sum_r P(r) P(x|r) P(y|r)such that H(R) < 1 bit.
r = 0 / 1 with probabilities p0 , p1, where p1 = 1/(2(1-f)) P(x|r=0) = { 1 , 0 } P(y|r=0) = { 1 , 0 } P(x|r=1) = { f , (1-f) } P(y|r=1) = { f , (1-f) }
H(R) = H_2(p1) = H_2( 1/(2(1-f)) )
We confirmed, by brute-force plotting of the functions
s(a,b,f) = -1.0/2.0*(2*f-1.0)/(-f+1.0-b-a+2*a*b) d(a,b,f) = (a-1+f)/(2.0*a-1) c(a,b,f) = (f-1+b)/(-1.0+2*b),that this is the optimal solution using a binary latent variable "r" to generate a single pair (x,y).