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 "Kullback朙eibler 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) Kullback朙eibler divergence. Wikipedia, The Free Encyclopedia.
%
% See also: MUTUALINFO, ENTROPY
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
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