In probability theory, the chain rule (also called the general product rule[1][2]) permits the calculation of any member of the joint distribution of a set of random variables using only conditional probabilities. The rule is useful in the study of Bayesian networks, which describe a probability distribution in terms of conditional probabilities.
This rule is illustrated in the following example. Urn 1 has 1 black ball and 2 white balls and Urn 2 has 1 black ball and 3 white balls. Suppose we pick an urn at random and then select a ball from that urn. Let event be choosing the first urn: . Let event be the chance we choose a white ball. The chance of choosing a white ball, given that we have chosen the first urn, is . Event would be their intersection: choosing the first urn and a white ball from it. The probability can be found by the chain rule for probability:
.
More than two events[]
For more than two events the chain rule extends to the formula
which by induction may be turned into
.
Example[]
With four events (), the chain rule is
Chain rule for random variables[]
Two random variables[]
For two random variables , to find the joint distribution, we can apply the definition of conditional probability to obtain:
More than two random variables[]
Consider an indexed collection of random variables . To find the value of this member of the joint distribution, we can apply the definition of conditional probability to obtain:
Repeating this process with each final term creates the product:
Example[]
With four variables (), the chain rule produces this product of conditional probabilities:
Footnotes[]
^Schum, David A. (1994). The Evidential Foundations of Probabilistic Reasoning. Northwestern University Press. p. 49. ISBN978-0-8101-1821-8.
^Klugh, Henry E. (2013). Statistics: The Essentials for Research (3rd ed.). Psychology Press. p. 149. ISBN978-1-134-92862-0.