Conditional independence

In probability theory, conditional independence describes situations in which 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:

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: ${\displaystyle (A\perp \!\!\!\perp B\mid C)}$.

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.

Let ${\displaystyle A}$, ${\displaystyle B}$, and ${\displaystyle C}$ be events. ${\displaystyle A}$ and ${\displaystyle B}$ are said to be conditionally independent given ${\displaystyle C}$ if and only if ${\displaystyle P(C)>0}$ and${\displaystyle P(A\mid B,C)=P(A\mid C)}$. This property is symmetric (more on this below) and often written as ${\displaystyle (A\perp \!\!\!\perp B\mid C)}$, which should be read as${\displaystyle ((A\perp \!\!\!\perp B)\vert C)}$.

Equivalently, conditional independence may be stated as ${\displaystyle P(A,B|C)=P(A|C)P(B|C)}$ where ${\displaystyle P(A,B|C)}$ is the joint probability of ${\displaystyle A}$ and ${\displaystyle B}$ given ${\displaystyle C}$. This alternate formulation states that ${\displaystyle A}$ and ${\displaystyle B}$ are independent events, given ${\displaystyle C}$.

It demonstrates that ${\displaystyle (A\perp \!\!\!\perp B\mid C)}$ is equivalent to ${\displaystyle (B\perp \!\!\!\perp A\mid C)}$.

Each cell represents a possible outcome. The events ${\displaystyle \color {red}R}$, ${\displaystyle \color {blue}B}$ and ${\displaystyle \color {gold}Y}$ are represented by the areas shaded red, blue and yellow respectively. The overlap between the events ${\displaystyle \color {red}R}$ and ${\displaystyle \color {blue}B}$ is shaded purple.

The probabilities of these events are shaded areas with respect to the total area. In both examples ${\displaystyle \color {red}R}$ and ${\displaystyle \color {blue}B}$ are conditionally independent given ${\displaystyle \color {gold}Y}$ because: