Considering a joint distribution's factorization, when will $A$ and $B$ be independent?

Question

Consider a joint distribution known to factorize as:

$$P(A,B,C) = P(C\mid A,B)P(B)P(A)$$

Given this, motivate mathematically under what circumstances $A$ and $B$ will be statistically independent?

Answer 1

I will assume $\Pr(A\cap B)\ne0$.

In that case we have $$ \Pr(A\cap B\cap C) = \Pr(C\mid A\cap B)\Pr(A\cap B). \tag 1 $$ This question invites us to suppose that $$ \Pr(A\cap B\cap C) = \Pr(C\mid A\cap B)\Pr(A)\Pr(B). \tag 2 $$ If $(1)$ and $(2)$ are both true then we have $$ \Pr(C\mid A\cap B)\Pr(A\cap B) = \Pr(C\mid A\cap B)\Pr(A)\Pr(B). \tag 3 $$ If in addition we have $\Pr(C\mid A\cap B)\ne0,$ then we can divide both sides of $(3)$ by $\Pr(C\mid A\cap B),$ yielding $$ \Pr(A\cap B) = \Pr(A)\Pr(B), $$ and we conclude that $A$ and $B$ are independent.

However, it is possible that $\Pr(C\mid A\cap B)=0$ and $(2)$ holds while $A$ and $B$ are not independent. Here's how: Suppose there are three equally probable outcomes: $\Omega = \{1,2,3\},$ and $A= \{1,2\}$ and $B=\{2,3\}$ and $C=\varnothing.$ Then $(1)$ holds since both sides of $(1)$ are $0$, even though $A$ and $B$ are not independent.