Chain rule (probability)

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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.

Consider an indexed set of sets A_1, \ldots , A_n. To find the value of this member of the joint distribution, we can apply the definition of conditional probability to obtain:

\mathrm  P(A_n, \ldots , A_1)  = \mathrm P(A_n | A_{n-1}, \ldots , A_1) \cdot\mathrm P( A_{n-1}, \ldots , A_1)

Repeating this process with each final term creates the product:

\mathrm  P\left(\bigcap_{k=1}^n A_k\right)  = \prod_{k=1}^n  \mathrm P\left(A_k \,\Bigg|\, \bigcap_{j=1}^{k-1} A_j\right)

With four variables, the chain rule produces this product of conditional probabilities:

 \mathrm P(A_4, A_3, A_2, A_1) = \mathrm P(A_4 \mid A_3, A_2, A_1)\cdot \mathrm P(A_3 \mid A_2, A_1)\cdot \mathrm P(A_2 \mid A_1)\cdot \mathrm P(A_1)

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. The chance of choosing a white ball, given that we've chosen the first urn, is P(B|A) = 2/3. 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:

 \mathrm P(A, B)=\mathrm P(B \mid A) \mathrm P(A) = 2/3 \times 1/2 = 1/3.