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.