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. Exact inference using Markov chain Monte Carlo is computationally expensive, particularly for long time series data. Variational Bayes methods are useful when exact inference is infeasible. These 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隐状态变量的难以处理的积分。使用马尔可夫链蒙特卡洛进行精确推断计算成本高昂，尤其对于长时间序列数据。当精确推断不可行时，变分贝叶斯方法具有实用价值。这些方法通过优化寻找一个简单且易处理的分布来近似参数的后验密度。本文提出了一种新颖的序贯变分贝叶斯方法，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对数似然的梯度和海森矩阵，这两者对广泛类型的模型均存在闭式解。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能够提供参数后验分布的良好近似，并且与哈密顿蒙特卡洛等渐近精确方法相比具有极高的计算效率。