# Is the product of two factorial distributions also factorial?

For a distribution to be factorial : $$P(X) = P(x_1)P(x_2)P(x_3)\cdots P(x_d)$$ My question is :

will $$P(X\mid Y)P(Y\mid Z)$$ be factorial ?

for $P(X\mid Y)$ and $P(Y\mid Z)$ are factorial

Actually the problem I really want to solve is the entropy of the above equation

$$H = \sum_{X,Y}P(X\mid Y)P(Y\mid Z)\log{P(X\mid Y)P(Y\mid Z)}$$

where I know both distribution are factorial, that is

$$P(X\mid Y)=P(X_1\mid Y)P(X_2\mid Y)\cdots P(X_d\mid Y)$$ $$P(Y\mid Z)=P(Y_1\mid Z)P(Y_2\mid Z)\cdots P(Y_d\mid Z)$$

and I am wondering if I can simplify that.

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