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，我们证明了乘积空间因子上的诱导链可视为关于Kullback-Leibler散度的信息投影。这一视角导出了Han-Shearer型不等式及马尔可夫链熵率的次模性，并应用于大偏差与混合时间比较的语境中。作为马尔可夫链蒙特卡洛（MCMC）与近似推断中的具体算法应用，我们基于提升MCMC、交换算法与分解滤波提供了三个示例，以证明投影采样器相比原始采样器能改善混合性能。基于交换算法的投影采样器在每一步平稳状态下对最高温度坐标进行重采样，我们证明该做法相比原始交换算法，能以与温度数量及底层状态空间维度相关的乘性因子加速混合时间。通过对双峰目标分布的简单数值实验，我们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。在滤波问题中，我们提出的分解滤波方案能够扩展至高维情形，其每步计算成本与维度呈线性关系，代价是可利用独立性距离追踪的近似误差；相比之下，精确滤波器每步计算成本与维度呈指数关系。