that all three are valid Bayesian networks, and the desired conditional independence relations can be deduced using d-separation.](bc_bayes)

One-shot Distributed Lossy Source Coding
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The one-shot distributed lossy source coding setting is described as follows. Let $(X_{1},X_{2})\sim P_{X_{1},X_{2}}$. Upon observing $X_{j}$, encoder $j$ produces $M_{j}\in[1:\mathsf{L}_{j}]$, $j=1,2$. The decoder observes $M_{1},M_{2}$ and recovers $\hat{Z}_{1}\in\mathcal{Z}_{1}$, $\hat{Z}_{2}\in\mathcal{Z}_{2}$ with probability of excess distortion $P_{e}=\mathbf{P}\{\mathsf{d}_{1}(X_{1},\hat{Z}_{1})>\mathsf{D}_{1}\;\mathrm{or}\;\mathsf{d}_{2}(X_{2},\hat{Z}_{2})>\mathsf{D}_{2}\}$, where $\mathsf{d}_{j}:\mathcal{X}_{j}\times\mathcal{Z}_{j}\to\mathbb{R}_{\ge0}$ is a distortion measure for $j=1,2$.

We show a one-shot version of the Berger-Tung inner bound [@berger1978multiterminal; @tung1978multiterminal].

\[thm:dlsc\]Fix any $P_{U_{1}|X_{1}}$, $P_{U_{2}|X_{2}}$ and functions $z_{j}:\mathcal{U}_{1}\times\mathcal{U}_{2}\to\mathcal{Z}_{j}$, $j=1,2$. There exists a code for distributed lossy source coding with sources $P_{X_{1}},P_{X_{2}}$ and message sizes $\mathsf{L}_{1},\mathsf{L}_{2}$, with probability of excess distortion $$\begin{aligned}
P_{e} & \le\mathbf{E}\biggl[\min\biggl\{\mathbf{1}\{\mathsf{d}_{1}(X_{1},Z_{1})>\mathsf{D}_{1}\;\mathrm{or}\;\mathsf{d}_{2}(X_{2},Z_{2})>\mathsf{D}_{2}\}+\mathsf{L}_{1}^{-1}2^{\iota_{U_{1};X_{1}|U_{2}}(U_{1};X_{1}|U_{2})}\nonumber \\
 & \;\;\;\;+\left(\mathsf{L}_{1}^{-1}\mathsf{L}_{2}^{-1}2^{\iota_{U_{1},U_{2};X_{1},X_{2}}(U_{1},U_{2};X_{1},X_{2})}+\mathsf{L}_{2}^{-1}2^{\iota_{U_{2};X_{2}|U_{1}}(U_{2};X_{2}|U_{1})}\right)\left(\log(\mathsf{L}_{2}2^{-\iota_{U_{2};X_{2}|U_{1}}(U_{2};X_{2}|U_{1})}+1)+1\right)^{2},\,1\biggr\}\biggr]\label{eq:dlsc_pe1}\end{aligned}$$ if all the information density terms are defined, where $(X_{1},X_{2},U_{1},U_{2},Z_{1},Z_{2})\sim P_{X_{1},X_{2}}P_{U_{1}|X_{1}}P_{U_{2}|X_{2}}\delta_{z_{1}(U_{1},U_{2})}\delta_{z_{2}(U_{1},U_{2})}$. As a result, for $\gamma>0$, $$\begin{aligned}
P_{e} & \le\mathbf{P}\biggl\{\mathsf{d}_{1}(X_{1},Z_{1})>\mathsf{D}_{1}\;\mathrm{or}\;\mathsf{d}_{2}(X_{2},Z_{2})>\mathsf{D}_{2}\;\;\mathrm{or}\;\;\log\mathsf{L}_{1}<\iota_{U_{1};X_{1}|U_{2}}(U_{1};X_{1}|U_{2})+\gamma\nonumber \\
 & \;\;\;\;\mathrm{or}\;\;\log\mathsf{L}_{2}<\iota_{U_{2};X_{2}|U_{1}}(U_{2};X_{2}|U_{1})+\gamma\;\;\mathrm{or}\;\;\log\mathsf{L}_{1}\mathsf{L}_{2}<\iota_{U_{1},U_{2};X_{1},X_{2}}(U_{1},U_{2};X_{1},X_{2})+\gamma\biggr\}\nonumber \\
 & \;\;\;+2^{-\gamma}\left(4\mathbf{E}[(\iota_{U_{1};U_{2}}(U_{1};U_{2}))^{2}]+4\gamma^{2}+29\right).\label{eq:dlsc_pe2}\end{aligned}$$

The logarithmic term in (or the last term in ) results in an $O(n^{-1}\log n)$ penalty on the rate in the finite blocklength regime, and does not affect the second order result. Ignoring the last term in , the error event in is a strict subset of that in [@yassaee2013oneshot eqn (47)]. This is because the error event 