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.^[1]

Venn diagram illustrating the relation between information entropies, mutual information and variation of information.

Definition

Suppose we have two partitions $X$ and $Y$ of a set $A$ into disjoint subsets, namely $X = \{X_{1}, X_{2}, ..,, X_{k}\}$, $Y = \{Y_{1}, Y_{2}, ..,, Y_{l}\}$. Let $n = \Sigma_{i} |X_{i}| = \Sigma_{j} |Y_{j}|=|A|$, $p_{i} = |X_{i}| / n$ , $q_{j} = |Y_{j}| / n$, $r_{ij} = |X_i\cap Y_{j}| / n$. Then the variation of information between the two partitions is:

$VI(X; Y ) = - \sum_{i,j} r_{ij} \left[\log(r_{ij}/p_i)+\log(r_{ij}/q_j) \right]$.

This is equivalent to the shared information distance between the random variables i and j with respect to the uniform probability measure on $A$defined by $\mu(B):=|B|/n$ for $B\subseteq A$.

Identities

The variation of information satisfies

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

where $H(X)$ is the entropy of $X$, and $I(X, Y)$ is mutual information between $X$ and $Y$ with respect to the uniform probability measure on $A$. This can be rewritten as

$VI(X; Y ) = H(X,Y) - I(X, Y)$,

where $H(X,Y)$ is the joint entropy of $X$ and $Y$, or

$VI(X; Y ) = H(X|Y) + H(Y|X)$,

where $H(X|Y)$ and $H(Y|X)$ are the respective conditional entropies.

References

1. Alexander Kraskov, Harald Stögbauer, Ralph G. Andrzejak, and Peter Grassberger, "Hierarchical Clustering Based on Mutual Information", (2003) ArXiv q-bio/0311039

Further reading

• Arabie, P.; Boorman, S. A. (1973). "Multidimensional scaling of measures of distance between partitions". Journal of Mathematical Psychology 10: 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: 173–187. doi:10.1007/978-3-540-45167-9_14.
• 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.

External links

• C++ implementation with MATLAB mex files