derived from ), and outputs $X=x(U_{1},U_{2})$. It can be verified that is satisfied.

The decoding functions are $\hat{M}_{1}=(\tilde{M}_{1})_{P_{U_{1}|Y_{1}}(\cdot|Y_{1})\times P_{M_{1}}}$, $\hat{M}_{2}=(\tilde{M}_{2})_{P_{U_{2}|Y_{2}}(\cdot|Y_{2})\times P_{M_{2}}}$. We have the following almost surely: $$\begin{aligned}
 & \mathbf{P}\biggl\{(\tilde{U}_{1},\tilde{M}_{1})_{P_{U_{1}|Y_{1}}(\cdot|Y_{1})\times P_{M_{1}}}\neq(U_{1},M_{1})\,\biggl|\,U_{1},U_{2},Y_{1},Y_{2},M_{1},K\biggr\}\\
 & \stackrel{(a)}{=}\mathbf{P}\biggl\{(\tilde{U}_{1},\tilde{M}_{1})_{P_{U_{1}|Y_{1}}(\cdot|Y_{1})\times P_{M_{1}}}\neq(U_{1},M_{1})\,\biggl|\,U_{1},Y_{1},M_{1},K\biggr\}\\
 & \stackrel{(b)}{\le}K\frac{dP_{U_{1}}\times\delta_{M_{1}}}{dP_{U_{1}|Y_{1}}(\cdot|Y_{1})\times P_{M_{1}}}(U_{1},M_{1})\\
 & \le\mathsf{L}_{1}\mathsf{J}2^{-\iota_{U_{1};Y_{1}}(U_{1};Y_{1})},\end{aligned}$$ where (a) is by $(U_{2},Y_{2})\leftrightarrow(U_{1},Y_{1},M_{1},K)\leftrightarrow\{(\bar{U}_{1,i},\bar{M}_{1,i}),T_{1,i}\}_{i}$ (see Figure \[fig:bc\_bayes\] middle), and (b) is by the conditional generalized Poisson matching lemma on $(X,J,U,Y,Q_{U|Y})\leftarrow(M_{1},\,K,\,(U_{1},M_{1}),\,Y_{1},\,P_{U_{1}|Y_{1}}\times P_{M_{1}})$, since $P_{U_{1},M_{1}|M_{1},K}=P_{U_{1}}\times\delta_{M_{1}}$, $(M_{1},K)\perp\!\!\!\perp\{(\bar{U}_{1,i},\bar{M}_{1,i}),T_{1,i}\}_{i}$, and $Y_{1}\leftrightarrow(U_{1},M_{1},K)\leftrightarrow\{(\bar{U}_{1,i},\bar{M}_{1,i}),T_{1,i}\}_{i}$, which can be deduced from and $\check{U}_{1j}=(\tilde{U}_{1})_{P_{U_{1}}\times\delta_{M_{1}}}(j)$ (see Figure \[fig:bc\_bayes\] middle).

