We develop results on geometric ergodicity of Markov chains and apply these and other recent results in Markov chain theory to multidimensional Hastings and metropolis algorithms. For those based on random walk candid...
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We develop results on geometric ergodicity of Markov chains and apply these and other recent results in Markov chain theory to multidimensional Hastings and metropolis algorithms. For those based on random walk candidate distributions, we find sufficient conditions for moments and moment generating functions to converge at a geometric rate to a prescribed distribution pi. By phrasing the conditions in terms of the curvature of the densities we show that the results apply to all distributions with positive densities in a large class which encompasses many commonly-used statistical forms. From these results we develop central limit theorems for the metropolis algorithm. Converse results, showing non-geometric convergence rates for chains where the rejection rate is not bounded away from unity, are also given;these show that the negative-definiteness property is not redundant.
This paper considers the problem of scaling the proposal distribution of a multidimensional random walk metropolis algorithm in order to maximize the efficiency of the algorithm. The main result is a weak convergence ...
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This paper considers the problem of scaling the proposal distribution of a multidimensional random walk metropolis algorithm in order to maximize the efficiency of the algorithm. The main result is a weak convergence result as the dimension of a sequence of target densities, n, converges to infinity. When the proposal variance is appropriately scaled according to n, the sequence of stochastic processes formed by the first component of each Markov chain converges to the appropriate limiting Langevin diffusion process. The limiting diffusion approximation admits a straightforward efficiency maximization problem, and the resulting asymptotically optimal policy is related to the asymptotic acceptance rate of proposed moves for the algorithm. The asymptotically optimal acceptance rate is 0.234 under quite general conditions. The main result is proved in the case where the target density has a symmetric product form. Extensions of the result are discussed.
It is shown that Markov chains for sampling from combinatorial sets in the form of experimental designs can be made more efficient by using syzygies on gradient vectors. Examples are presented. (C) 2011 Elsevier Inc. ...
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It is shown that Markov chains for sampling from combinatorial sets in the form of experimental designs can be made more efficient by using syzygies on gradient vectors. Examples are presented. (C) 2011 Elsevier Inc. All rights reserved.
The metropolis-adjusted Langevin algorithm (MALA) is a metropolis Hastings method for approximate sampling from continuous distributions. We derive upper bounds for the contraction rate in Kantorovich-Rubinstein-Wasse...
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The metropolis-adjusted Langevin algorithm (MALA) is a metropolis Hastings method for approximate sampling from continuous distributions. We derive upper bounds for the contraction rate in Kantorovich-Rubinstein-Wasserstein distance of the MALA chain with semi-implicit Euler proposals applied to log-concave probability measures that have a density w.r.t. a Gaussian reference measure. For sufficiently "regular" densities, the estimates are dimension-independent, and they hold for sufficiently small step sizes h that do not depend on the dimension either. In the limit h down arrow 0, the bounds approach the known optimal contraction rates for overdamped Langevin diffusions in a convex potential. A similar approach also applies to metropolis Hastings chains with Ornstein-Uhlenbeck proposals. In this case, the resulting estimates are still independent of the dimension but less optimal, reflecting the fact that MALA is a higher order approximation of the diffusion limit than metropolis Hastings with Ornstein-Uhlenbeck proposals.
A phi-irreducible and aperiodic Markov chain with stationary probability distribution will converge to its stationary distribution from almost all starting points. The property of Harris recurrence allows us to replac...
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A phi-irreducible and aperiodic Markov chain with stationary probability distribution will converge to its stationary distribution from almost all starting points. The property of Harris recurrence allows us to replace "almost all" by "all," which is potentially important when running Markov chain Monte Carlo algorithms. Full-dimensional metropolis-Hastings algorithms are known to be Harris recurrent. In this paper, we consider conditions under which metropolis-within-Gibbs and trans-dimensional Markov chains are or are not Harris recurrent. We present a simple but natural two-dimensional counter-example showing how Harris recurrence can fail, and also a variety of positive results which guarantee Harris recurrence. We also present some open problems. We close with a discussion of the practical implications for MCMC algorithms.
We prove sharp rates of convergence to the Ewens equilibrium distribution for a family of metropolis algorithms based on the random transposition shuffle on the symmetric group, with starting point at the identity. Th...
