Approximate Smoothing and Parameter Estimation in High-Dimensional State-Space Models

TL;DR

This paper introduces approximate smoothing algorithms for high-dimensional state-space models using blocking strategies and particle filters, effectively addressing the curse of dimensionality and enabling scalable parameter estimation.

Contribution

It develops recursive, parallelizable approximate smoothers with bounded bias and variance that do not grow with model dimension, facilitating high-dimensional inference.

Findings

Bias of the blocked smoother is uniformly bounded over time and dimension.

Variance of the particle smoother remains unaffected by model dimension.

Successfully applied to 100-dimensional state-space model for maximum likelihood estimation.

Abstract

We present approximate algorithms for performing smoothing in a class of high-dimensional state-space models via sequential Monte Carlo methods ("particle filters"). In high dimensions, a prohibitively large number of Monte Carlo samples ("particles") -- growing exponentially in the dimension of the state space -- is usually required to obtain a useful smoother. Using blocking strategies as in Rebeschini and Van Handel (2015) (and earlier pioneering work on blocking), we exploit the spatial ergodicity properties of the model to circumvent this curse of dimensionality. We thus obtain approximate smoothers that can be computed recursively in time and in parallel in space. First, we show that the bias of our blocked smoother is bounded uniformly in the time horizon and in the model dimension. We then approximate the blocked smoother with particles and derive the asymptotic variance of idealised versions of our blocked particle smoother to show that variance is no longer adversely effected by the dimension of the model. Finally, we employ our method to successfully perform maximum-likelihood estimation via stochastic gradient-ascent and stochastic expectation--maximisation algorithms in a 100-dimensional state-space model.