
Independence
I have a joint probability distribution on the variables x and y given the parameters a and b:
$\displaystyle p(x,ya,b)$
x and y are assumed to be independent. $\displaystyle p(xa)$ and $\displaystyle p(yb)$ are multinomial distributions separately. Therefore, a is defined for x and b is defined for y. Therefore the equation can be written as given below (because of the independence between x and y):
$\displaystyle p(xa,b) p(ya,b)$
Is it also possible to derive this equation as given below: ?
$\displaystyle p(xa) p(yb)$
Because b does not mean anything for x, and a does not mean anything for x.