function is $$\begin{gathered}
\symProbabilityMass(\symRegVal_1,\symRegVal_2)
=
\symProbability(
\symRegValVariate_1 \leq \symRegVal_1
\wedge
\symRegValVariate_2 \leq \symRegVal_2
)
-
\symProbability(
\symRegValVariate_1 \leq \symRegVal_1-1
\wedge
\symRegValVariate_2 \leq \symRegVal_2
)
\\
-\symProbability(
\symRegValVariate_1 \leq \symRegVal_1
\wedge
\symRegValVariate_2 \leq \symRegVal_2-1
)
+\symProbability(
\symRegValVariate_1 \leq \symRegVal_1-1
\wedge
\symRegValVariate_2 \leq \symRegVal_2-1
).\end{gathered}$$ Since the values for different registers are independent under the Poisson model, the joint probability mass function for all registers in both sketches is $\symProbabilityMass(\boldsymbol{\symRegVal}_1,\boldsymbol{\symRegVal}_2)
=
\prod_{\symIndexI = 1}^{\symNumReg}
\symProbabilityMass(\symRegVal_{1\symIndexI},\symRegVal_{2\symIndexI})$. The estimates $\symPoissonRateEstimate_\symSetASuffix$, $\symPoissonRateEstimate_\symSetBSuffix$, and $\symPoissonRateEstimate_\symSetXSuffix$ can be finally obtained by maximizing the log-likelihood function given by $$\begin{gathered}
\label{equ:log_likelihood_pair}
\log \symLikelihood(
\symPoissonRate_\symSetASuffix,
\symPoissonRate_\symSetBSuffix,
\symPoissonRate_\symSetXSuffix
\vert
\boldsymbol{\symRegValVariate}_1,
\boldsymbol{\symRegValVariate}_2
)
=
\sum_{\symIndexI = 1}^{\symNumReg}
\log(\symProbabilityMass(\symRegValVariate_{1\symIndexI},\symRegValVariate_{2\symIndexI}))
=
\\
\sum_{\symRegVal=1}^{\symRegRange}
\log\left(1-e^{-\frac{\symPoissonRate_\symSetASuffix+\symPoissonRate_\symSetXSuffix}{\symNumReg 2^{\symRegVal}}}\right)
\symCountVariate^{<}_{1\symRegVal}
+
\log\left(1-e^{-\frac{\symPoissonRate_\symSetBSuffix+\symPoissonRate_\symSetXSuffix}{\symNumReg 2^{\symRegVal}}}\right)
\symCountVariate^{<}_{2\symRegVal}
\\
+
\sum_{\symRegVal=1}^{\symRegRange+1}
\log\left(1-e^{-\frac{\symPoissonRate_\symSetASuffix}{\symNumReg 2^{\min\left(\symRegVal,\symRegRange\right)}}}\right)
\symCountVariate^{>}_{1\symRegVal}
+
\log\left(1-e^{-\frac{\symPoissonRate_\symSetBSuffix}{\symNumReg 2^{\min\left(\symRegVal,\symRegRange\right)}}}\right)
\symCountVariate^{>}_{2\symRegVal}
\\
+
\sum_{\symRegVal=1}^{\symRegRange+1}
\log\left(
1
-e^{-\frac{\symPoissonRate_\symSetASuffix+\symPoissonRate_\symSetXSuffix}{\symNumReg 2^{\min\left(\symRegVal,\symRegRange\right)}}}
-
e^{-\frac{\symPoissonRate_\symSetBSuffix+\symPoissonRate_\symSetXSuffix}{\symNumReg 2^{\min\left(\symRegVal,\symRegRange\right)}}}
+
e^{-\frac{\symPoissonRate_\symSetASuffix+\symPoissonRate_\symSetBSuffix+\symPoissonRate_\symSetXSuffix}{\symNumReg 2^{\min\left(\symRegVal,\symRegRange\right)}}}
\right)
\symCountVariate^{=}_{\symRegVal}
\\
-
\frac{\symPoissonRate_\symSetASuffix}{\symNumReg}
\sum_{\symRegVal=0}^{\symRegRange}
\frac{
  \symCountVariate^{<}_{1\symRegVal}+
  \symCountVariate^{=}_{\symRegVal}+
  \symCountVariate^{>}_{1\symRegVal}
}{2^{\symRegVal}}
-
\frac{\symPoissonRate_\symSetBSuffix}{\symNumReg}
\sum_{\symRegVal=0}^{\symRegRange}
\frac{
  \symCountVariate^{<}_{2\symRegVal}+
  \symCountVariate^{=}_{\symRegVal}+
  \symCountVariate^{>}_{2\symRegVal}
}{2^{\symRegVal}}
\\
-
\frac{\symPoissonRate_\symSetXSuffix}{\symNumReg}
\sum_{\symRegVal=0}^{\symRegRange}
\frac{
  \symCountVariate^{<}_{1\symRegVal}+
  \symCountVariate^{=}_{\symRegVal}+
  \symCountVariate^{<}_{2\symRegVal}
}{2^{\symRegVal}}\end{gathered}$$ where $\symCountVariate^{<}_{1\symRegVal}$, $\symCountVariate^{>}_{1\symRegVal}$, $\symCountVariate^{<}_{2\symRegVal}$, $\symCountVariate^{>}_{2\symRegVal}$, and $\symCountVariate^{=}_{\symRegVal}$ are defined as $$\label{equ:sufficient_joint_statistic}
\begin{aligned}
\symCountVariate^{<}_{1\symRegVal}&:=\left|\lbrace\symIndexI\vert\symRegVal=\symRegValVariate_{1\symIndexI}<
\symRegValVariate_{2\symIndexI}\rbrace\right|,&
\symCountVariate^{>}_{1\symRegVal}&:=\left|\lbrace\symIndexI\vert\symRegVal=\symRegValVariate_{1\symIndexI}>
\symRegValVariate_{2\symIndexI}\rbrace\right|,
\\
\symCountVariate^{<}_{2\symRegVal}&:=\left|\lbrace\symIndexI\vert\symRegVal=\symRegValVariate_{2\symIndexI}<
\symRegValVariate_{1\symIndexI}\rbrace\right|,&
\symCountVariate^{>}_{2\symRegVal}&:=\left|\lbrace\symIndexI\vert\symRegVal=\symRegValVariate_{2\symIndexI}>
\symRegValVariate_{1\symIndexI}\rbrace\right|,
\\
\symCountVariate^{=}_{\symRegVal}&:=\left|\lbrace\symIndexI\vert\symRegVal=\symRegValVariate_{1\symIndexI}=
\symRegValVariate_{2\symIndexI}\rbrace\right|
\end{aligned}$$ for $0\leq \symRegVal \leq \symRegRange+1$. These $5(\symRegRange+2)$ values 