In probability theory, two events A and B are conditionally independent given a third event C precisely if the occurrence or non-occurrence of A and B are independent events in their conditional probability distribution given C. In other words, Probability theory is the mathematical study of probability. ... In mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. ...

Two random variables X and Y are conditionally independent given an event C if they are independent in their conditional probability distribution given C. Two random variables X and Y are conditionally independent given a third random variable W if for any measurable set S of possible values of W, X and Y are conditionally independent given the event [W ∈ S]. A random variable can be thought of as the numeric result of operating a non-deterministic mechanism or performing a non-deterministic experiment to generate a random result. ...

Conditional independence of more than two events, or of more than two random variables, is defined analogously.

Uses in Bayesian statistics

Let p be the proportion of voters who will vote "yes" in an upcoming referendum. In taking an opinion poll, one chooses n voters randomly from the population. For i = 1, ..., n, let X_i = 1 or 0 according as the ith chosen voter will or will not vote "yes".

In a frequentist approach to statistical inference one would not attribute any probability distribution to p (unless the probabilities could be somehow interpreted as relative frequencies of occurrence of some event or as proportions of some population) and one would say that X₁, ..., X_n are independent random variables. Statistical regularity has motivated the development of the relative frequency concept of probability. ...

By contrast, in a Bayesian approach to statistical inference, one would assign a probability distribution to p regardless of the non-existence of any such "frequency" interpretation, and one would construe the probabilities as degrees of belief that p is in any interval to which a probability is assigned. In that model, the random variables X₁, ..., X_n are not independent, but they are conditionally independent given the value of p. In particular, if a large number of the Xs are observed to be equal to 1, that would imply a high conditional probability, given that observation, that p is near 1, and thus a high conditional probability, given that observation, that the next X to be observed will be equal to 1. Bayesianism is the philosophical tenet that the mathematical theory of probability applies to the degree of plausibility of a statement. ...

See also

de Finetti's theorem[[[[]] In probability theory, de Finettis theorem explains why exchangeable observations are conditionally independent given some (usually) unobservable quantity to which an epistemic probability distribution would then be assigned. ...