Bayesian analysis for reversible Markov chains

Diaconis, Persi; Rolles, Silke W. W.

doi:10.1214/009053606000000290

Cited by 62 publications

(84 citation statements)

References 12 publications

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“…Then τ (1/e) = Θ(m) but τ 2 (1/e) = Θ(m log m). More interesting examples include the lamplighter walks [64] and a non-reversible walk of [19]. In each of these cases the walk stays in a small set of vertices for a long time, and so even as variation distance falls the L 2 distance stays large.…”

Section: Blocking Conductancementioning

confidence: 99%

“…We give now a more interesting example of this; a chain constructed by Diaconis, Holmes and Neal [19] specifically for the purpose of speeding mixing. − 1), −(m − 2), .…”

Section: Sharpness Of Boundsmentioning

confidence: 99%

“…,(m − 1), m} with transitions P(i, i + 1) = 1 − 1/m and P(i, −i) = 1/m. Diaconis, Holmes and Neal [19] show that τ ( ) = Θ(m log(1/ )) and τ 2 ( ) = Θ(m log(m/ )), both much faster than the time Θ(m 2 log(1/ )) required by a simple random walk on a cycle.…”

Section: Sharpness Of Boundsmentioning

confidence: 99%

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Mathematical Aspects of Mixing Times in Markov Chains

Montenegro

Tetali

2006

FNT in Theoretical Computer Science

213

222

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In the past few years we have seen a surge in the theory of finite Markov chains, by way of new techniques to bounding the convergence to stationarity. This includes functional techniques such as logarithmic Sobolev and Nash inequalities, refined spectral and entropy techniques, and isoperimetric techniques such as the average and blocking conductance and the evolving set methodology. We attempt to give a more or less self-contained treatment of some of these modern techniques, after reviewing several preliminaries. We also review classical and modern lower bounds on mixing times. There have been other important contributions to this theory such as variants on coupling techniques and decomposition methods, which are not included here; our choice was to keep the analytical methods as the theme of this presentation. We illustrate the strength of the main techniques by way of simple examples, a recent result on the Pollard Rho random walk to compute the discrete logarithm, as well as with an improved analysis of the Thorp shuffle.

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Section: Blocking Conductancementioning

confidence: 99%

“…We give now a more interesting example of this; a chain constructed by Diaconis, Holmes and Neal [19] specifically for the purpose of speeding mixing. − 1), −(m − 2), .…”

Section: Sharpness Of Boundsmentioning

confidence: 99%

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Mathematical Aspects of Mixing Times in Markov Chains

Montenegro

Tetali

2006

FNT in Theoretical Computer Science

213

222

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“…The idea of the Markov chain lifting was first investigated in [8], [9] to accelerate convergence. A lifted chain is constructed by creating multiple replica states corresponding to each state in the original chain, such that the transition probabilities and stationary probabilities of the new chain conform to those of the original chain.…”

Section: Fast Distributed Consensus Via Lifting Markov Chainsmentioning

confidence: 99%

“…However, it is observed by Diaconis et al [8] and later by Chen et al. [9] that certain nonreversible chains mix substantially faster than corresponding reversible chains, by overcoming the diffusive behavior of reversible random walks.…”

Section: Introductionmentioning

confidence: 99%

Location-Aided Fast Distributed Consensus in Wireless Networks

Dai

Zhang

2010

IEEE Trans. Inform. Theory

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Existing works on distributed consensus explore linear iterations based on reversible Markov chains, which contribute to the slow convergence of the algorithms. It has been observed that by overcoming the diffusive behavior of reversible chains, certain nonreversible chains lifted from reversible ones mix substantially faster than the original chains. In this paper, we investigate the idea of accelerating distributed consensus via lifting Markov chains, and propose a class of Location-Aided Distributed Averaging (LADA) algorithms for wireless networks, where nodes' coarse location information is used to construct nonreversible chains that facilitate distributed computing and cooperative processing. First, two general pseudo-algorithms are presented to illustrate the notion of distributed averaging through chain-lifting. These pseudo-algorithms are then respectively instantiated through one LADA algorithm on grid networks, and one on general wireless networks. For a k × k grid network, the proposed LADA algorithm achieves an ǫ-averaging time of O(k log(ǫ −1 )). Based on this algorithm, in a wireless network with transmission range r, an ǫ-averaging time of O(r −1 log(ǫ −1 )) can be attained through a centralized algorithm. Subsequently, we present a fully-distributed LADA algorithm for wireless networks, which utilizes only the direction information of neighbors to construct nonreversible chains. It is shown that this distributed LADA algorithm achieves the same scaling law in averaging time as the centralized scheme in wireless networks for all r satisfying the connectivity requirement. The constructed chain attains the optimal scaling law in terms of an important mixing metric, the fill time, among all chains lifted from one with an approximately uniform stationary distribution on geometric random graphs. Finally, we propose a cluster-based LADA (C-LADA) algorithm, which, requiring no central coordination, provides the additional benefit of reduced message complexity compared with the distributed LADA algorithm.

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