KLDIV KullbackLeibler or JensenShannon divergence between two distributions.
KLDIV(X,P1,P2) returns the KullbackLeibler divergence between two distributions specified over the M variable values in vector X. P1 is a lengthM vector of probabilities representing distribution 1, and P2 is a lengthM 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 KullbackLeibler 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 zerovalued 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 KullbackLeibler divergence, given by [KL(P1,P2)+KL(P2,P1)]/2. See Johnson and Sinanovic (2001).
KLDIV(X,P1,P2,'js') returns the JensenShannon 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 socalled "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);
KL =
19.4899
Note also that we avoided the logofzero warning by adding 'eps' to all probability values in P2. We didn't need to renormalize because we're still within the sumtoone tolerance.
REFERENCES:
1) Cover, T.M. and J.A. Thomas. "Elements of Information Theory," Wiley, 1991.
2) Johnson, D.H. and S. Sinanovic. "Symmetrizing the KullbackLeibler 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): 99121.
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
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Liang Jia (view profile)
Subhas Das (view profile)
how can i compare two histograms using this function?
KAO YUAN LIN (view profile)
Wajahat (view profile)
Thanks a lot. Its a good implementation
Tomasz Galka (view profile)
Omid G (view profile)
Dushyant Kumar (view profile)
Thanks for writing KLD for two discrete random variables and thanks for sharing.
Yuval Aviel (view profile)
Bao: See "A 'log of zero' warning will be thrown for zerovalued probabilities. " in the text above.
Do Quoc Bao (view profile)
Anyone has taken care about the log of 0? Because in thi formula, we have Log(pi)  Log(qi) !!!!
This implementation may be fine for two discrete multinomial distribution. A very good notes about this:
http://www.snl.salk.edu/~shlens/pub/notes/kl.pdf
I have not find any general solution this problem.
This is NOT an implementation of the KullbackLeibler divergence (KLD) for probability densities P1 and P2. It may be an implementation for KLD for 2 discrete random variables. In this case the vector X is unnecessary,X in fact only used by the code for uniqueness of the Pvalues