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.  
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 .
Explicit information content
We can rewrite this definition in terms that explicitly highlight the information content of this metric.
The set of all partitions of a set form a compact Lattice where the partial order induces two operations, the meet and the join , where the maximum is the partition with only one block, i.e., all elements grouped together, and the minimum is , the partition consisting of all elements as singletons. The meet of two partitions and is easy to understand as that partition formed by all pair intersections of one block of, , of and one, , of . It then follows that and .
Let's define the entropy of a partition as
where . Clearly, and . The entropy of a partition is a monotonous function on the lattice of partitions in the sense that .
Then the VI distance between and is given by
The difference is a pseudo-metric as doesn't necessarily imply that . From the definition of , it is .
If in the Hasse diagram we draw an edge from every partition to the maximum and assign it a weight equal to the VI distance between the given partition and , we can interpret the VI distance as basically an average of differences of edge weights to the maximum
For as defined above, it holds that the joint information of two partitions coincides with the entropy of the meet
and we also have that coincides with the conditional entropy of the meet (intersection) relative to .
The variation of information satisfies
where is the joint entropy of and , or
where and are the respective conditional entropies.
The variation of information can also be bounded, either in terms of the number of elements:
Or with respect to a maximum number of clusters, :
- P. Arabie, S.A. Boorman, S. A., "Multidimensional scaling of measures of distance between partitions", Journal of Mathematical Psychology (1973) , vol. 10, 2, pp. 148–203, doi: 10.1016/0022-2496(73)90012-6
- W.H. Zurek, Nature, vol 341, p119 (1989); W.H. Zurek, Physics Review A, vol 40, p4731 (1989)
- Marina Meila, "Comparing Clusterings by the Variation of Information", Learning Theory and Kernel Machines (2003), vol. 2777 , pp. 173–187, doi:10.1007/978-3-540-45167-9_14, Lecture Notes in Computer Science, ISBN 978-3-540-40720-1
- Arabie, P.; Boorman, S. A. (1973). "Multidimensional scaling of measures of distance between partitions". Journal of Mathematical Psychology. 10 (2): 148–203. doi:10.1016/0022-2496(73)90012-6.
- Meila, Marina (2003). "Comparing Clusterings by the Variation of Information". Learning Theory and Kernel Machines. Lecture Notes in Computer Science. 2777: 173–187. doi:10.1007/978-3-540-45167-9_14. ISBN 978-3-540-40720-1.
- Meila, M. (2007). "Comparing clusterings—an information based distance". Journal of Multivariate Analysis. 98 (5): 873–895. doi:10.1016/j.jmva.2006.11.013.
- Kingsford, Carl (2009). "Information Theory Notes" (PDF). Retrieved 22 September 2009.
- Kraskov, Alexander; Harald Stögbauer; Ralph G. Andrzejak; Peter Grassberger (2003). "Hierarchical Clustering Based on Mutual Information". arXiv:q-bio/0311039.