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 is the hypothesis, and and are observations, conditional independence can be stated as an equality: where is the probability of given both and . Since the probability of given is the same as the probability of given both and , this equality expresses that contributes nothing to the certainty of . In this case, and are said to be conditionally independent given , written symbolically as: . In the language of causal equality notation, two functions and which both depend on a common variable are described as conditionally independent using the notation , which is equivalent to the notation . 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. Conditional independence of events Let , , and be events. and are said to be conditionally independent given if and only if and: This property is often written: , which should be read . Equivalently, conditional independence may be stated as: where is the joint probability of and given . This alternate formulation states that and are independent events, given . It demonstrates that is equivalent to . Proof of the equivalent definition iff (definition of conditional probability) iff (multiply both sides by ) iff (divide both sides by ) iff (definition of conditional probability) Examples Coloured boxes Each cell represents a possible outcome. The events , and are represented by the areas shaded , and respectively. The overlap between the events and is shaded . The probabilities of these events are shaded areas with respect to the total area. In both examples and are conditionally independent given because: but not conditionally independent given because: Proximity and delays Let events A and B be defined as the probability that person A and person B will be home in time for dinner where both people are randomly sampled from the entire world. Events A and B can be assumed to be independent i.e. knowledge that A is late has minimal to no change on the probability that B will be late.