Zero variance Markov chain Monte Carlo for Bayesian estimators

Mira, Antonietta; Solgi, Reza; Imparato, Daniele

doi:10.1007/s11222-012-9344-6

Cited by 66 publications

(131 citation statements)

References 43 publications

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“…The results marked (b) echo the conclusions of Mira et al . (), that ZV control variates are effective in many cases where f is well approximated by a low degree polynomial and π is a Gaussian or gamma density. However, when f is not well approximated by a low degree polynomial, or when π takes a more complex form, as in cases marked (c), ZV control variates can be outperformed by CFs, which have the potential to decrease variance dramatically.…”

Section: Methodsmentioning

confidence: 98%

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Control Functionals for Monte Carlo Integration

Oates

Girolami

Chopin

2016

Journal of the Royal Statistical Society Series B: Statistical Methodology

138

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: Methodsmentioning

confidence: 98%

“…Lemma 1 generalizes equation of Mira et al . () and implies that

H_{0}

consists of only valid CFs, i.e.

ψ \in {scriptH}_{0} \Rightarrow μ false(italicψ false) = 0

.…”

Section: Methodsmentioning

confidence: 99%

Control Functionals for Monte Carlo Integration

Oates

Girolami

Chopin

2016

Journal of the Royal Statistical Society Series B: Statistical Methodology

138

217

View full text Add to dashboard Cite

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“…In a different context, and partly motivated by considerations from statistical mechanics, Assaraf and Caffarel (1999) introduced a family of control variates that they called ‘zero‐variance estimators’. The Assaraf–Cafarel estimators have been adapted and applied to problems in statistics by Mira et al. (2003, 2010) and Dalla Valle and Leisen (2010).…”

Section: Introductionmentioning

confidence: 99%

Control Variates for Estimation Based on Reversible Markov Chain Monte Carlo Samplers

Δελλαπόρτας

Kontoyiannis

2011

Journal of the Royal Statistical Society Series B: Statistical Methodology

106

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A general methodology is introduced for the construction and effective application of control variates to estimation problems involving data from reversible Markov chain Monte Carlo samplers. We propose the use of a specific class of functions as control variates, and we introduce a new consistent estimator for the values of the coefficients of the optimal linear combination of these functions. For a specific Markov chain Monte Carlo scenario, the form and proposed construction of the control variates is shown to provide an exact solution of the associated Poisson equation. This implies that the estimation variance in this case (in the central limit theorem regime) is exactly zero. The new estimator is derived from a novel, finite dimensional, explicit representation for the optimal coefficients. The resulting variance reduction methodology is primarily (though certainly not exclusively) applicable when the simulated data are generated by a random-scan Gibbs sampler. Markov chain Monte Carlo examples of Bayesian inference problems demonstrate that the corresponding reduction in the estimation variance is significant, and that in some cases it can be quite dramatic. Extensions of this methodology are discussed and simulation examples are presented illustrating the utility of the methods proposed. All methodological and asymptotic arguments are rigorously justified under essentially minimal conditions.

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“…In vanilla Monte Carlo, this approach is known as control variates and is a useful method for variance reduction by using a known analytical approximation of the target. In the p m h algorithm, we can make use of zv (Mira et al, 2013;Papamarkou et al, 2014) to achieve the same objective as in Example 4.1. In principle, it is straight-forward to directly apply zv approaches for any the pmh algorithms proposed in Papers B-D to further accelerate the inference.…”

Section: Outlook and Extensionsmentioning

confidence: 99%

“…Remember that the p m h 0 algorithm makes use of a random walk proposal such as (3.32), which does not include information about the gradient and Hessian of the target. Furthermore, we make use of a post-processing of the Markov chain known as zero-variance ( zv; Mira et al, 2013;Papamarkou et al, 2014) to decrease the variance in the estimates. The z v approach is based on the idea of control variates in vanilla Monte Carlo sampling, see Robert and Casella (2004).…”

mentioning

confidence: 99%

Accelerating Monte Carlo methods for Bayesian inference in dynamical models

Dahlin

2016

Linköping Studies in Science and Technology. Dissertations

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Cover illustration: A Markov chain generated by the Metropolis-Hastings algorithm with an autoregressive proposal on a manifold given by a parametric function.This thesis was typeset using the L A T E X typesetting system originally developed by Leslie Lamport, based on T E X created by Donald Knuth. The text is set in Garamond and Cabin. The source code is set in Inconso ata. All plots are made using R (R Core together with colors from the RColorBrewer package (Neuwirth, 2014 Denna avhandling tillägnas min familj! AbstractMaking decisions and predictions from noisy observations are two important and challenging problems in many areas of society. Some examples of applications are recommendation systems for online shopping and streaming services, connecting genes with certain diseases and modelling climate change. In this thesis, we make use of Bayesian statistics to construct probabilistic models given prior information and historical data, which can be used for decision support and predictions. The main obstacle with this approach is that it often results in mathematical problems lacking analytical solutions. To cope with this, we make use of statistical simulation algorithms known as Monte Carlo methods to approximate the intractable solution. These methods enjoy well-understood statistical properties but are often computational prohibitive to employ.The main contribution of this thesis is the exploration of different strategies for accelerating inference methods based on sequential Monte Carlo ( smc) and Markov chain Monte Carlo ( mcmc). That is, strategies for reducing the computational effort while keeping or improving the accuracy. A major part of the thesis is devoted to proposing such strategies for the mcmc method known as the particle Metropolis-Hastings ( pmh) algorithm. We investigate two strategies: (i) introducing estimates of the gradient and Hessian of the target to better tailor the algorithm to the problem and (ii) introducing a positive correlation between the point-wise estimates of the target.Furthermore, we propose an algorithm based on the combination of s m c and Gaussian process optimisation, which can provide reasonable estimates of the posterior but with a significant decrease in computational effort compared with pmh. Moreover, we explore the use of sparseness priors for approximate inference in over-parametrised mixed effects models and autoregressive processes. This can potentially be a practical strategy for inference in the big data era. Finally, we propose a general method for increasing the accuracy of the parameter estimates in non-linear state space models by applying a designed input signal.v Populärvetenskaplig sammanfattningBorde Riksbanken höja eller sänka reporäntan vid sitt nästa möte för att nå inflationsmålet? Vilka gener är förknippade med en viss sjukdom? Hur kan Netflix och Spotify veta vilka filmer och vilken musik som jag vill lyssna på härnäst?Dessa tre problem är exempel på frågor där statistiska modeller kan vara användbara för att ge hjälp och underlag för be...

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Zero variance Markov chain Monte Carlo for Bayesian estimators

Cited by 66 publications

References 43 publications

Control Functionals for Monte Carlo Integration

Control Functionals for Monte Carlo Integration

Control Variates for Estimation Based on Reversible Markov Chain Monte Carlo Samplers

Accelerating Monte Carlo methods for Bayesian inference in dynamical models

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