wolffd@0: function [alpha, beta, gamma, loglik, xi_summed, gamma2] = fwdback(init_state_distrib, ...
wolffd@0:    transmat, obslik, varargin)
wolffd@0: % FWDBACK Compute the posterior probs. in an HMM using the forwards backwards algo.
wolffd@0: %
wolffd@0: % [alpha, beta, gamma, loglik, xi, gamma2] = fwdback(init_state_distrib, transmat, obslik, ...)
wolffd@0: %
wolffd@0: % Notation:
wolffd@0: % Y(t) = observation, Q(t) = hidden state, M(t) = mixture variable (for MOG outputs)
wolffd@0: % A(t) = discrete input (action) (for POMDP models)
wolffd@0: %
wolffd@0: % INPUT:
wolffd@0: % init_state_distrib(i) = Pr(Q(1) = i)
wolffd@0: % transmat(i,j) = Pr(Q(t) = j | Q(t-1)=i)
wolffd@0: %  or transmat{a}(i,j) = Pr(Q(t) = j | Q(t-1)=i, A(t-1)=a) if there are discrete inputs
wolffd@0: % obslik(i,t) = Pr(Y(t)| Q(t)=i)
wolffd@0: %   (Compute obslik using eval_pdf_xxx on your data sequence first.)
wolffd@0: %
wolffd@0: % Optional parameters may be passed as 'param_name', param_value pairs.
wolffd@0: % Parameter names are shown below; default values in [] - if none, argument is mandatory.
wolffd@0: %
wolffd@0: % For HMMs with MOG outputs: if you want to compute gamma2, you must specify
wolffd@0: % 'obslik2' - obslik(i,j,t) = Pr(Y(t)| Q(t)=i,M(t)=j)  []
wolffd@0: % 'mixmat' - mixmat(i,j) = Pr(M(t) = j | Q(t)=i)  []
wolffd@0: %
wolffd@0: % For HMMs with discrete inputs:
wolffd@0: % 'act' - act(t) = action performed at step t
wolffd@0: %
wolffd@0: % Optional arguments:
wolffd@0: % 'fwd_only' - if 1, only do a forwards pass and set beta=[], gamma2=[]  [0]
wolffd@0: % 'scaled' - if 1,  normalize alphas and betas to prevent underflow [1]
wolffd@0: % 'maximize' - if 1, use max-product instead of sum-product [0]
wolffd@0: %
wolffd@0: % OUTPUTS:
wolffd@0: % alpha(i,t) = p(Q(t)=i | y(1:t)) (or p(Q(t)=i, y(1:t)) if scaled=0)
wolffd@0: % beta(i,t) = p(y(t+1:T) | Q(t)=i)*p(y(t+1:T)|y(1:t)) (or p(y(t+1:T) | Q(t)=i) if scaled=0)
wolffd@0: % gamma(i,t) = p(Q(t)=i | y(1:T))
wolffd@0: % loglik = log p(y(1:T))
wolffd@0: % xi(i,j,t-1)  = p(Q(t-1)=i, Q(t)=j | y(1:T))  - NO LONGER COMPUTED
wolffd@0: % xi_summed(i,j) = sum_{t=}^{T-1} xi(i,j,t)  - changed made by Herbert Jaeger
wolffd@0: % gamma2(j,k,t) = p(Q(t)=j, M(t)=k | y(1:T)) (only for MOG  outputs)
wolffd@0: %
wolffd@0: % If fwd_only = 1, these become
wolffd@0: % alpha(i,t) = p(Q(t)=i | y(1:t))
wolffd@0: % beta = []
wolffd@0: % gamma(i,t) = p(Q(t)=i | y(1:t))
wolffd@0: % xi(i,j,t-1)  = p(Q(t-1)=i, Q(t)=j | y(1:t))
wolffd@0: % gamma2 = []
wolffd@0: %
wolffd@0: % Note: we only compute xi if it is requested as a return argument, since it can be very large.
wolffd@0: % Similarly, we only compute gamma2 on request (and if using MOG outputs).
