We analyze the information geometric structure of time reversibility for parametric families of irreducible transition kernels of Markov chains. We define and characterize reversible exponential families of Markov kernels, and show that irreducible and reversible Markov kernels form both a mixture family and, perhaps surprisingly, an exponential family in the set of all stochastic kernels. We propose a parametrization of the entire manifold of reversible kernels, and inspect reversible geodesics. We define information projections onto the reversible manifold, and derive closed-form expressions for the e-projection and m-projection, along with Pythagorean identities with respect to information divergence, leading to some new notion of reversiblization of Markov kernels. We show the family of edge measures pertaining to irreducible and reversible kernels also forms an exponential family among distributions over pairs. We further explore geometric properties of the reversible family, by comparing them with other remarkable families of stochastic matrices. Finally, we show that reversible kernels are, in a sense we define, the minimal exponential family generated by the m-family of symmetric kernels, and the smallest mixture family that comprises the e-family of memoryless kernels.
翻译:我们分析了马尔科夫链链中不可降低的过渡内核的准临界家庭的时间可逆性的信息几何结构。我们定义和定性马尔科夫内核的可逆指数型家庭,并表明不可减少和可逆的马尔科夫内核是混合家庭,也许令人惊讶的是,在所有碎裂内核的组合中,不可逆的马尔科夫内核是指数型家庭。我们建议对可逆内核的全方位进行可逆内核反射,并检查可逆的大地测量。我们将信息预测确定为可逆的元体,为电子预测和移动内核的可逆性指数型家庭定出封闭式的表达方式,以及Pythagore在信息差异方面的特性,导致对马尔科夫内核的再现性新概念。我们展示了与不可逆和可逆内核的内核细胞组成的边缘结构,我们进一步探索可逆家庭的几何特性,把它们与其他非凡的可变式家庭内部结构加以比较,我们用最小的内核的内核的内核决定了家庭最小的内核的内核。最后,我们展示了可变式家庭内核的内核的内核的内核的内核的内核的内核。