points of a Poisson process with intensity measure $P_{X}\times P_{M}\times\lambda_{\mathbb{R}_{\ge0}}$ (where $P_{M}$ is $\mathrm{Unif}[1:\mathsf{L}]$) independent of $M$. The encoding function is $m\mapsto\tilde{X}_{P_{X}\times\delta_{m}}$ (i.e., $X=\tilde{X}_{P_{X}\times\delta_{M}}$), and the decoding function is $y\mapsto\tilde{M}_{P_{X|Y}(\cdot|y)\times P_{M}}$ (i.e., $\hat{M}=\tilde{M}_{P_{X|Y}(\cdot|Y)\times P_{M}}$). Note that the encoding and decoding functions also depend on the common randomness $\{(\bar{X}_{i},\bar{M}_{i}),T_{i}\}_{i\in\mathbb{N}}$, which will be fixed later. We have $(M,X,Y)\sim P_{M}\times P_{X}P_{Y|X}$. $$\begin{aligned}
 & \mathbf{P}\left\{ M\neq\tilde{M}_{P_{X|Y}(\cdot|Y)\times P_{M}}\right\} \\
 & \le\mathbf{P}\left\{ (X,M)\neq(\tilde{X},\tilde{M})_{P_{X|Y}(\cdot|Y)\times P_{M}}\right\} \\
 & =\mathbf{E}\left[\mathbf{P}\left\{ \left.(X,M)\neq(\tilde{X},\tilde{M})_{P_{X|Y}(\cdot|Y)\times P_{M}}\,\right|\,M,X,Y\right\} \right]\\
 & \stackrel{(a)}{\le}\mathbf{E}\left[1-\left(1+\frac{dP_{X}\times\delta_{M}}{dP_{X|Y}(\cdot|Y)\times P_{M}}(X,M)\right)^{-1}\right]\\
 & =\mathbf{E}\left[1-(1+\mathsf{L}2^{-\iota_{X;Y}(X;Y)})^{-1}\right],\end{aligned}$$ where (a) is by the conditional Poisson matching lemma (Lemma \[lem:cond\_phidiv\]) on $(X,U,Y,Q_{U|Y})\leftarrow(M,(X,M),Y,P_{X|Y}\times P_{M})$ (note that $P_{X,M|M}=P_{X}\times\delta_{M}$). Therefore there exists a fixed $\{(\bar{x}_{i},\bar{m}_{i}),t_{i}\}_{i\in\mathbb{N}}$ such that conditioned on $\{(\bar{X}_{i},\bar{M}_{i}),T_{i}\}_{i\in\mathbb{N}}=\{(\bar{x}_{i},\bar{m}_{i}),t_{i}\}_{i\in\mathbb{N}}$, the average probability of error is bounded by $\mathbf{E}\left[1-(1+\mathsf{L}2^{-\iota_{X;Y}(X;Y)})^{-1}\right]$.

Compared to the scheme in [@yassaee2013oneshot], we use the Poisson process $\{(\bar{X}_{i},\bar{M}_{i}),T_{i}\}$ to create a codebook, instead of the conventional i.i.d. random codebook in [@yassaee2013oneshot]. While the codewords for different $m$’s are still i.i.d., we attach a bias $T_{i}$ to each codeword. Our scheme does not use a stochastic decoder as in [@yassaee2013oneshot], but rather a biased maximum likelihood decoder $\tilde{M}_{P_{X|Y}(\cdot|y)\times P_{M}}=\bar{M}_{K}$ where $K=\arg\max_{i}T_{i}^{-1}(dP_{X|Y}(\cdot|y)/dP_{X})(\bar{X}_{i})$. In the following sections, we will demonstrate how our approach can lead to simpler proofs and sharper bounds compared to 