ is the probability of ${\displaystyle A}$ given both ${\displaystyle B}$ and ${\displaystyle C}$. Since the probability of ${\displaystyle A}$ given ${\displaystyle C}$ is the same as the probability of ${\displaystyle A}$ given both ${\displaystyle B}$ and ${\displaystyle C}$, this equality expresses that ${\displaystyle B}$ contributes nothing to the certainty of ${\displaystyle A}$. In this case, ${\displaystyle A}$ and ${\displaystyle B}$ are said to be conditionally independent given ${\displaystyle C}$, written symbolically as: ${\displaystyle (A\perp \!\!\!\perp B\mid C)}$. In the language of causal equality notation, two functions ${\displaystyle f(y)}$ and ${\displaystyle g(y)}$ which both depend on a common variable ${\displaystyle y}$ are described as conditionally independent using the notation ${\displaystyle f\left(y\right)~{\overset {\curvearrowleft \curvearrowright }{=}}~g\left(y\right)}$, which is equivalent to the notation ${\displaystyle P(f\mid g,y)=P(f\mid y)}$.

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.