In probability theory and information theory, the variation of information or shared information distance is a measure of the distance between two clusterings (partitions of elements). It is closely related to mutual information; indeed, it is a simple linear expression involving the mutual information. Unlike the mutual information, however, the variation of information is a true metric, in that it obeys the triangle inequality.
Venn diagram illustrating the relation between information entropies, mutual information and variation of information.
Suppose we have two partitions and of a set into disjoint subsets, namely , . Let , , , . Then the variation of information between the two partitions is:
This is equivalent to the shared information distance between the random variables i and j with respect to the uniform probability measure on defined by for .
The variation of information satisfies
where is the entropy of , and is mutual information between and with respect to the uniform probability measure on . This can be rewritten as
where is the joint entropy of and , or
where and are the respective conditional entropies.
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