Estimating the Parameters of a Hidden Stochastic Model

Abstract

Having discussed the algorithms for estimating the realization of the hidden process {x _t } from the observations {y _t }, we will now introduce some methods for estimating the parameters in the model by which one wants to interpret the observations. In the first case, this will be a hidden Markov model, in the second case, a state space model will be assumed. In each case, all of the parameters will be collected in the vector θ. As an estimator we use the maximum-likelihood estimator,so that we have to find the maximum of the likelihood ρ(y _1…N |θ) with respect to θ. In general, the likelihood is the sum over all possible paths x_1…N of the hidden process

$$L = \rho ({y_{1...N}}|\theta ) = \sum\limits_{{x_{1...N}}} {\rho ({y_{1...N}},{x_{1...N}}|\theta )} = \sum\limits_{{x_{1...N}}} {\rho ({y_{1...N}}|{x_{1...N}})\rho ({x_{1...N}}|\theta )} $$

(12.1)

This is a sum over m ^N terms, when m is the number of discrete states. In nearly every case, this summation cannot be done.