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 go to infinity, they tend to have high computational cost, particularly for observations of a long time series. Variational Bayes (VB) methods are a useful alternative when inference with MCMC methods is computationally expensive. VB methods approximate the posterior density of the parameters by a simple and tractable distribution found through optimisation. In this paper, we propose a novel sequential variational Bayes approach that makes use of the Whittle likelihood for computationally efficient parameter inference in this class of state space models. Our algorithm, which we call 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 for a wide class of models. Through several examples using a linear Gaussian state space model and a univariate/bivariate non-Gaussian stochastic volatility model, we show that R-VGA-Whittle provides good approximations to posterior distributions of the parameters and is very computationally efficient when compared to asymptotically exact methods such as Hamiltonian Monte Carlo.

翻译：线性非高斯状态空间模型的参数推断具有挑战性，因为似然函数包含关于潜状态变量的难以处理的积分。虽然马尔可夫链蒙特卡罗（MCMC）方法在样本量趋于无穷时能提供后验分布的精确样本，但其计算成本往往很高，尤其对于长时间序列的观测数据。当MCMC方法的推断计算成本过高时，变分贝叶斯（VB）方法是一种有用的替代方案。VB方法通过优化找到一个简单且易于处理的分布，以近似参数的后验密度。本文提出了一种新颖的序列变分贝叶斯方法，该方法利用Whittle似然在此类状态空间模型中实现计算高效的参数推断。我们的算法称为基于Whittle似然的递归变分高斯近似（R-VGA-Whittle），通过在频域处理数据来更新变分参数。在每次迭代中，R-VGA-Whittle需要Whittle对数似然的梯度和Hessian矩阵，这对于一大类模型均存在闭式解。通过在线性高斯状态空间模型及单变量/双变量非高斯随机波动率模型上的多个示例，我们证明R-VGA-Whittle能够为参数的后验分布提供良好的近似，并且与哈密顿蒙特卡罗等渐近精确方法相比具有极高的计算效率。