# Chain rule (probability)

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.

## Chain rule for events

#### Example

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

## References

- Schum, David A. (1994).
*The Evidential Foundations of Probabilistic Reasoning*. Northwestern University Press. p. 49. ISBN 978-0-8101-1821-8. - Klugh, Henry E. (2013).
*Statistics: The Essentials for Research*(3rd ed.). Psychology Press. p. 149. ISBN 1-134-92862-9. - Russell, Stuart J.; Norvig, Peter (2003),
*Artificial Intelligence: A Modern Approach*(2nd ed.), Upper Saddle River, New Jersey: Prentice Hall, ISBN 0-13-790395-2, p. 496. - "The Chain Rule of Probability",
*developerWorks*, Nov 3, 2012.