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 a concrete algorithmic application, we introduce a projection sampler based on the swapping algorithm, which resamples the highest-temperature coordinate at stationarity at each step. 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.

翻译：本文分析了多元马尔可夫链转移矩阵的可因子分解性与几何结构。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型不等式与马尔可夫链熵率的次模性，并在大偏差及混合时间比较中产生应用。作为具体的算法应用，我们提出一种基于交换算法的投影采样器，其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，相较于原始交换算法，该实践能以与温度数量及底层状态空间维度相关的乘法因子加速混合时间。