The paper introduces a rate-distortion framework for Markov chains and demonstrates how various MCMC algorithms can be viewed as specific instances within this framework. The key insights are:
The authors define an "entropic distance to independence" of a given Markov chain P on a finite product state space, denoted as Iπ
f(P), which measures how far P is from being a product chain. They show that Iπ
f(P) is zero if and only if P is a product chain under suitable assumptions.
The authors derive a Pythagorean identity for the KL divergence, which implies that the product chain with transition matrix ⊗d
i=1P (i)
π (the ith marginal transition matrix of P with respect to the stationary distribution π) is the unique closest product chain to P.
The authors generalize the notion of "leave-one-out" and "leave-S-out" transition matrices, and investigate the factorizability of P with respect to partitions or cliques of a given graph. This leads to comparisons of mixing and hitting time parameters between P and its information projections.
The authors formulate a rate-distortion optimization problem for a source Markov chain M, and demonstrate that many common MCMC algorithms, such as Metropolis-Hastings, Glauber dynamics, Feynman-Kac path models, swapping algorithm, and simulated annealing, can be shown to be optimal chains under suitable source chain and cost function.
The authors analyze the geometric structure of irreducible multivariate Markov chains induced from the information divergence rate, establishing connections to exponential families and mixture families.
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by Michael C.H.... ที่ arxiv.org 04-22-2024
https://arxiv.org/pdf/2404.12589.pdfสอบถามเพิ่มเติม