we applied the same approach also for the estimation of set operation result sizes. Assume two given sketches with register values $\boldsymbol{\symRegValVariate}_1=(\symRegValVariate_{11},\ldots,\symRegValVariate_{1\symNumReg})$ and $\boldsymbol{\symRegValVariate}_2=(\symRegValVariate_{21},\ldots,\symRegValVariate_{2\symNumReg})$ representing sets $\symSetS_1$ and $\symSetS_2$, respectively. The goal is to find estimates for the cardinalities of the pairwise disjoint sets $\symSetA = \symSetS_1\setminus\symSetS_2$, $\symSetB = \symSetS_2\setminus\symSetS_1$, and $\symSetX = \symSetS_1\cap\symSetS_2$. The Poisson approximation allows us to assume that pairwise distinct elements are inserted into the sketches representing $\symSetS_1$ and $\symSetS_2$ at rates $\symPoissonRate_\symSetASuffix$ and $\symPoissonRate_\symSetBSuffix$, respectively. Furthermore, we assume that further unique elements are inserted into both sketches simultaneously at rate $\symPoissonRate_\symSetXSuffix$. We expect that good estimates $\symPoissonRateEstimate_\symSetASuffix$, $\symPoissonRateEstimate_\symSetBSuffix$, and $\symPoissonRateEstimate_\symSetXSuffix$ for the rates are also good estimates for the cardinalities $\vert\symSetA\vert$, $\vert\symSetB\vert$, and $\vert\symSetX\vert$.

Joint Log-Likelihood Function
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In order to apply the method, we need to find the joint probability distribution of both sketches. Under the Poisson model the individual registers are independent and identically distributed. Therefore, we first derive the joint probability distribution for a single register that has value $\symRegValVariate_1$ in the first sketch that represents $\symSetS_1$ and value $\symRegValVariate_2$ in the second that represents $\symSetS_2$. The first one can be thought to be constructed by merging two sketches representing $\symSetA$ and 