A method for high-dimensional smoothing

• Published in: Journal of the Korean Statistical Society Vol. 48; no. 1; pp. 50 - 67
• Main Authors: Xu, Yaxian, Jasra, Ajay
• Format: Journal Article
• Language: English
• Published: Singapore Elsevier B.V 01.03.2019; Springer Singapore; 한국통계학회
• Subjects: Applied Statistics; Bayesian Inference; Forward smoothing; High dimensions; Sequential Monte Carlo; Smoothing; Statistical Theory and Methods; Statistics; Statistics and Computing/Statistics Programs; 통계학; Sequential Monte Carlo; secondary 60G35; High dimensions; Smoothing; Forward smoothing; primary 65C05
• ISSN: 1226-3192; 2005-2863
• DOI: 10.1016/j.jkss.2018.08.004

We consider the problem of the computation of smoothed additive functionals, which are some integrals with respect to the joint smoothing distribution. It is a key issue in inference for general state-space models as these quantities appear naturally for maximum likelihood parameter inference. The computation of smoothed additive functionals is very challenging as exact computations are not possible for non-linear non-Gaussian state-space models. It becomes even more difficult when the hidden state lies in a high dimensional space because traditional numerical methods suffer from the curse of dimensionality. We propose a new algorithm to efficiently calculate the smoothed additive functionals in an online manner for a specific family of high-dimensional state-space models in discrete time, which is named the Space–Time Forward Smoothing (STFS) algorithm. The cost of this algorithm is at least O(N2d2T), which is polynomial in d. T and N denote the number of time steps and the number of particles respectively, while d is the dimension of the hidden state space. Its superior performance over other existing methods is illustrated by various simulation studies. Moreover, STFS algorithm is successfully applied to perform Maximum Likelihood estimation for static model parameters both in an online and an offline manner.