KLDIV Kullback-Leibler or Jensen-Shannon divergence between two distributions.
KLDIV(X,P1,P2) returns the Kullback-Leibler divergence between two distributions specified over the M variable values in vector X. P1 is a length-M vector of probabilities representing distribution 1, and P2 is a length-M vector of probabilities representing distribution 2. Thus, the probability of value X(i) is P1(i) for distribution 1 and P2(i) for distribution 2. The Kullback-Leibler divergence is given by:
KL(P1(x),P2(x)) = sum[P1(x).log(P1(x)/P2(x))]
If X contains duplicate values, there will be an warning message, and these values will be treated as distinct values. (I.e., the actual values do not enter into the computation, but the probabilities for the two duplicate values will be considered as probabilities corresponding to two unique values.) The elements of probability vectors P1 and P2 must each sum to 1 +/- .00001.
A "log of zero" warning will be thrown for zero-valued probabilities. Handle this however you wish. Adding 'eps' or some other small value to all probabilities seems reasonable. (Renormalize if necessary.)
KLDIV(X,P1,P2,'sym') returns a symmetric variant of the Kullback-Leibler divergence, given by [KL(P1,P2)+KL(P2,P1)]/2. See Johnson and Sinanovic (2001).
KLDIV(X,P1,P2,'js') returns the Jensen-Shannon divergence, given by [KL(P1,Q)+KL(P2,Q)]/2, where Q = (P1+P2)/2. See the Wikipedia article for "Kullback–Leibler divergence". This is equal to 1/2 the so-called "Jeffrey divergence." See Rubner et al. (2000).
EXAMPLE: Let the event set and probability sets be as follow:
X = [1 2 3 3 4]';
P1 = ones(5,1)/5;
P2 = [0 0 .5 .2 .3]' + eps;
Note that the event set here has duplicate values (two 3's). These will be treated as DISTINCT events by KLDIV. If you want these to be treated as the SAME event, you will need to collapse their probabilities together before running KLDIV. One way to do this is to use UNIQUE to find the set of unique events, and then iterate over that set, summing probabilities for each instance of each unique event. Here, we just leave the duplicate values to be treated independently (the default):
KL = kldiv(X,P1,P2);
Note also that we avoided the log-of-zero warning by adding 'eps' to all probability values in P2. We didn't need to renormalize because we're still within the sum-to-one tolerance.
1) Cover, T.M. and J.A. Thomas. "Elements of Information Theory," Wiley, 1991.
2) Johnson, D.H. and S. Sinanovic. "Symmetrizing the Kullback-Leibler distance." IEEE Transactions on Information Theory (Submitted).
3) Rubner, Y., Tomasi, C., and Guibas, L. J., 2000. "The Earth Mover's distance as a metric for image retrieval." International Journal of Computer Vision, 40(2): 99-121.
4) <a href="http://en.wikipedia.org/wiki/Kullback%E2%80%93Leibler_divergence">Kullback–Leibler divergence</a>. Wikipedia, The Free Encyclopedia.
See also: MUTUALINFO, ENTROPY