wolffd@0: %
wolffd@0: % Examples:
wolffd@0: %
wolffd@0: % [alpha, beta, gamma, loglik] = fwdback(pi, A, multinomial_prob(sequence, B));
wolffd@0: %
wolffd@0: % [B, B2] = mixgauss_prob(data, mu, Sigma, mixmat);
wolffd@0: % [alpha, beta, gamma, loglik, xi, gamma2] = fwdback(pi, A, B, 'obslik2', B2, 'mixmat', mixmat);
wolffd@0: 
wolffd@0: if nargout >= 5, compute_xi = 1; else compute_xi = 0; end
wolffd@0: if nargout >= 6, compute_gamma2 = 1; else compute_gamma2 = 0; end
wolffd@0: 
wolffd@0: [obslik2, mixmat, fwd_only, scaled, act, maximize, compute_xi, compute_gamma2] = ...
wolffd@0:    process_options(varargin, ...
wolffd@0:        'obslik2', [], 'mixmat', [], ...
wolffd@0:        'fwd_only', 0, 'scaled', 1, 'act', [], 'maximize', 0, ...
wolffd@0:                    'compute_xi', compute_xi, 'compute_gamma2', compute_gamma2);
wolffd@0: 
wolffd@0: [Q T] = size(obslik);
wolffd@0: 
wolffd@0: if isempty(obslik2)
wolffd@0:  compute_gamma2 = 0;
wolffd@0: end
wolffd@0: 
wolffd@0: if isempty(act)
wolffd@0:  act = ones(1,T);
wolffd@0:  transmat = { transmat } ;
wolffd@0: end
wolffd@0: 
wolffd@0: scale = ones(1,T);
wolffd@0: 
wolffd@0: % scale(t) = Pr(O(t) | O(1:t-1)) = 1/c(t) as defined by Rabiner (1989).
wolffd@0: % Hence prod_t scale(t) = Pr(O(1)) Pr(O(2)|O(1)) Pr(O(3) | O(1:2)) ... = Pr(O(1), ... ,O(T))
wolffd@0: % or log P = sum_t log scale(t).
wolffd@0: % Rabiner suggests multiplying beta(t) by scale(t), but we can instead
wolffd@0: % normalise beta(t) - the constants will cancel when we compute gamma.
wolffd@0: 
wolffd@0: loglik = 0;
wolffd@0: 
wolffd@0: alpha = zeros(Q,T);
wolffd@0: gamma = zeros(Q,T);
wolffd@0: if compute_xi
wolffd@0:  xi_summed = zeros(Q,Q);
wolffd@0: else
wolffd@0:  xi_summed = [];
wolffd@0: end
wolffd@0: 
wolffd@0: %%%%%%%%% Forwards %%%%%%%%%%
wolffd@0: 
wolffd@0: t = 1;
wolffd@0: alpha(:,1) = init_state_distrib(:) .* obslik(:,t);
wolffd@0: if scaled
wolffd@0:  %[alpha(:,t), scale(t)] = normaliseC(alpha(:,t));
wolffd@0:  [alpha(:,t), scale(t)] = normalise(alpha(:,t));
wolffd@0: end
wolffd@0: assert(approxeq(sum(alpha(:,t)),1))
wolffd@0: for t=2:T
wolffd@0:  %trans = transmat(:,:,act(t-1))';
wolffd@0:  trans = transmat{act(t-1)};
wolffd@0:  if maximize
wolffd@0:    m = max_mult(trans', alpha(:,t-1));
wolffd@0:    %A = repmat(alpha(:,t-1), [1 Q]);
wolffd@0:    %m = max(trans .* A, [], 1);
wolffd@0:  else
wolffd@0:    m = trans' * alpha(:,t-1);
wolffd@0:  end
wolffd@0:  alpha(:,t) = m(:) .* obslik(:,t);
wolffd@0:  if scaled
wolffd@0:    %[alpha(:,t), scale(t)] = normaliseC(alpha(:,t));
wolffd@0:    [alpha(:,t), scale(t)] = normalise(alpha(:,t));
wolffd@0:  end
wolffd@0:  if compute_xi & fwd_only  % useful for online EM
wolffd@0:    %xi(:,:,t-1) = normaliseC((alpha(:,t-1) * obslik(:,t)') .* trans);
wolffd@0:    xi_summed = xi_summed + normalise((alpha(:,t-1) * obslik(:,t)') .* trans);
