Monte Carlo Methods in Statistical Mechanics: Foundations and New Algorithms

Sokal, Alan D.

doi:10.1007/978-1-4899-0319-8_6

Cited by 313 publications

(384 citation statements)

References 104 publications

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“…On the same observables, we have studied the autocorrelation times at different time steps, with the windowing method [8]. We build the connected autocorrelation function Finally, we present a rough quantitative comparison of the relative speed of the two algorithms used for these simulations (namely, HMD and the Lüscher local bosonic one adopted in [3]); an extensive study of the scaling of the autocorrelation times with the different parameters relevant to the simulations has yet to be performed and, moreover, it is likely that further improvements can be implemented.…”

Section: Preliminary Resultsmentioning

confidence: 99%

Hybrid Molecular Dynamics for lattice supersymmetry

Donini

Guagnelli

1996

Physics Letters B

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Section: Preliminary Resultsmentioning

confidence: 99%

Hybrid Molecular Dynamics for lattice supersymmetry

Donini

Guagnelli

1996

Physics Letters B

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“…. λ K > -1 are the eigenvalues of P and columns of E are the corresponding right eigenvectors, normalized so that E T BE = I (9,19,20). We calculate the eigenvalues and eigenvectors of P as follows.…”

Section: Methodsmentioning

confidence: 99%

Searching for efficient Markov chain Monte Carlo proposal kernels

Yang

Rodríguez

2013

Proc. Natl. Acad. Sci. U.S.A.

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Markov chain Monte Carlo (MCMC) or the Metropolis-Hastings algorithm is a simulation algorithm that has made modern Bayesian statistical inference possible. Nevertheless, the efficiency of different Metropolis-Hastings proposal kernels has rarely been studied except for the Gaussian proposal. Here we propose a unique class of Bactrian kernels, which avoid proposing values that are very close to the current value, and compare their efficiency with a number of proposals for simulating different target distributions, with efficiency measured by the asymptotic variance of a parameter estimate. The uniform kernel is found to be more efficient than the Gaussian kernel, whereas the Bactrian kernel is even better. When optimal scales are used for both, the Bactrian kernel is at least 50% more efficient than the Gaussian. Implementation in a Bayesian program for molecular clock dating confirms the general applicability of our results to generic MCMC algorithms. Our results refute a previous claim that all proposals had nearly identical performance and will prompt further research into efficient MCMC proposals.Bayesian inference | mixing | convergence rate M arkov chain Monte Carlo (MCMC) algorithms can be used to simulate a probability distribution π(x) that is known only up to a factor, that is, with only πðyÞ πðxÞ known; they are especially important in Bayesian inference where π is the posterior distribution. In a Metropolis-Hastings (MH) algorithm (1, 2), a proposal density q(yjx), with x, y ∈ χ, is used to generate a new state y given the current state x. The proposal is accepted with probability α(x, y). If the proposal is accepted, the new state becomes y; otherwise it stays at x. The algorithm generates a discrete-time Markov chain with state space χ and transition law P having transition probability density pðx; yÞ = qð yjxÞ · αðx; yÞ; y ≠ x; 1 − Z χ qðyjxÞ · αðx; yÞdy; y = x:The acceptance probability α is chosen so that the detailed balance condition is satisfied: π(x)p(x, y) = π(y)p(y, x), for all x, y ∈ χ. The MH choice of α is αðx; yÞ = min & 1; πðyÞ πðxÞ × qðxjyÞ qðyjxÞThe proposal kernel q(yjx) can be very general; as long as it specifies an irreducible aperiodic Markov chain, the algorithm will generate a reversible Markov chain with stationary distribution π. Here we assume that χ is a subset of R k and both π(y) and q(yjx) are densities on χ.Given a sample x 1 , x 2 , . . ., x n simulated from P, the expectation of any function f(x) over πcan be approximated by the time average over the sampleĨThis is an unbiased estimate of I, and converges to I according to the central limit theorem, independent of the initial state x 0 (ref. 3, p. 99). The asymptotic variance ofĨ iswhere V f = var π {f(X)} is the variance of f(X) over π, ρ k = corr{f(x i ), f(x i + k )} is the lag-k autocorrelation (ref. 4, pp. 87-92), and the variance ratio E = V f /ν is the efficiency: an MCMC sample of size N is as informative about I as an independent sample of size NE. Thus, NE is known as the effective sample size. Given π an...

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“…As only local moves are applied, consecutive conformations are highly correlated. To determine the relaxation time of a ring polymer, we evaluated the autocorrelation function C(t) 28 of the squared radius of gyration A(t) = R 2 g (t) and from C(t) we estimated the integrated autocorrelation time τ using Sokal's windowing procedure 14,29 . Two subsequent conformations are considered independent after 5τ Monte Carlo steps (MCS).…”

Section: B Monte-carlo Algorithmmentioning

confidence: 99%

Topological interactions between ring polymers: Implications for chromatin loops

Bohn

Heermann²

2010

The Journal of Chemical Physics

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Chromatin looping is a major epigenetic regulatory mechanism in higher eukaryotes. Besides its role in transcriptional regulation, chromatin loops have been proposed to play a pivotal role in the segregation of entire chromosomes. The detailed topological and entropic forces between loops still remain elusive. Here, we quantitatively determine the potential of mean force between the centers of mass of two ring polymers, i.e. loops. We find that the transition from a linear to a ring polymer induces a strong increase in the entropic repulsion between these two polymers. On top, topological interactions such as the non-catenation constraint further reduce the number of accessible conformations of close-by ring polymers by about 50%, resulting in an additional effective repulsion. Furthermore, the transition from linear to ring polymers displays changes in the conformational and structural properties of the system. In fact, ring polymers adopt a markedly more ordered and aligned state than linear ones. The forces and accompanying changes in shape and alignment between ring polymers suggest an important regulatory function of such a topology in biopolymers. We conjecture that dynamic loop formation in chromatin might act as a versatile control mechanism regulating and maintaining different local states of compaction and order.

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Monte Carlo Methods in Statistical Mechanics: Foundations and New Algorithms

Cited by 313 publications

References 104 publications

Hybrid Molecular Dynamics for lattice supersymmetry

Hybrid Molecular Dynamics for lattice supersymmetry

Searching for efficient Markov chain Monte Carlo proposal kernels

Topological interactions between ring polymers: Implications for chromatin loops

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