Chain Rule (probabilities)

Chain rule is successive application of the product rule:

$$\begin{array}{rl}P(a,b,c)& =P(a,b)P(c|a,b)\\ & =P(a)P(b|a)P(c|a,b)\end{array}$$

In general:

$$\begin{array}{rl}P(X_1,\dots ,X_n)& =P(X_1,\dots ,X_{n-1})P(X_n|X_1,\dots ,X_{n-1})\\ & =P(x_1,\dots ,X_{n-2})P(X_{n-1}|X_1,\dots ,X_{n-2})P(x_n|X_1,\dots ,X_{n-1})\\ & =\dots \\ & =\prod _{i=1}^{n}P(X_i|X_1,\dots ,X_{i-1})\end{array}$$