Fast evaluation of the likelihood of an HMM: ion channel currents with filtering and colored noise

Hidden Markov models (HMMs) have been used in the study of single-channel recordings of ion channel currents for restoration of idealized signals from noisy recordings and for estimation of kinetic parameters. A key to their effectiveness from a computational point of view is that the number of operations to evaluate the likelihood, posterior probabilities and the most likely state sequence is proportional to the product of the square of the dimension of the state space and the length of the series. However, when the state space is quite large, computations can become infeasible. This can happen when the record has been lowpass filtered and when the noise is colored. In this paper, we present an approximate method that can provide very substantial reductions in computational cost at the expense of only a very small error. We describe the method and illustrate through examples the gains that can be made in evaluating the likelihood.