I have a joint probability distribution on the variables x and y given the parameters a and b:

x and y are assumed to be independent. and 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):

Is it also possible to derive this equation as given below: ?

Because b does not mean anything for x, and a does not mean anything for x.