This paper analyzes the factorizability and geometry of transition matrices of multivariate Markov chains. Specifically, we demonstrate that the induced chains on factors of a product space can be regarded as information projections with respect to the Kullback-Leibler divergence. This perspective yields Han-Shearer type inequalities and submodularity of the entropy rate of Markov chains, as well as applications in the context of large deviations and mixing time comparison. As concrete algorithmic applications in Markov chain Monte Carlo (MCMC) and approximate inference, we provide three illustrations based on lifted MCMC, swapping algorithm and factored filtering to demonstrate projection samplers improve mixing over the original samplers. The projection sampler based on the swapping algorithm resamples the highest-temperature coordinate at stationarity at each step, and we prove that such practice accelerates the mixing time by multiplicative factors related to the number of temperatures and the dimension of the underlying state space when compared with the original swapping algorithm. Through simple numerical experiments on a bimodal target distribution, we show that the projection samplers mix effectively, in contrast to lifted MCMC and the swapping algorithm, which mix less well. In filtering, our proposed factored filtering scheme is able to scale to high dimensions with linear-in-dimension computational cost per step at the price of an approximation error that can be tracked using the distance to independence, compared with the exponential-in-dimension cost per step of the exact filter.

翻译：本文分析了多元马尔可夫链转移矩阵的可分解性与几何结构。具体而言，我们证明了在乘积空间的因子上诱导出的链可视为关于Kullback-Leibler散度的信息投影。这一视角导出了马尔可夫链熵率的Han-Shearer型不等式与次模性，并应用于大偏差理论与混合时间比较。在马尔可夫链蒙特卡洛（MCMC）与近似推断的具体算法应用中，我们通过提升MCMC、交换算法与因式滤波三种示例，展示了投影采样器相较于原始采样器在混合性能上的改进。基于交换算法的投影采样器在每步平稳状态下重新采样最高温度坐标，我们证明该操作相较于原始交换算法，可在混合时间上实现与温度数量及底层状态空间维度相关的乘性加速。基于双峰目标分布的简单数值实验表明，投影采样器能有效混合，而提升MCMC与交换算法的混合效果较差。在滤波场景中，我们提出的因式滤波方案能以每步计算代价随维度线性增长为代价，实现高维扩展，其近似误差可通过距离独立性进行追踪，而精确滤波器的每步计算代价随维度呈指数增长。