\atop n_{xc}-i\right)
    \left(n_{\neg xq} \atop j
    \right)\left(n_{\neg x\neg q)} \atop n_{\neg xc}-j\right)}{\left(n_x \atop n_{xc} \right)\left(n_{\neg x} \atop
    n_{\neg xc}\right)}.\end{gathered}$$

One problem is to select a threshold value $\alpha\in ]0,1]$ such that rules with $p_B>\alpha$ can be safely pruned out as specious (i.e., the null hypothesis of conditional independence is kept). A classical threshold in significance testing is $\alpha=0.05$ but a smaller or larger value can be used to prune out more or less rules as specious.

A corresponding information-theoretic test for speciousness can be constructed using conditional mutual information. With a short-hand notation $p_x=P(X)$ it is defined as

$$\begin{gathered}
\label{condMI}
MI(Q\rightarrow C|X\rightarrow C)\\
=MI(Q\rightarrow C|X)+MI(Q\rightarrow C|\neg X)\\
=\log \frac{p_x^{p_x}p_{xqc}^{p_{xqc}}p_{xq\neg c}^{p_{xq\neg c}}p_{x\neg qc}^{p_{x\neg qc}}p_{x\neg q\neg c}^{p_{x\neg q\neg c}}}{p_{xq}^{p_{xq}}p_{x\neg q}^{p_{x\neg q}}p_{xc}^{p_{xc}}p_{x\neg c}^{p_{x\neg c}}}\\
+\log \frac{p_{\neg x}^{p_{\neg x}}p_{\neg xqc}^{p_{\neg xqc}}p_{\neg xq\neg c}^{p_{\neg xq\neg c}}p_{\neg x\neg qc}^{p_{\neg x\neg qc}}p_{\neg x\neg q\neg c}^{p_{\neg x\neg q\neg c}}}{p_{\neg xq}^{p_{\neg xq}}p_{\neg x\neg q}^{p_{\neg x\neg q}}p_{\neg xc}^{p_{\neg xc}}p_{\neg x\neg c}^{p_{\neg x\neg c}}}.
$$

This measure is computationally easy to evaluate and it has also other attractive properties which help in the search. However, there is one drawback since $MI$ is always non-negative and, therefore, negative and positive conditional dependence cannot be separated. As a solution, we have modified the 