Estimation by direct maximization of the likelihood

If the Markov chain is assumed to be stationary (in which case δ = δΓ), we can choose to use instead

α0 = δ

and αt = αt−1ΓP(xt) for t = 1, 2, . . . , T.

We shall first consider the stationary case. The number of operations involved is of order Tm2, making the eval-

uation of the likelihood quite feasible even for large T . Parameter estimation can therefore be performed by numerical maximization of the likelihood with respect to the parameters.