Hidden Markov and State Space Models Asymptotic Analysis of Exact and Approximate Methods for Prediction, Filtering, Smoothing and Statistical Inference

摘要

State space and hidden Markov models can both be subsumed under the same mathematical structure. On a suitable probability space (Ω, А, P) are defined (X_1, Y_1, X_2, Y_2,…, X_n, Y_n,…) a sequence of random "variables" taking values in a product space П_(j=1)~∞ (х_j * у_j) with an appropriate sigma field. The joint behavior under P is that the X_j are stationary Markovian and that given (X_1, X_2,…) the Y_j are independent and further that Y_j is independent of all X_i : i ≠ j given X_j. If Н is finite these are referred to as Hidden Markov models. The general case though focussing on χ Euclidean is referred to as state space models. Essentially we observe only the Y's and want to infer statistical properties of the X's given the Y's. The fundamental problems of filtering, smoothing prediction are to give algorithms for computing exactly or approximately the conditional distribution of X_t given (Y_1,…, Y_t) (Filtering), the conditional distribution of X_t given Y_1,…, Y_T, T > t (Smoothing) and the conditional distribution of X_(t+1),…, X_T given Y_1,…, Y_t (Prediction). If as is usually the case P is unknown and is assumed to belong to a smooth parametric family of probabilities {P_θ : θ ∈ R~d}, we face the further problem of efficiently estimating θ using Y_1,…, Y_T (computation of the likelihood, and maximum likelihood estimation, etc.). State space models have long played an important role in signal processing. The Gaussian case can be treated algorithmically using the famous Kalman filter [6]. Similarly since the 1970s there has been extensive application of Hidden Markov models in speech recognition with prediction being the most important goal. The basic theoretical work here, in the case χ and у finite (small) providing both algorithms and asymptotic analysis for inference is that of Baum and colleagues [1]. During the last 30-40 years these general models have proved of great value in applications ranging from genomics to finance-see for example [7].