Factorization (Probability distribution)

The factorization of a probability distribution $$P$$ is a set of probability distributions whose multiplication gives $$P$$.

In general, the factorization of a 2-RV probability distro is: $$P(X,Y) = P(Y|X)P(X) = P(X|Y)P(Y)$$

However, if $$X,Y$$ are independent, then the following is also a factorization: $$P(X,Y)=P(X)P(Y)$$.

Therefore, if $$X,Y$$ are independent, then $$P(X,Y)$$ has three factorizations: $$P(X|Y)P(Y), P(Y|X)P(X)$$ and $$P(X)P(Y)$$.