# KLDIV

### David Fass (view profile)

23 Nov 2006 (Updated )

Kullback-Leibler or Jensen-Shannon divergence between two distributions.

kldiv.m
```function KL = kldiv(varValue,pVect1,pVect2,varargin)
%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 "KullbackLeibler 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);
%                 KL =
%                      19.4899
%
%             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.
%
%   REFERENCES:
%   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="matlab:web('http://en.wikipedia.org/wiki/Kullback%E2%80%93Leibler_divergence','-browser')">KullbackLeibler divergence</a>. Wikipedia, The Free Encyclopedia.
%

if ~isequal(unique(varValue),sort(varValue)),
warning('KLDIV:duplicates','X contains duplicate values. Treated as distinct values.')
end
if ~isequal(size(varValue),size(pVect1)) || ~isequal(size(varValue),size(pVect2)),
error('All inputs must have same dimension.')
end
% Check probabilities sum to 1:
if (abs(sum(pVect1) - 1) > .00001) || (abs(sum(pVect2) - 1) > .00001),
error('Probablities don''t sum to 1.')
end

if ~isempty(varargin),
switch varargin{1},
case 'js',
logQvect = log2((pVect2+pVect1)/2);
KL = .5 * (sum(pVect1.*(log2(pVect1)-logQvect)) + ...
sum(pVect2.*(log2(pVect2)-logQvect)));

case 'sym',
KL1 = sum(pVect1 .* (log2(pVect1)-log2(pVect2)));
KL2 = sum(pVect2 .* (log2(pVect2)-log2(pVect1)));
KL = (KL1+KL2)/2;

otherwise
error(['Last argument' ' "' varargin{1} '" ' 'not recognized.'])
end
else
KL = sum(pVect1 .* (log2(pVect1)-log2(pVect2)));
end

```