# Gini index/criterion/impurity/coefficient according to Breiman, Sen, Bishop, and Duda

once again a student (me) is lost in the sea of Gini... I am currently trying to figure out, where the Gini based formula for feature selection proposed by Cehovin and Bosnic  comes from:

$Gini(A)=\sum_{j}p(j)\sum_{k}p(k|j)^2 - \sum_{k}p(k)^2$,

with $p(j)$ being the probability that the feature $A$ takes value $j$, $p(k|j)$ being the probability that the sample is of class $k$ given feature $A$ has value $j$, and $p(k)$ being the probability that a random sample belongs to class $k$.

Is it right, that ...

• The formula uses the Gini "criterion" according to Breiman et al. ($1-\sum_k p_{ik}^2$)?
• The right side (right of the $-$) is the Gini "criterion"  of the original dataset.
• The left side (left side of $-$) refers the weighted average of the Gini "criterions" (averaged by the probability of feature $A$ taking the respective value $j$, see $\sum_{j}p(j))$. The average is calculated across all "sub" data sets, which each sub data set holding only samples of one distinct value $j$ of feature $A$. Thus, $\sum_{k}p(k|j)^2$ essentially refers the Gini "criterion" for the sub data set.
• What is the meaning of the $\sum_{j \neq k}$ formulation e.g. Breiman et al. 2 are using?
• All of this (Gini "criterion" Breiman , Gini "index" Bishop , Gini "impurity" Duda ) has (mathematically) absolutely nothing to do with the Gini "coefficient" proposed by Sen .

Do I understand the meaning of this formula correctly? Can somebody shed some light on this confusion topic?

Many thanks

 Cehovin and Bosnic, Empirical evaluation of feature selection methods in classification

 Breiman, Classification and Regression Trees (CART)

 Bishop, Pattern Recognition and Machine Learning

 Duda, Pattern Recognition

 Sen, On Economic Inequqlity