For $\symPrecision=22$, $\symPrecision=10$ we see that the the estimator does not have the small oscillating bias as the improved estimator. Since all observed improvements are not very relevant in practice, the improved estimator should be preferred over the method, because it leads to a simpler and faster algorithm.

Joint Estimation {#sec:cardinality_estimation_set_intersections}
================

While the union of two sets that are represented by sketches can be straightforwardly obtained by taking the register-wise maximums, the computation of cardinalities for other set operations like intersections and relative complements is more challenging. The conventional approach uses the inclusion-exclusion principle $$\label{equ:conventional_approach}
\begin{aligned}
&\left\vert\symSetS_1 \setminus \symSetS_2\right\vert
=
\left\vert\symSetS_1 \cup \symSetS_2\right\vert
-
\left\vert\symSetS_2\right\vert,
\quad
\left\vert\symSetS_2 \setminus \symSetS_1\right\vert
=
\left\vert\symSetS_1 \cup \symSetS_2\right\vert
-
\left\vert\symSetS_1\right\vert,
\\
&\left\vert\symSetS_1 \cap \symSetS_2\right\vert
=  
\left\vert\symSetS_1\right\vert  
+
\left\vert\symSetS_2\right\vert
-
\left\vert\symSetS_1 \cup \symSetS_2\right\vert.
\end{aligned}$$ It allows to express set operation cardinalities in terms of the union cardinality. However, this approach can lead to very large estimation errors, especially if the result is small compared to the operand cardinalities [@Dasgupta2015]. In the worst case, the estimate could be negative without artificial restriction to nonnegative values.

Motivated by the good results we have obtained for a single sketch using the method in combination with the Poisson approximation, 