MATLAB: Hidden Markov Models - Computing Alpha

Formula

\[\mathbb{P}(H_T = s_i, v_1,...,v_T) = \alpha_T = \mathbb{P}(V_t|s_j)\sum_{i=1}^N \alpha_{T-1}(i)p_{i,j}\]

Implementation

function a = alpha_dynamic(M,p,B,v)
% ALPHA DYNAMIC(M,p,B,v) calculates the matrix of alpha's the
% hmm with transition matrix M, emission matrix B, and initial
% probabilities p, given the observations v

    [np, ~]     = size(p);
    [N, ~]      = size(M);
    T           = length(v);
    %initialise alphas
    a           = zeros(N,T);
    % making sure p is a column
    if np == 1
        p = p'; 
    end
    % alpha 1 from initial probability
    a(:,1) = p.*B(:,v(1));
    %subsequent alphas
    for t = 2:T
        a(:,t) = M' * a(:,t-1) .* B(:,v(t));
    end
end