Also, almost surely, $$\begin{aligned}
 & \mathbf{P}\left\{ \left.(\tilde{U}_{2},\tilde{M}_{2})_{P_{U_{2}|Y_{2}}(\cdot|Y_{2})\times P_{M_{2}}}\neq(U_{2},M_{2})\,\right|\,U_{1},U_{2},Y_{1},Y_{2},M_{2}\right\} \\
 & \stackrel{(a)}{=}\mathbf{P}\left\{ \left.(\tilde{U}_{2},\tilde{M}_{2})_{P_{U_{2}|Y_{2}}(\cdot|Y_{2})\times P_{M_{2}}}\neq(U_{2},M_{2})\,\right|\,U_{1},U_{2},Y_{2},M_{2}\right\} \\
 & \stackrel{(b)}{\le}\mathbf{E}\left[\left.\frac{d(\mathsf{J}^{-1}\sum_{j=1}^{\mathsf{J}}P_{U_{2}|U_{1}}(\cdot|\check{U}_{1j}))\times\delta_{M_{2}}}{dP_{U_{2}|Y_{2}}(\cdot|Y_{2})\times P_{M_{2}}}(U_{2},M_{2})\,\right|\,U_{1},U_{2},Y_{2},M_{2}\right]\\
 & =\mathbf{E}\left[\left.\mathsf{L}_{2}\mathsf{J}^{-1}\sum_{j=1}^{\mathsf{J}}2^{\iota_{U_{1};U_{2}}(\check{U}_{1j};U_{2})-\iota_{U_{2};Y_{2}}(U_{2};Y_{2})}\,\right|\,U_{1},U_{2},Y_{2},M_{2}\right]\\
 & =\mathbf{E}\left[\left.\mathsf{L}_{2}\mathsf{J}^{-1}2^{-\iota_{U_{2};Y_{2}}(U_{2};Y_{2})}\Bigl(2^{\iota_{U_{1};U_{2}}(U_{1};U_{2})}+\sum_{j\in[1:\mathsf{J}]\backslash K}2^{\iota_{U_{1};U_{2}}(\check{U}_{1j};U_{2})}\Bigr)\,\right|\,U_{1},U_{2},Y_{2},M_{2}\right]\\
 & =\mathbf{E}\left[\left.\mathsf{L}_{2}\mathsf{J}^{-1}2^{-\iota_{U_{2};Y_{2}}(U_{2};Y_{2})}\Bigl(2^{\iota_{U_{1};U_{2}}(U_{1};U_{2})}+\sum_{j=1}^{\mathsf{J}-1}2^{\iota_{U_{1};U_{2}}(\check{U}_{1,j+\mathbf{1}\{j\ge K\}};U_{2})}\Bigr)\,\right|\,U_{1},U_{2},Y_{2},M_{2}\right]\\
 & \stackrel{(c)}{\le}\mathsf{L}_{2}\mathsf{J}^{-1}2^{-\iota_{U_{2};Y_{2}}(U_{2};Y_{2})}(2^{\iota_{U_{1};U_{2}}(U_{1};U_{2})}+\mathsf{J}-1),\end{aligned}$$ where (a) is by $Y_{1}\leftrightarrow(U_{1},U_{2},Y_{2},M_{2})\leftrightarrow\{(\bar{U}_{2,i},\bar{M}_{2,i}),T_{2,i}\}_{i}$ (see Figure \[fig:bc\_bayes\] right), (b) is by the conditional Poisson matching lemma on $((\{\check{U}_{1j}\}_{j},M_{2}),\,(U_{2},M_{2}),\,Y_{2},\,P_{U_{2}|Y_{2}}\times P_{M_{2}})$, and (c) is because $\{\check{U}_{1,j+\mathbf{1}\{j\ge K\}}\}_{j\in[1:\mathsf{J}-1]}$ (the $\check{U}_{1j}$’s not selected as $U_{1}$) are independent of $(U_{1},U_{2},Y_{2},M_{2})$, $\mathbf{E}[2^{\iota_{U_{1};U_{2}}(\check{U}_{1,j+\mathbf{1}\{j\ge K\}};U_{2})}\,|\,U_{2}]=1$, and Jensen’s inequality. Hence, $$\begin{aligned}
 & \mathbf{P}\{(M_{1},M_{2})\neq(\hat{M}_{1},\hat{M}_{2})\}\\
 & =\mathbf{E}\left[\mathbf{P}\left\{ \left.(M_{1},M_{2})\neq(\hat{M}_{1},\hat{M}_{2})\,\right|\,U_{1},U_{2},Y_{1},Y_{2}\right\} \right]\\
 & \le\mathbf{E}\left[\min\left\{ \mathbf{P}\left\{ \left.M_{1}\neq\hat{M_{1}}\,\right|\,U_{1},U_{2},Y_{1},Y_{2}\right\} +\mathbf{P}\left\{ \left.M_{2}\neq\hat{M_{2}}\,\right|\,U_{1},U_{2},Y_{1},Y_{2}\right\} ,\,1\right\} \right]\\
 & \le\mathbf{E}\biggl[\min\Bigl\{\mathsf{L}_{1}\mathsf{J}2^{-\iota_{U_{1};Y_{1}}(U_{1};Y_{1})}+\mathsf{L}_{2}\mathsf{J}^{-1}2^{-\iota_{U_{2};Y_{2}}(U_{2};Y_{2})}(2^{\iota_{U_{1};U_{2}}(U_{1};U_{2})}+\mathsf{J}-1),\,1\Bigr\}\biggr].\end{aligned}$$ Therefore there exist fixed realizations of the Poisson processes attaining the desired bound.

![\[fig:bc\_bayes\]Left: The Bayesian network described in . Middle: The Bayesian network deduced from and $\check{U}_{1j}=(\tilde{U}_{1})_{P_{U_{1}}\times\delta_{M_{1}}}(j)$. Right: The Bayesian network describing the encoding scheme. Note 