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We prove sharp rates of convergence to the Ewens equilibrium distribution for a family of metropolis algorithms based on the random transposition shuffle on the symmetric group, with starting point at the identity. The proofs rely heavily on the theory of symmetric Jack polynomials, developed initially by Jack [Proc. Roy. Soc. Edinburgh Sect. A 69 (1970/1971) 1-18], Macdonald [Symmetric Functions and Hall Polynomials (1995) New York] and Stanley [Adv. Math. 77 (1989) 76-115]. This completes the analysis started by Diaconis and Hanlon in [Contemp. Math. 138 (1992) 99-117]. In the end we also explore other integrable Markov chains that can be obtained from symmetric function theory.
Urban street networks of unplanned or self-organized cities typically exhibit astonishing scale-free patterns. This scale-freeness can be shown, within the maximum entropy formalism (MaxEnt), as the manifestation of a...
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Urban street networks of unplanned or self-organized cities typically exhibit astonishing scale-free patterns. This scale-freeness can be shown, within the maximum entropy formalism (MaxEnt), as the manifestation of a fluctuating system that preserves on average some amount of information. Monte Carlo methods that can further this perspective are cruelly missing. Here we adapt to self-organized urban street networks the metropolis algorithm. The "coming to equilibrium" distribution is established with MaxEnt by taking scale-freeness as prior hypothesis along with symmetry-conservation arguments. The equilibrium parameter is the scaling;its concomitant extensive quantity is, assuming our lack of knowledge, an amount of information. To design an ergodic dynamics, we disentangle the state-of-the-art street generating paradigms based on non-overlapping walks into layout-at-junction dynamics. Our adaptation reminisces the single-spin-flip metropolis algorithm for Ising models. We thus expect metropolis simulations to reveal that self-organized urban street networks, besides sustaining scale-freeness over a wide range of scalings, undergo a crossover as scaling varies-literature argues for a small-world crossover. Simulations for Central London are consistent against the state-of-the-art outputs over a realistic range of scaling exponents. Our illustrative Watts-Strogatz phase diagram with scaling as rewiring parameter demonstrates a small-world crossover curving within the realistic window 2-3;it also shows that the state-of-the-art outputs underlie relatively large worlds. Our metropolis adaptation to self-organized urban street networks thusly appears as a scaling variant of the Watts-Strogatz model. Such insights may ultimately allow the urban profession to anticipate self-organization or unplanned evolution of urban street networks.
In a recent paper [I. Wegener, Simulated Annealing beats metropolis in combinatorial optimization, in: L. Caires, G.F. Italiano, L. Monteiro, C. Palamidessi, M. Yung (Eds.), Proc. ICALP 2005, in: LNCS, vol. 3580, 2005...
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In a recent paper [I. Wegener, Simulated Annealing beats metropolis in combinatorial optimization, in: L. Caires, G.F. Italiano, L. Monteiro, C. Palamidessi, M. Yung (Eds.), Proc. ICALP 2005, in: LNCS, vol. 3580, 2005, pp. 589-601] Wegener gave a first natural example of a combinatorial optimization problem where for certain instances a Simulated Annealing algorithm provably performs better than the metropolis algorithm for any fixed temperature. Wegener's example deals with a special instance of the Minimum Spanning Tree problem. In this short note we show that Wegener's technique as well can be used to prove a similar result for another important problem in combinatorial optimization, namely the Traveling Salesman Problem. The main task is to construct a suitable TSP instance for which SA outperforms MA when using the well known 2-Opt local search heuristic. (C) 2007 Elsevier B.V. All rights reserved.
In this paper, we study the convergence of metropolis-type algorithms used in modeling statistical systems with a fluctuating number of particles located in a finite volume. We justify the use of metropolis algorithms...
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In this paper, we study the convergence of metropolis-type algorithms used in modeling statistical systems with a fluctuating number of particles located in a finite volume. We justify the use of metropolis algorithms for a particular class of Such statistical systems. We prove a theorem oil the geometric ergodicity of the Markov process modeling the behavior of an ensemble with a fluctuating number of particles in a finite Volume whose interaction is described by a potential bounded below and decreasing according to the law r(-3-a), alpha >= 0, as r -> 0.
Given an everywhere positive probability measure pi on a finite state space E and the associated energy function H, this note gives convergence results for time-inhomogeneous metropolis chains which are used to simula...
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Given an everywhere positive probability measure pi on a finite state space E and the associated energy function H, this note gives convergence results for time-inhomogeneous metropolis chains which are used to simulate re or minimize H under some constraints. (C) 2000 Published by Elsevier Science B.V. All rights reserved.
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