spm_kl_wishart.m

function [kl] = spm_kl_wishart (q,Q,p,P)
% KL divergence between two Wishart densities
% FORMAT [kl] = spm_kl_wishart (q,Q,p,P)
%
% Calculate KL (Q||P) = <log Q/P> where avg is wrt Q
% between two Wishart densities Q and P
%
% q,Q      Parameters of first density
% p,P      Parameters of first density
%___________________________________________________________________________
% Copyright (C) 2008 Wellcome Trust Centre for Neuroimaging

% Will Penny 
% $Id: spm_kl_wishart.m 1276 2008-03-28 18:29:19Z guillaume $

logdetQ=log(det(Q));
logdetP=log(det(P));

% Get dimension 
d=size(Q,1);

LqQ=0;
for i=1:d,
  LqQ=LqQ+psi((q+1-i)/2);
end  
LqQ=LqQ+d*log(2)-logdetQ;

LpP=0;
for i=1:d,
  LpP=LpP+psi((p+1-i)/2);
end  
LpP=LpP+d*log(2)-logdetP;

logZqQ=0.5*q*d*log(2)-0.5*q*logdetQ+spm_lg_gamma(d,q/2);
logZpP=0.5*p*d*log(2)-0.5*p*logdetP+spm_lg_gamma(d,p/2);

Qinv=inv(Q);
kl=0.5*(q-d-1)*LqQ-0.5*(p-d-1)*LpP-0.5*q*d+0.5*q*trace(P*Qinv);
kl=kl+logZpP-logZqQ;