We have n = 197 samples divided into four categories: y = (125,18,20,34). A genetic model for the population cell probabilities is given by(21+4,41,41,4) and the sampling model is a multinomial distribution: p(y)=y1!y2!y3!y4!n!(21+4)y1(41)y2(41)y3(4)y4, where n = y1+y2+y3+y4. We're supposed to assume the prior distribution for to be Uniform(0,1). To find the posterior distribution of , a Gibbs sampling algorithm can be implemented by splitting the first category into two (y0,y1-y0) with probabilities (1/2,/4). Here y0 can be viewed as another parameter (or a latent variable). Thus,p(,y0y)=y0!(y1y0)!y2!y3!y4!n!(21)y0+(4)y1y0(41)y2(41)y3(4)y4

Derive the full conditional distributions of and y0.

I know to get the full conditional distributions we'll need to do the joint distribution of , y0, and y by multiplying the functions for all three of those variables. But with being Uniform and y0 having a probability of 1/2 it seems like both of those terms would drop out of the proportional joint distribution and that the full conditional distributions of and y0 will just be the parts of the distribution of y that contain these terms. Is this correct?