- Chain rule (probability)
-
In probability theory, the chain rule permits the calculation of any member of the joint distribution of a set of random variables using only conditional probabilities.
Consider an indexed set of sets . 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
For example:
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 A be choosing the first urn: P(A) = P(~A) = 1/2. Let event B be the chance we choose a white ball. Chance of choosing a white ball, given that we've chose the first urn, is P(B|A) = 2/3. Chance of choosing a white ball, given that we've chosen the second urn is P(B|~A) = 3/4. Event A, B 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:
-
- .
References
- 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, http://aima.cs.berkeley.edu/, p. 496.
Categories: -
Wikimedia Foundation. 2010.