spm_KL_dir.m

function [d] = spm_KL_dir(q,p)
% KL divergence between two Dirichlet distributions
% FORMAT [d] = spm_kl_dir(lambda_q,lambda_p)
%
% Calculate KL(Q||P) = <log Q/P> where avg is wrt Q between two Dirichlet 
% distributions Q and P
%
% lambda_q   -   concentration parameter matrix of Q
% lambda_p   -   concentration parameter matrix of P
%
% This routine uses an efficient computation that handles arrays, matrices 
% or vectors. It returns the sum of divergences over columns.
%
% See also: spm_kl_dirichlet.m (for row vectors)
%__________________________________________________________________________

% Will Penny 
% Copyright (C) 2018-2022 Wellcome Centre for Human Neuroimaging


%  KL divergence based on log beta functions
%--------------------------------------------------------------------------
d = spm_betaln(p) - spm_betaln(q) + sum((q - p).*spm_psi(q),1);
d = sum(d(:));

return

% NOTES: numerical check on KL of Dirichlet distributions
%==========================================================================
p  = rand(6,1) + 1;
q  = rand(6,1) + p;
p0 = sum(p);
q0 = sum(q);

d  = q - p;
KL = spm_betaln(p) - spm_betaln(q) + d'*spm_psi(q)
kl = gammaln(q0)   - sum(gammaln(q)) - gammaln(p0) + sum(gammaln(p)) + ...
    d'*(spm_psi(q) - spm_psi(q0))