function [hmm,Fp,Fpi,entHidden]=hmm_mstep(hmm,Gamma,Xi);% [hmm,Fp,Fpi,entHidden]=hmm_mstep(hmm,Gamma,Xi);%% Learns the HMM-specific parameters in a 1-dimensional HMM % using the variational Bayes framework.%% Called from 'learn_mix1d'.%%% -----------% Input% -----------%% Necessary parameters%% hmm        Current model% Gamma      State probabilities% Xi         Pairwise transition probabilities%%%% -----------% Output% -----------%% hmm        HMM model. P: transition matrix%                       pi: initial state probabilities%                       posts.eta: posterior params for P%                       posts.lambda: posterior params for pi% Fp         Contribution to energy from trans. matrix% Fpi        Contribution to energy from initial state probs.% entHidden  Entropy of hidden states%%% --------------------------------------------------------------%% Original code by Iead Rezek% Paper: Learning Ensemble Hidden Markov Models for Biosignal %        Analysis (www.robots.ox.ac.uk/~parg)%% Modified by Rizwan Choudrey for use in vbICA model% Thesis: Variational Methods for Bayesian Independent%         Component Analysis (www.robots.ox.ac.uk/~parg)      [K,T] = size(Gamma);      %-----------------------Transition matrix------------------------------  sxi=sum(Xi);  sxi=reshape(sxi,K,K);  eta0 = hmm.priors.eta_0;  eta = eta0+sxi;                                  %'iota_{cd}' in thesis  sumeta = repmat(sum(eta,2),1,K);  trueP=eta./sumeta;                               %'R' in thesis      %contribution to free energy  if length(eta0(1,:))~=K      eta0 = repmat(eta0,1,K);  end  bit1 = sum(gammaln(eta) - gammaln(eta0),2);  bit2 = gammaln(sum(eta,2))-gammaln(sum(eta0,2));  Fp = sum(bit1-bit2);  %-----------------------Transition matrix------------------------------        %----------------------Initial state probs-----------------------------  lambda0 = hmm.priors.lambda_0;  lambda = lambda0+Gamma(:,1)';  sumlambda = sum(lambda);  truePi = lambda./sum(lambda);       %contribution to free energy  if length(lambda0)~=K      lambda0 = repmat(lambda0,1,K);  end  bit1 = sum(gammaln(lambda) - gammaln(lambda0));  bit2 = gammaln(sum(lambda))-gammaln(sum(lambda0));  Fpi = sum(bit1-bit2);  %----------------------Initial state probs-----------------------------        %---------------------Contribution to Energy---------------------------  entStart = -sum(Gamma(:,1).*log(Gamma(:,1)+eps));  for i=1:K    Xi3d = reshape(Xi,[T-1,K,K]);    sxi = sum(Xi3d(:,:,i),2);    Psi(:,:,i) = Xi3d(:,:,i)./(repmat(sxi,1,K)+eps);  end  entXi = -sum(sum(sum(Xi3d.*log(Psi+eps))));  entHidden = entStart+entXi;  %---------------------Contribution to Energy---------------------------        hmm.P = trueP;  hmm.pi = truePi;  hmm.posts.eta = eta;  hmm.posts.lambda = lambda;