% Example of fixed lag smoothingrand('state', 1);S = 2;O = 2;T = 7;data = sample_discrete([0.5 0.5], 1, T);transmat = mk_stochastic(rand(S,S));obsmat = mk_stochastic(rand(S,O));obslik = mk_dhmm_obs_lik(data, obsmat);prior = [0.5 0.5]';[alpha0, beta0, gamma0, xi0] = forwards_backwards(prior, transmat, obslik);w = 3;alpha1 = zeros(S, T);gamma1 = zeros(S, T);xi1 = zeros(S, S, T-1);t = 1;b = obsmat(:, data(t));olik_win = b; % window of conditional observation likelihoodsalpha_win = normalise(prior .* b);alpha1(:,t) = alpha_win;for t=2:T  [alpha_win, olik_win, gamma_win, xi_win] = ...      fixed_lag_smoother(w, alpha_win, olik_win, obsmat(:, data(t)), transmat);  alpha1(:,max(1,t-w+1):t) = alpha_win;  gamma1(:,max(1,t-w+1):t) = gamma_win;  xi1(:,:,max(1,t-w+1):t-1) = xi_win;ende = 1e-1;assert(approxeq(alpha0, alpha1, e));assert(approxeq(gamma0(:, T-w+1:end), gamma1(:, T-w+1:end), e));assert(approxeq(xi0(:,:,T-w+1:end), xi1(:,:,T-w+1:end), e));