Fixed Precision MCMC Estimation by Median of Products of Averages

Niemiro, Wojciech; Pokarowski, Piotr

doi:10.1239/jap/1245676089

Cited by 17 publications

(29 citation statements)

References 33 publications

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“…It turns out that the estimates are sharp depending on the available information of the eigenvalues. Similar estimates can be found in [Ald87] and [NP09]. However, the suggestion of the burn-in and the lower bound seem to be new.…”

Section: Introduction and Resultssupporting

confidence: 79%

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Explicit error bounds for Markov chain Monte Carlo

Rudolf

2012

Dissertationes Math.

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We prove explicit, i.e. non-asymptotic, error bounds for Markov chain Monte Carlo methods. The problem is to compute the expectation of a function f with respect to a measure π. Different convergence properties of Markov chains imply different error bounds. For uniformly ergodic and reversible Markov chains we prove a lower and an upper error bound with respect to f 2 . If there exists an L2-spectral gap, which is a weaker convergence property than uniform ergodicity, then we show an upper error bound with respect to f p for p > 2. Usually a burn-in period is an efficient way to tune the algorithm. We provide and justify a recipe how to choose the burn-in period. The error bounds are applied to the problem of the integration with respect to a possibly unnormalized density. More precise, we consider the integration with respect to logconcave densities and the integration over convex bodies. By the use of the Metropolis algorithm based on a ball walk and the hit-and-run algorithm it is shown that both problems are polynomial tractable.

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Section: Introduction and Resultssupporting

confidence: 79%

“…In [NP09] an explicit error bound is published which holds also for non-reversible Markov chains with an absolute ℓ 2 -spectral gap, i.e. β = P ℓ 0 2 →ℓ 0 2 < 1.…”

Section: Examplesmentioning

confidence: 99%

Explicit error bounds for Markov chain Monte Carlo

Rudolf

2012

Dissertationes Math.

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“…The explanatory variables are x ∈ R p the density and z ∈ R p the density adjusted for resin content. i=0 are chosen accordingly to the formulas (34) and (36). For each phase, the step size γ i is set equal to 10 −2 (κ i σ 2 i m i )/(dL 2 i ), the burn-in period N i to 10 3 (κ i γ i ) −1 and the number of samples n i to 10 4 m where m i , L i , κ i are defined in (13) and (14).…”

Section: Numerical Experimentsmentioning

confidence: 99%

Normalizing constants of log-concave densities

Brosse¹,

Durmus²,

Moulines³

2018

Electron. J. Statist.

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We derive explicit bounds for the computation of normalizing constants Z for log-concave densities π = e −U /Z w.r.t. the Lebesgue measure on R d . Our approach relies on a Gaussian annealing combined with recent and precise bounds on the Unadjusted Langevin Algorithm [15]. Polynomial bounds in the dimension d are obtained with an exponent that depends on the assumptions made on U. The algorithm also provides a theoretically grounded choice of the annealing sequence of variances. A numerical experiment supports our findings. Results of independent interest on the mean squared error of the empirical average of locally Lipschitz functions are established.MSC 2010 subject classifications: Primary 65C05, 60F25, 62L10; secondary 65C40, 60J05, 74G10, 74G15.

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“…12.19, p.165] the authors obtained for another error term a comparable bound where the chain starts deterministically. Very recently in [NP09] a similar result concerning the integration error for f ∈ ℓ ∞ was shown where the Markov chain is not necessarily reversible.…”

Section: Main Theoremmentioning

confidence: 58%

Error bounds for computing the expectation by Markov chain Monte Carlo

Rudolf¹

2010

Monte Carlo Methods and Applications

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We study the error of reversible Markov chain Monte Carlo methods for approximating the expectation of a function. Explicit error bounds with respect to different norms of the function are proven. By the estimation the well known asymptotical limit of the error is attained, i.e. there is no gap between the estimate and the asymptotical behavior. We discuss the dependence of the error on a burn-in of the Markov chain. Furthermore we suggest and justify a specific burn-in for optimizing the algorithm

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Fixed Precision MCMC Estimation by Median of Products of Averages

Cited by 17 publications

References 33 publications

Explicit error bounds for Markov chain Monte Carlo

Explicit error bounds for Markov chain Monte Carlo

Normalizing constants of log-concave densities

Error bounds for computing the expectation by Markov chain Monte Carlo

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