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))