On the Poisson equation for Metropolis–Hastings chains

Mijatović, Aleksandar; Vogrinc, Jure

doi:10.3150/17-bej932

Cited by 25 publications

(31 citation statements)

References 29 publications

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“…As such they achieve at most a constant factor reduction in estimator variance. Intuitively, one would like to increase the number m of basis functions to increase in line with the number n. Mijatović and Vogrinc (2015) explored this approach within the Metropolis-Hastings method. However, their solution requires the user to partition the state space, which limits its wider appeal.…”

Section: Introductionmentioning

confidence: 99%

Control Functionals for Monte Carlo Integration

Oates

Girolami

Chopin

2016

Journal of the Royal Statistical Society Series B: Statistical Methodology

141

217

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A non-parametric extension of control variates is presented.These leverage gradient information on the sampling density to achieve substantial variance reduction. It is not required that the sampling density be normalized. The novel contribution of this work is based on two important insights: a trade-off between random sampling and deterministic approximation and a new gradient-based function space derived from Stein's identity. Unlike classical control variates, our estimators improve rates of convergence, often requiring orders of magnitude fewer simulations to achieve a fixed level of precision. Theoretical and empirical results are presented, the latter focusing on integration problems arising in hierarchical models and models based on non-linear ordinary differential equations.

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Section: Introductionmentioning

confidence: 99%

Control Functionals for Monte Carlo Integration

Oates

Girolami

Chopin

2016

Journal of the Royal Statistical Society Series B: Statistical Methodology

141

217

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“…In order to illustrate the construction from Remark 26, let g : (R d × R d ) → R d×d be a field of orthogonal matrices (see (62)) and put…”

Section: The General Casementioning

confidence: 99%

Constructing Sampling Schemes via Coupling: Markov Semigroups and Optimal Transport

Nüsken¹,

Pavliotis²

2019

SIAM/ASA J. Uncertainty Quantification

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In this paper we develop a general framework for constructing and analysing coupled Markov chain Monte Carlo samplers, allowing for both (possibly degenerate) diffusion and piecewise deterministic Markov processes. For many performance criteria of interest, including the asymptotic variance, the task of finding efficient couplings can be phrased in terms of problems related to optimal transport theory. We investigate general structural properties, proving a singularity theorem that has both geometric and probabilistic interpretations. Moreover, we show that those problems can often be solved approximately and support our findings with numerical experiments. For the particular objective of estimating the variance of a Bayesian posterior, our analysis suggests using novel techniques in the spirit of antithetic variates. Addressing the convergence to equilibrium of coupled processes we furthermore derive a modified Poincaré inequality.

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“…This idea is motivated by the observation that the methods listed above are (in effect) solving a misspecified regression problem, since in general f does not belong to the linear span of the statistics {ψ i } k i=1 . The recent work by Mijatović and Vogrinc (2015); Oates et al (2017) alleviates model misspecification by increasing the number k of statistics alongside the number n of samples so that the limiting space spanned by the statistics {ψ i } ∞ i=1 is dense in a class of functions that contains the test function f of interest. Both methods provide a non-parametric alternative to classical control variates whose error is o P (n − 1 2 ).…”

Section: Introductionmentioning

confidence: 99%

“…Both methods provide a non-parametric alternative to classical control variates whose error is o P (n − 1 2 ). Of these two proposed solutions, Mijatović and Vogrinc (2015) is not considered here since it is unclear how to proceed when Π is known only up to a normalisation constant. On the other hand the control functional method of Oates et al (2017) is straight-forward to implement when gradients {∇ log π(x i )} n i=1 are provided.…”

Section: Introductionmentioning

confidence: 99%

Convergence rates for a class of estimators based on Stein’s method

Oates¹,

Cockayne²,

Briol³

et al. 2019

Bernoulli

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Gradient information on the sampling distribution can be used to reduce the variance of Monte Carlo estimators via Stein's method. An important application is that of estimating an expectation of a test function along the sample path of a Markov chain, where gradient information enables convergence rate improvement at the cost of a linear system which must be solved. The contribution of this paper is to establish theoretical bounds on convergence rates for a class of estimators based on Stein's method. Our analysis accounts for (i) the degree of smoothness of the sampling distribution and test function, (ii) the dimension of the state space, and (iii) the case of non-independent samples arising from a Markov chain. These results provide insight into the rapid convergence of gradient-based estimators observed for low-dimensional problems, as well as clarifying a curse-of-dimension that appears inherent to such methods.

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On the Poisson equation for Metropolis–Hastings chains

Cited by 25 publications

References 29 publications

Control Functionals for Monte Carlo Integration

Control Functionals for Monte Carlo Integration

Constructing Sampling Schemes via Coupling: Markov Semigroups and Optimal Transport

Convergence rates for a class of estimators based on Stein’s method

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