Conductance and the rapid mixing property for Markov chains: the approximation of permanent resolved

1988 ◽  
Author(s):  
Mark Jerrum ◽  
Alistair Sinclair
Keyword(s):  
Markov Chains ◽  
Rapid Mixing ◽  
Mixing Property

scholarly journals Rapid Mixing of Several Markov Chains for a Hard-Core Model

2003 ◽  
pp. 663-675 ◽  
Author(s):  
Ravi Kannan ◽  
Michael W. Mahoney ◽  
Ravi Montenegro
Keyword(s):  
Markov Chains ◽  
Hard Core ◽  
Core Model ◽  
Rapid Mixing ◽  
Hard Core Model

scholarly journals Simple conditions for metastability of continuous Markov chains

2021 ◽  
Vol 58 (1) ◽  
pp. 83-105
Author(s):  
Oren Mangoubi ◽  
Natesh Pillai ◽  
Aaron Smith
Keyword(s):  
Markov Chains ◽  
Spectral Gap ◽  
Sufficient Conditions ◽  
Mixture Distribution ◽  
Exact Formula ◽  
Companion Paper ◽  
Monte Carlo Algorithm ◽  
Mixing Times ◽  
Mixing Property ◽  
Walk Algorithm

AbstractA family $\{Q_{\beta}\}_{\beta \geq 0}$ of Markov chains is said to exhibit metastable mixing with modes$S_{\beta}^{(1)},\ldots,S_{\beta}^{(k)}$ if its spectral gap (or some other mixing property) is very close to the worst conductance $\min\!\big(\Phi_{\beta}\big(S_{\beta}^{(1)}\big), \ldots, \Phi_{\beta}\big(S_{\beta}^{(k)}\big)\big)$ of its modes for all large values of $\beta$. We give simple sufficient conditions for a family of Markov chains to exhibit metastability in this sense, and verify that these conditions hold for a prototypical Metropolis–Hastings chain targeting a mixture distribution. The existing metastability literature is large, and our present work is aimed at filling the following small gap: finding sufficient conditions for metastability that are easy to verify for typical examples from statistics using well-studied methods, while at the same time giving an asymptotically exact formula for the spectral gap (rather than a bound that can be very far from sharp). Our bounds from this paper are used in a companion paper (O. Mangoubi, N. S. Pillai, and A. Smith, arXiv:1808.03230) to compare the mixing times of the Hamiltonian Monte Carlo algorithm and a random walk algorithm for multimodal target distributions.

Markov models—Markov chains

2019 ◽  
Vol 16 (8) ◽  
pp. 663-664 ◽  
Author(s):  
Jasleen K. Grewal ◽  
Martin Krzywinski ◽  
Naomi Altman
Keyword(s):  
Markov Chains ◽  
Markov Models
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