Adjusted mutual information has been proposed by Vinh et al. that takes into account the number of samples in each category: 
$AMI(U,V) = \frac{I(U,V) - E(I(U,V))}{\frac{1}{2}(H(U)+H(V))-E(I(U,V))}$

University of Michigan - Ann Arbor

Independence tests are always included in the algorithms, either to avoid testing for causal relationships if the variables are independent, or to maximize the accuracy of the predictions as a class in the challenge was dedicated to independent pairs. 

The independence test statistics used consist in mainly two types: the correlation-based and the kernel-based tests. Firstly, the correlation-based tests consist in the well-known statistic tests such as the Pearson’s correlation and the Spearman’s correlation, but also tests based on mutual information. The challenges contained all types of data, including continuous data. In order to compute mutual information out of the empirical distributions, algorithms binned their continuous variables prior to computing mutual information based features. Here we will consider U , V , obtained by binning X and Y . Examples of these features are 
 - mutual information
 - normalized mutual information 
 - adjusted mutual information.

Statistical Features of the Distribution

 Isabelle Guyon, Alexander Statnikov, Berna Bakir Batu (2019)
Cause Effect Pairs in Machine Learning (The Springer Series on Challenges in Machine Learning)
https://www.amazon.com/Effect-Machine-Learning-Springer-Challenges/dp/3030218090

Cause Effect Pairs in Machine Learning

Mutual information is a quantity measuring dependence in information theory, basing itself on how the knowledge of one variable reduces the uncertainty on the other variable. In our case, this quantity can be expressed as: 
$I(U,V) = \sum_{i=1}^{|U|} \sum_{j=1}^{|V|} \frac{|U_i \cap V_j |}{N} \log \frac{|U_i \cap V_j |}{|U_i||V_j|}$ 
where $U$, $V$ represent the input variables and $U_i$ , $V_j$ represent the categories of the variables.

Mutual Information

Normalized mutual information is a variation of the mutual information score, normalized to range from 0, representing no mutual information, to 1, representing perfect correlation. 
$NMI(U,V) = \frac{2I(U,V)}{H(U)+H(V)}$
where H represents the entropy. However, this score does not account for chance.

Learn Before

Related