represent a sufficient statistic for estimating $\symPoissonRate_\symSetASuffix$, $\symPoissonRate_\symSetBSuffix$, and $\symPoissonRate_\symSetXSuffix$ and greatly reduce the number of terms and also the evaluation costs of the log-likelihood function. The derived formula is the generalization of to two sketches and therefore has a similar structure.

Numerical Optimization
----------------------

The estimates $\symPoissonRateEstimate_\symSetASuffix$, $\symPoissonRateEstimate_\symSetBSuffix$, and $\symPoissonRateEstimate_\symSetXSuffix$ are obtained by maximizing . Since the three parameters are all nonnegative, this is a constrained optimization problem. The transformation $\symPoissonRate=e^{\symPhi}$ helps to get rid of these constraints and also translates relative accuracy limits into absolute ones, because $\Delta\symPhi = \Delta\symPoissonRate/\symPoissonRate$. Many optimizer implementations allow the definition of absolute limits rather than relative ones. Quasi-Newton methods are commonly used to find the maximum of such multi-dimensional functions. They all require the computation of the gradient which can be straightforwardly derived for . Among these methods the algorithm [@Press2007] is very popular. We used the implementation provided by the Dlib C++ library [@King2009] for our experiments. Good initial guess values are important to ensure fast convergence for any optimization algorithm. An obvious choice are the cardinality estimates obtained by application of the inclusion-exclusion principle . However, in order to ensure that their logarithms are all defined, we require that the 