function [LL, prior, transmat, obsmat, gamma] = learn_hmm(data, prior, transmat, obsmat, max_iter, thresh, ...						  verbose, act, adj_prior, adj_trans, adj_obs, dirichlet)% LEARN_HMM Find the ML parameters of an HMM with discrete outputs using EM.%% [LL, PRIOR, TRANSMAT, OBSMAT] = LEARN_HMM(DATA, PRIOR0, TRANSMAT0, OBSMAT0) computes ML% estimates of the following parameters, where, for each time t, Q(t) is the hidden state, and% Y(t) is the observation%   prior(i) = Pr(Q(1) = i)%   transmat(i,j) = Pr(Q(t+1)=j | Q(t)=i)%   obsmat(i,o) = Pr(Y(t)=o | Q(t)=i)% PRIOR0 is the initial estimate of PRIOR, etc.%% Row l of DATA is the observation sequence for example l. If the sequences are of% different lengths, you can pass in a cell array, so DATA{l} is a vector.% If there is only one sequence, the estimate of prior will be poor.% If all the sequences are of length 1, transmat cannot be estimated.%% LL is the "learning curve": a vector of the log likelihood values at each iteration.%% There are several optional arguments, which should be passed in the following order%   LEARN_HMM(DATA, PRIOR, TRANSMAT, OBSMAT, MAX_ITER, THRESH, VERBOSE)% These have the following meanings%   max_iter = max. num EM steps to take (default 10)%   thresh = threshold for stopping EM (default 1e-4)%   verbose = 0 to suppress the display of the log lik at each iteration (Default 1).%% If the transition matrix is non-stationary (e.g., as in a POMDP),% then TRANSMAT should be a cell array, where T{a}(i,j) = Pr(Q(t+1)=j|Q(t)=i,A(t)=a).% The last arg should specify the sequence of actions in the same form as DATA:%   LEARN_HMM(DATA, PRIOR, TRANSMAT, OBSMAT, MAX_ITER, THRESH, VERBOSE, As)% The action at time 1 is ignored.%% If you want to clamp some of the parameters at fixed values, set the corresponding adjustable% argument to 0 (default: everything is adjustable)%   LEARN_HMM(..., VERBOSE, As, ADJ_PRIOR, ADJ_TRANS, ADJ_OBS)%% To avoid 0s when estimating OBSMAT, specify a non-zero equivalent sample size (e.g., 0.01) for% the Dirichlet prior: LEARN_HMM(..., ADJ_OBS, DIRICHLET)%% When there is a single sequence, the smoothed posteriors using the penultimate set of% parameters are returned in GAMMA:%   [LL, PRIOR, TRANSMAT, OBSMAT, GAMMA] = LEARN_HMM(...)% This can be useful for online learning and decision making.if ~exist('max_iter'), max_iter = 10; endif ~exist('thresh'), thresh = 1e-4; endif ~exist('verbose'), verbose = 1; endif ~exist('adj_prior'), adj_prior = 1; endif ~exist('adj_trans'), adj_trans = 1; endif ~exist('adj_obs'), adj_obs = 1; endif ~exist('dirichlet'), dirichlet = 0; endif ~exist('act'),  act = [];  A = 0;else  A = length(transmat);endprevious_loglik = -inf;loglik = 0;converged = 0;num_iter = 1;LL = [];if ~iscell(data)  data = num2cell(data, 2); % each row gets its own cellendif ~isempty(act) & ~iscell(act)  act = num2cell(act, 2);endnumex = length(data);while (num_iter <= max_iter) & ~converged  % E step  [loglik, exp_num_trans, exp_num_visits1, exp_num_emit, gamma] = ...      compute_ess(prior, transmat, obsmat, data, act, dirichlet);  if verbose, fprintf(1, 'iteration %d, loglik = %f\n', num_iter, loglik); end  num_iter =  num_iter + 1;  % M step  if adj_prior    prior = normalise(exp_num_visits1);  end  if adj_trans & ~isempty(exp_num_trans)    if isempty(act)      transmat = mk_stochastic(exp_num_trans);    else      for a=1:A	transmat{a} = mk_stochastic(exp_num_trans{a});      end    end  end  if adj_obs    obsmat = mk_stochastic(exp_num_emit);  end    converged = em_converged(loglik, previous_loglik, thresh);  previous_loglik = loglik;  LL = [LL loglik];end%%%%%%%%%%%function [loglik, exp_num_trans, exp_num_visits1, exp_num_emit, gamma] = ...    compute_ess(prior, transmat, obsmat, data, act, dirichlet)%% Compute the Expected Sufficient Statistics for a discrete Hidden Markov Model.%% Outputs:% exp_num_trans(i,j) = sum_l sum_{t=2}^T Pr(X(t-1) = i, X(t) = j| Obs(l))% exp_num_visits1(i) = sum_l Pr(X(1)=i | Obs(l))% exp_num_emit(i,o) = sum_l sum_{t=1}^T Pr(X(t) = i, O(t)=o| Obs(l))% where Obs(l) = O_1 .. O_T for sequence l.numex = length(data);[S O] = size(obsmat);if isempty(act)  exp_num_trans = zeros(S,S);  A = 0;else  A = length(transmat);  exp_num_trans = cell(1,A);  for a=1:A    exp_num_trans{a} = zeros(S,S);  endendexp_num_visits1 = zeros(S,1);exp_num_emit = dirichlet*ones(S,O);loglik = 0;estimated_trans = 0;for ex=1:numex  obs = data{ex};  T = length(obs);  olikseq = mk_dhmm_obs_lik(obs, obsmat);  if isempty(act)    [alpha, beta, gamma, xi, current_ll] = forwards_backwards(prior, transmat, olikseq);  else    [alpha, beta, gamma, xi, current_ll] = forwards_backwards(prior, transmat, olikseq, [], [], act{ex});  end  loglik = loglik +  current_ll;   if T > 1    estimated_trans = 1;    if isempty(act)      exp_num_trans = exp_num_trans + sum(xi,3);    else      % act(2) determines Q(2), xi(:,:,1) holds P(Q(1), Q(2))      A = length(transmat);      for a=1:A	ndx = find(act{ex}(2:end)==a);	if ~isempty(ndx)	  exp_num_trans{a} = exp_num_trans{a} + sum(xi(:,:,ndx), 3);	end      end    end  end    exp_num_visits1 = exp_num_visits1 + gamma(:,1);    if T < O    for t=1:T      o = obs(t);      exp_num_emit(:,o) = exp_num_emit(:,o) + gamma(:,t);    end  else    for o=1:O      ndx = find(obs==o);      if ~isempty(ndx)	exp_num_emit(:,o) = exp_num_emit(:,o) + sum(gamma(:, ndx), 2);      end    end  endendif ~estimated_trans  exp_num_trans = [];end