We study the problem of designing efficient exact MCMC algorithms for sampling from the full posterior distribution of high-dimensional (in the number of time steps and the dimension of the latent space) non-linear non-Gaussian latent dynamical models. Particle Gibbs, also known as conditional sequential Monte Carlo (SMC), constitutes the de facto golden standard to do so, but suffers from degeneracy problems when the dimension of the latent space increases. On the other hand, the routinely employed globally Gaussian-approximated (e.g., extended Kalman filtering) biased solutions are seldom used for this same purpose even though they are more robust than their SMC counterparts. In this article, we show how, by introducing auxiliary observation variables in the model, we can both implement efficient exact Kalman-based samplers for large state-space models, as well as dramatically improve the mixing speed of particle Gibbs algorithms when the dimension of the latent space increases. We demonstrate when and how we can parallelise these auxiliary samplers along the time dimension, resulting in algorithms that scale logarithmically with the number of time steps when implemented on graphics processing units (GPUs). Both algorithms are easily tuned and can be extended to accommodate sophisticated approximation techniques. We demonstrate the improved statistical and computational performance of our auxiliary samplers compared to state-of-the-art alternatives for high-dimensional (in both time and state space) latent dynamical systems.

翻译：本文研究如何为高维（时间步数与潜在空间维度）非线性非高斯潜在动态模型设计高效精确的MCMC算法，以实现全后验分布的采样。粒子吉布斯（亦称条件序贯蒙特卡罗方法）是该领域的实际黄金标准，但在潜在空间维度增加时会出现退化问题。另一方面，常规采用的全局高斯近似方法（如扩展卡尔曼滤波）虽存在偏差，且其稳健性优于序贯蒙特卡罗方法，却极少用于此类精确采样。本文通过引入辅助观测变量，实现了针对大规模状态空间模型的高效精确卡尔曼类采样器，并显著提升了粒子吉布斯算法在潜在空间维度增加时的混合速度。我们论证了这些辅助采样器在时间维度上实现并行化的条件与方法，当在图形处理器上实施时，算法复杂度随时间步数呈对数级增长。两种算法均易于调参，并可扩展至复杂近似技术。通过在高维（时间与状态空间）潜在动态系统上的实验，我们证明了所提辅助采样器相较于现有最优方法在统计性能与计算效率上的显著提升。