Recursive variational Gaussian approximation with the Whittle likelihood for linear non-Gaussian state space models

Abstract

Parameter inference for linear and non-Gaussian state space models is challenging because the likelihood function contains an intractable integral over the latent state variables. While Markov chain Monte Carlo (MCMC) methods provide exact samples from the posterior distribution as the number of samples goes to infinity, they tend to have high computational cost, particularly for observations of a long time series. When inference with MCMC methods is computationally expensive, variational Bayes (VB) methods are a useful alternative. VB methods approximate the posterior density of the parameters with a simple and tractable distribution found through optimisation. This work proposes a novel sequential VB approach that makes use of the Whittle likelihood for computationally efficient parameter inference in linear, non-Gaussian state space models. Our algorithm, called Recursive Variational Gaussian Approximation with the Whittle Likelihood (R-VGA-Whittle), updates the variational parameters by processing data in the frequency domain. At each iteration, R-VGA-Whittle requires the gradient and Hessian of the Whittle log-likelihood, which are available in closed form. Through several examples involving a linear Gaussian state space model; a univariate/bivariate stochastic volatility model; and a state space model with Student's t measurement error, where the latent states follow an autoregressive fractionally integrated moving average (ARFIMA) model, we show that R-VGA-Whittle provides good approximations to posterior distributions of the parameters, and that it is very computationally efficient when compared to asymptotically exact methods such as Hamiltonian Monte Carlo.