Comments on "Reducing computation in HMM evaluation"

| We show that HMM word recognition using Deller and Snider's \any path" procedure makes an assumption of independence that is not made by either the forward or Viterbi algorithms. We also point out that additional savings in execution time can be achieved by precomputation. I N 1], hidden Markov model (HMM) recognition is considered using a state-space-like formulation: x(t + 1) = Ax(t) + u(t)(t) (1) y(t) = Bx(t) (2) where the (i; j) th element of A is the probability of making a transition from state j to state i and the (k; j) th element of B is the probability of observing output k from state j 1. The following state-space transformation is used: x(t + 1) = Ax(t) + u(t)(t) (3) y(t) = Bx(t) (4) where x(t) = UPx(t) and u(t) = UPu(t) (5) If the columns of P ?1 are the normalized eigenvectors of A and if U is diagonal, then the new transition matrix, A, is diagonal, consisting of the system's eigenvalues: If the diagonal matrix, U, is chosen such that diagonalfU ?1 g = Pu(0), then u(0) will consist of all ones. At time t, the probability of the current output symbol, O t , is taken as the dot product of x(t) and the k th row of B, where k = O t : y Ot (t) = N X j=1 x j (t)b(O t ; j) = N X j=1 x j (t)b(O t ; j) (8) The entire observation sequence is taken into account by forming the products of the individual dot products: prob(O 1 ; :::; O T jmodel) = T Y t=1 y Ot (t) (9) 1 A and B are deened here as in 1], but these deenitions diier from standard HMM notation by a transpose. This formulation assumes that the observations are unconditionally independent. By contrast, both the forward algorithm and the Viterbi algorithm assume that the observations are independent only if the state path is known. The incorrect assumption of the any-path procedure can cause it to assign a positive probability to an observation sequence that is impossible according to the HMM paradigm. This behavior is illustrated with a simple example. Consider the following observation sequence, HMM parameters, and diagonalized model: (12) Using 1, 2, and 10, the any-path procedure without diago-nalization yields the following results: x(t)= 2 4 t = 1 t = …