Efficient Online Smoothing in General Hidden Markov Models
Johan Westerborn · Lund University Publications Student Papers (Lund University) · 2012
This thesis discusses the problem of estimating smoothed expectations of sums of additive functionals of sequences of hidden states in general hidden Markov models. To compute expectations of this sort, the smoothing distribution, i.e. the conditional distribution of the hidden states given the corresponding observations, needs to be approximated. This thesis proposes a new algorithm to achieve this in an efficient way. The method is based on an algorithm proposed by Del Moral et al. (2009) and contains as a key ingredient an accept-reject sampling step or a Metropolis-Hastings algorithm (depending on wether the transition density of the state sequence is bounded or not). The proposed algorithm, which replaces certain expectations occur- ring in the original algorithm by Monte Carlo estimates, allows for on- line computation of these expectations. When a single Monte Carlo draw is used in these estimates a degeneracy problem occurs, which can be avoided completely by simply using two or more draws. The new algorithm is tested on three different models: a linear and Gaus- sian state-space model, a stochastic volatility model, and a model with non-bounded transition density. For the same input the new algorithm produces an output similar to the method proposed by Del Moral et al. (2009), but at a fraction of the computation time.