The conditional mutual inclusive information-based network inference (CMI2NI) method improves the PCACMI method by considering the Kullback-Leibler divergences from the joint probability density function (PDF) of target variables to the interventional PDFs removing the dependency between two variables of interest. Instead of using CMI, CMI2NI uses the conditional mutual inclusive information (CMI2) as the measure of dependency between two variables of interest given other variables.

To be specific, consider three random variables $X$, $Y$ and $Z$. For these three random variables, the CMI2 between $X$ and $Y$ given $Z$ is defined as $$CMI2(X,Y|Z) = (D_{KL}(P||P_{X->Y}) + D_{KL}(P||P_{Y->X}))/2,$$ where $D_{KL}(f||g)$ is the Kullback-Leibler divergence from $f$ to $g$, $P$ is the joint PDF of $X$, $Y$ and $Z$, and $P_{X \to Y}$ is the interventional probability of $X$, $Y$ and $Z$ for removing the connection from $X$ to $Y$. With Gaussian assumption on the observed data, the CMI2 for two random variables $X$ and $Y$ given m-dimensional vector $Z$ can be expressed as $$CMI2(X,Y|Z) = \frac{1}{4}(tr(C^{-1}\Sigma) + tr(\tilde{C}^{-1}\tilde{\Sigma}) + logC_{0} +log\tilde{C}_{0}-2n),$$ where $\Sigma$ is the covariance matrix of $( X, Y, Z^T )^T$, $\tilde{\Sigma}$ is the covariance matrix of $( X, Y, Z^T )^T$, Σ XZ is the covariance matrix of $\Sigma_{X,Z}$, $( X, Z^T )^T$ is the covariance matrix of $( Y, Z^T )^T, n=m+2$, and $C$, $\tilde{C}$, $C_0$ and $\tilde{C_0}$ are defined with the elements of $\Sigma, \Sigma_{XZ}, \Sigma_{YZ}, \Sigma^{-1}, \Sigma^{-1}_{XZ}, \Sigma^{-1}_{YZ}$. As applied in PCACMI, CMI2NI adopts the path consistency algorithm (PCA) to efficiently calculate the CMI2 estimates. All steps of the PCA in CMI2NI are the same as one of PCACMI if we change the CMI to the CMI2. In the PCA steps of CMI2NI, two variables are regarded as independent if the corresponding CMI2 estimate is less than a given threshold $\alpha$.

Reference:
1. Donghyeon Yu, Johan Lim, Xinlei Wang, Faming Liang, and Guanghua Xiao. "Enhanced construction of gene regulatory networks using hub gene information." BMC bioinformatics 18.1 (2017): 186.
2. Zhang, Xiujun, Juan Zhao, Jin-Kao Hao, Xing-Ming Zhao, and Luonan Chen. "Conditional mutual inclusive information enables accurate quantification of associations in gene regulatory networks." Nucleic acids research 43.5 (2014): e31-e31.

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Note:
Change the $\alpha$ value $(\alpha > 0)$ to control the sparsity of network. The larger the $\alpha$, the more sparse the constructed network. If you don't know how to choose a value, use the default one.