$Y\supset X$ holds $P(Y)<P(X)$, and it is called a minimal generator if for all $Y\subset X$ holds $P(Y)>P(X)$. Now if $X$ and $Q$ are closed sets with the same minimal generator $Z=X\cap Q\neq \emptyset$, then $cov(X)=cov(Q)=cov(Z)$ and both $X\rightarrow C{=}c$ and $Q\rightarrow C{=}c$ are equivalent with $Z\rightarrow C{=}c$ and thus specious. However, specious rules of this type are also superfluous specializations explained in the previous subsection and are pruned as such. We recall that $Z=X\cap Q$ is not necessarily a generator at all (i.e., $P(X|Z)<1$ and $P(X|Q)<1$), even if $X$ and $Q$ were equivalent. In summary, all rules $Q\rightarrow C{=}c$ and $X\rightarrow C{=}c$, whose antecedents have the same minimal generator $Z$, are mutually equivalent and specious, but all mutually equivalent rules do not have the same minimal generator.

Yule-Simpson’s paradox
----------------------

Speciousness is closely related to Yule-Simpson’s paradox (see e.g., [@Alin2010; @simpson1951]). The paradox refers to the situation where the unconditional dependency between $Q$ and $C{=}c$ is positive, but becomes negative when conditioned on either $X$ or $\neg X$. In a wider definition, disappearing dependences are also included here, i.e., there is marginal independence but conditional dependence or vice versa, like in the original definition by Yule [@yule1903]. In this 