we have found an estimator $\symPoissonRateEstimate$ for the Poisson rate that is unbiased $\symExpectation(\symPoissonRateEstimate\vert\symPoissonRate) = \symPoissonRate$ for all $\symPoissonRate\geq 0$. This implies $\symExpectation(\symPoissonRateEstimate\vert\symCardinality) = \symCardinality$ as it is the only solution of $$\symExpectation(\symPoissonRateEstimate\vert\symPoissonRate) 
 = 
\sum_{\symCardinality=0}^\infty \symExpectation(\symPoissonRateEstimate\vert\symCardinality) e^{-\symPoissonRate}\frac{\symPoissonRate^\symCardinality}{\symCardinality!}
= \symPoissonRate
\quad
\text{for all}\ \symPoissonRate\geq 0.$$ Hence, the unbiased estimator $\symPoissonRateEstimate$ conditioned on $\symCardinality$ is also an unbiased estimator for $\symCardinality$, which motivates us to use $\symPoissonRateEstimate$ directly as estimator for the cardinality $\symCardinalityEstimate := \symPoissonRateEstimate$. As our results will show later, this Poisson approximation works well over the full cardinality range, even for estimators that are not exactly unbiased.

Original Estimation Approach {#sec:cardinality_estimation}
============================

The original cardinality estimator [@Flajolet2007] is based on the idea that the number of distinct element insertions a register needs to reach the value $\symRegVal$ is proportional to $\symNumReg  2^{\symRegVal}$. Given that, a rough cardinality estimate can be obtained by averaging the values $\lbrace \symNumReg 2^{\symRegValVariate_1},\ldots,\symNumReg2^{\symRegValVariate_\symNumReg}\rbrace$. The harmonic mean was found to work best as it is less sensitive to outliers. The result is the so-called raw estimator given by $$\label{equ:raw_estimator}
\symCardinalityRawEstimate
=
\alpha_\symNumReg
\frac{\symNumReg}
{\frac{1}{\symNumReg 2^{\symRegValVariate_1}}+\ldots+\frac{1}{\symNumReg2^{\symRegValVariate_\symNumReg}}}
= 
\frac{\symAlpha_\symNumReg \symNumReg^2}{\sum_{\symRegVal=0}^{\symRegRange+1}\symCountVariate_\symRegVal 2^{-\symRegVal}}.$$ Here $\alpha_\symNumReg$ is a bias correction factor [@Flajolet2007] which can be well approximated by $\symAlpha_\infty := 