chain rule (probability)

• Related
  • joint entropy, #independence (probability), Bayes theorem
• Statistic

Consider an indexed collection of random variables $X_1,\dots ,X_n$.

To find the value of this member of the joint distribution, we can apply the definition of conditional probability to obtain

$$\mathrm{P}\left(X_n,\dots ,X_1\right)=\mathrm{P}\left(X_n\mid X_{n-1},\dots ,X_1\right)\cdot \mathrm{P}\left(X_{n-1},\dots ,X_1\right)$$

II. Example

$$\begin{array}{rl}& \mathrm{P}\left(X_4,X_3,X_2,X_1\right)\\ & =\mathrm{P}\left(X_4\mid X_3,X_2,X_1\right)\cdot \mathrm{P}\left(X_3,X_2,X_1\right)\\ & =\mathrm{P}\left(X_4\mid X_3,X_2,X_1\right)\cdot \mathrm{P}\left(X_3\mid X_2,X_1\right)\cdot \mathrm{P}\left(X_2,X_1\right)\\ & =\mathrm{P}\left(X_4\mid X_3,X_2,X_1\right)\cdot \mathrm{P}\left(X_3\mid X_2,X_1\right)\cdot \mathrm{P}\left(X_2\mid X_1\right)\cdot \mathrm{P}\left(X_1\right)\end{array}$$

IV. References

• https://en.wikipedia.org/wiki/Chain_rule_(probability)