这个函数很重要：

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