wolffd@0:  end
wolffd@0:  assert(approxeq(sum(alpha(:,t)),1))
wolffd@0: end
wolffd@0: if scaled
wolffd@0:  if any(scale==0)
wolffd@0:    loglik = -inf;
wolffd@0:  else
wolffd@0:    loglik = sum(log(scale));
wolffd@0:  end
wolffd@0: else
wolffd@0:  loglik = log(sum(alpha(:,T)));
wolffd@0: end
wolffd@0: 
wolffd@0: if fwd_only
wolffd@0:  gamma = alpha;
wolffd@0:  beta = [];
wolffd@0:  gamma2 = [];
wolffd@0:  return;
wolffd@0: end
wolffd@0: 
wolffd@0: %%%%%%%%% Backwards %%%%%%%%%%
wolffd@0: 
wolffd@0: beta = zeros(Q,T);
wolffd@0: if compute_gamma2
wolffd@0:  M = size(mixmat, 2);
wolffd@0:  gamma2 = zeros(Q,M,T);
wolffd@0: else
wolffd@0:  gamma2 = [];
wolffd@0: end
wolffd@0: 
wolffd@0: beta(:,T) = ones(Q,1);
wolffd@0: %gamma(:,T) = normaliseC(alpha(:,T) .* beta(:,T));
wolffd@0: gamma(:,T) = normalise(alpha(:,T) .* beta(:,T));
wolffd@0: t=T;
wolffd@0: if compute_gamma2
wolffd@0:  denom = obslik(:,t) + (obslik(:,t)==0); % replace 0s with 1s before dividing
wolffd@0:  gamma2(:,:,t) = obslik2(:,:,t) .* mixmat .* repmat(gamma(:,t), [1 M]) ./ repmat(denom, [1 M]);
wolffd@0:  %gamma2(:,:,t) = normaliseC(obslik2(:,:,t) .* mixmat .* repmat(gamma(:,t), [1 M])); % wrong!
wolffd@0: end
wolffd@0: for t=T-1:-1:1
wolffd@0:  b = beta(:,t+1) .* obslik(:,t+1);
wolffd@0:  %trans = transmat(:,:,act(t));
wolffd@0:  trans = transmat{act(t)};
wolffd@0:  if maximize
wolffd@0:    B = repmat(b(:)', Q, 1);
wolffd@0:    beta(:,t) = max(trans .* B, [], 2);
wolffd@0:  else
wolffd@0:    beta(:,t) = trans * b;
wolffd@0:  end
wolffd@0:  if scaled
wolffd@0:    %beta(:,t) = normaliseC(beta(:,t));
wolffd@0:    beta(:,t) = normalise(beta(:,t));
wolffd@0:  end
wolffd@0:  %gamma(:,t) = normaliseC(alpha(:,t) .* beta(:,t));
wolffd@0:  gamma(:,t) = normalise(alpha(:,t) .* beta(:,t));
wolffd@0:  if compute_xi
wolffd@0:    %xi(:,:,t) = normaliseC((trans .* (alpha(:,t) * b')));
wolffd@0:    xi_summed = xi_summed + normalise((trans .* (alpha(:,t) * b')));
wolffd@0:  end
wolffd@0:  if compute_gamma2
wolffd@0:    denom = obslik(:,t) + (obslik(:,t)==0); % replace 0s with 1s before dividing
wolffd@0:    gamma2(:,:,t) = obslik2(:,:,t) .* mixmat .* repmat(gamma(:,t), [1 M]) ./ repmat(denom, [1 M]);
wolffd@0:    %gamma2(:,:,t) = normaliseC(obslik2(:,:,t) .* mixmat .* repmat(gamma(:,t), [1 M]));
wolffd@0:  end
wolffd@0: end
wolffd@0: 
wolffd@0: % We now explain the equation for gamma2
wolffd@0: % Let zt=y(1:t-1,t+1:T) be all observations except y(t)
wolffd@0: % gamma2(Q,M,t) = P(Qt,Mt|yt,zt) = P(yt|Qt,Mt,zt) P(Qt,Mt|zt) / P(yt|zt)
wolffd@0: %                = P(yt|Qt,Mt) P(Mt|Qt) P(Qt|zt) / P(yt|zt)
wolffd@0: % Now gamma(Q,t) = P(Qt|yt,zt) = P(yt|Qt) P(Qt|zt) / P(yt|zt)
wolffd@0: % hence
wolffd@0: % P(Qt,Mt|yt,zt) = P(yt|Qt,Mt) P(Mt|Qt) [P(Qt|yt,zt) P(yt|zt) / P(yt|Qt)] / P(yt|zt)
wolffd@0: %                = P(yt|Qt,Mt) P(Mt|Qt) P(Qt|yt,zt) / P(yt|Qt)