R_{0}+R_{b} & \geq & H(X)-p_{0}I(X;W|A=0)-p_{0}I(X;W|A=0)\label{eq:R0+Rb_lossless_ex}\end{aligned}$$  where the mutual information terms are evaluated with respect to the joint pmf $$\begin{aligned} p(x,y,z,a)=p(x,a)p(y,z|a,x),\label{eq:joint-BC_lossless