Variation of information

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. Even more, it is a universal metric, in that if any other distance measure two items close-by, then the variation of information will also judge them close.[1]

Background

Definition

Suppose we have two clusterings (a division of a set into several subsets) X and Y where X = \{X_{1}, X_{2}, ..,, X_{k}\}, p_{i} = |X_{i}| / n, n = \Sigma_{k} |X_{i}|. Then the variation of information between two clusterings is:

VI(X; Y ) = H(X) + H(Y) - 2I(X, Y)

where H(X) is entropy of X and I(X, Y) is mutual information between X and Y.

This is completely equivalent to the shared information distance.

References

Further reading

  • expand by hand

External links

  • C++ implementation with MATLAB mex files
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