function [alpha, obslik, gamma, xi] = fixed_lag_smoother(d, alpha, obslik, obsvec, transmat, act)% FIXED_LAG_SMOOTHER Computed smoothed posterior estimates within a window given previous filtered window.% [alpha, obslik, gamma, xi] = fixed_lag_smoother(d, alpha, obslik, obsvec, transmat, act)%% d >= 2 is the desired window width.% Actually, we use d=min(d, t0), where t0 is the current time.%% alpha(:, t0-d:t0-1) - length d window, excluding t0 (Columns indexed 1..d)% obslik(:, t0-d:t0-1) - length d window% obsvec              - likelihood vector for current observation% transmat            - transition matrix% If we specify the optional 'act' argument, transmat{a} should be a cell array, and% act(t0-d:t0)          - length d window, last column = current action%% Output:% alpha(:, t0-d+1:t0)     - last column = new filtered estimate% obslik(:, t0-d+1:t0)    - last column = obsvec% xi(:, :, t0-d+1:t0-1)   - 2 slice smoothed window% gamma(:, t0-d+1:t0)     - smoothed window%% As usual, we define (using T=d)% alpha(i,t) = Pr(Q(t)=i | Y(1:t))% gamma(i,t) = Pr(Q(t)=i | Y(1:T))% xi(i,j,t) = Pr(Q(t)=i, Q(t+1)=j | Y(1:T))%% obslik(i,t) = Pr(Y(t) | Q(t)=i) % transmat{a}(i,j) = Pr(Q(t)=j | Q(t-1)=i, A(t)=a)[S n] = size(alpha); d = min(d, n+1); if d < 2  error('must keep a window of length at least 2');endif ~exist('act')  act = ones(1, n+1);  transmat = { transmat };end% pluck out last d-1 components from the historyalpha = alpha(:, n-d+2:n); obslik = obslik(:, n-d+2:n); % Extend window by 1t = d;obslik(:,t) = obsvec;xi = zeros(S, S, d-1);xi(:,:,t-1) = normalise((alpha(:,t-1) * obslik(:,t)') .* transmat{act(t)});alpha(:,t) = sum(xi(:,:,t-1), 1)';% Now smooth backwards inside the windowbeta = ones(S, d);T = d;%fprintf('smooth from %d to 1, i.e., %d to %d\n', d, t0, t0-d+1);gamma(:,T) = alpha(:,T);for t=T-1:-1:1  b = beta(:,t+1) .* obslik(:,t+1);   beta(:,t) = normalise(transmat{act(t)} * b);  gamma(:,t) = normalise(alpha(:,t) .* beta(:,t));  xi(:,:,t) = normalise((transmat{act(t)} .* (alpha(:,t) * b')));end