# Name for decomposition of joint probability into product of conditional probabilities

Consider a set of events $$A_1, \dots, A_n$$.

By definition we have

$$\mathbb{P}[A_1 \cap (A_2 \cap \dots \cap A_n)] = \mathbb{P} [(A_2 \cap \dots \cap A_n) | A_1] \mathbb{P}(A_1).$$

Applying the definition again,

$$\mathbb{P} [A_2 \cap (A_3 \cap \dots \cap A_n) | A_1] = \mathbb{P} [ (A_3 \cap \dots \cap A_n) | A_1, A_2] P(A_2|A_1),$$

which together with the first equality implies

$$\mathbb{P}[A_1 \cap (A_2 \cap \dots \cap A_n)] = \mathbb{P} [ (A_3 \cap \dots \cap A_n) | A_1, A_2] \mathbb{P}(A_2|A_1) \mathbb{P}(A_1).$$

Continuing the argument recursively yields the following often used decomposition:

$$\mathbb{P}[A_1 \cap A_2 \cap \dots \cap A_n] = \mathbb{P}(A_1) \mathbb{P}(A_2|A_1) \dots \mathbb{P}(A_{n}|A_1, A_2, \dots, A_{n-1}) .$$

My question is, is there a standard terminology to refer to this decomposition (say as standard as the "Law of Total Probability" terminology)?

• Closely related to the prediction-error decomposition $p(X_n,X_{n-1},...,X_2,X_1) =$ $p(X_n|X_{n-1},...,X_2,X_1) \cdot p(X_{n-1}|X_{n-2}...,X_2,X_1)\cdot ...\cdot p(X_2|X_1) \cdot p(X_1)$ May 24, 2019 at 7:52

That is repetitive implementation of the definition of conditional probability as you describe, and is referred as Chain Rule. More generally, it can be written as $$P\left(\bigcap_{k=1}^n A_k\right)=\prod_{k=1}^n P\left(A_k\bigg\vert\bigcap_{m=1}^{k-1}A_j\right)$$