Improving Stochastic Relaxation for Gussian Random Fields

Barone, Piero; Frigessi, Arnoldo

doi:10.1017/s0269964800001674

Cited by 41 publications

(42 citation statements)

References 10 publications

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“…An early reference for variance reduction for Markov chain samplers is Green and Han (1992), who exploited an idea of Barone and Frigessi (1989) and constructed antithetic variables that may achieve variance reduction in simple settings but do not appear to be widely applicable. Andradòttir et al.…”

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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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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“…Let

ϑ = {(ϑ_{1}, \dots, ϑ_{n})}^{normalT}

be a parameter vector with normal full conditional distributions

ϑ_{i} | {bold-italicϑ}_{- i} \sim N false(μ_{i}, σ_{i}^{2} false)

, where the conditional mean μ i and the conditional variance

σ_{i}^{2}

may depend on

{bold-italicϑ}_{- i} = false{ϑ_{j} : j = 1, \dots, n, j \neq i false}

. Adler () and Barone and Frigessi () introduced an overrelaxation method where the update on ϑ is performed by using Gibbs sampling, and where the new value

ϑ_{i}^{'}

for each margin of ϑ is generated as

ϑ_{i}^{'} = false(1 + κ false) μ_{i} - κ ϑ_{i} + u σ_{i} \sqrt false(1 - κ^{2} false), i = 1, \dots, n,

with

u \sim N (0, 1)

being a standard normal random variable. Equation enables the introduction of dependence between successive samples via the constant antithetic parameter κ , which is required to be in the open interval (−1,1) so that the Markov chain is ergodic and produces π ( ϑ ) as its stationary distribution.…”

Section: A Deterministic Proposal Distributionmentioning

confidence: 99%

“…Variance reduction in estimating

E [f (ϑ)]

is achieved through the antithetic variable method (Hammersley and Morton, ) by setting κ >0 so that the estimation bias in the previous sample is corrected in the opposite direction. The rate of convergence for the overrelaxation method in expression (11) was studied in Barone and Frigessi (), whereas Green and Han () established that the asymptotic variance of the estimator for

E [f (ϑ)]

by using this strategy for linear f is proportional to (1− κ )/(1+ κ ).…”

Section: A Deterministic Proposal Distributionmentioning

confidence: 99%

“…Although the chain update proposal is deterministic, the convergence properties are not compromised when this is embedded within a larger system of MCMC sampling. Our proposed methodology is motivated by the overrelaxation algorithm (Adler, 1981;Barone and Frigessi, 1990) and is similar to the idea that is built within the framework of HMC sampling in Pakman and Paninski (2014). However, our proposed sampler is different from these methods in two main respects.…”

Section: Introductionmentioning

confidence: 99%

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Efficient Data Augmentation for Multivariate Probit Models with Panel Data: An Application to General Practitioner Decision Making about Contraceptives

Chin

Gunawan

Fiebig

et al. 2020

Journal of the Royal Statistical Society Series C: Applied Statistics

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Summary The paper considers the problem of estimating a multivariate probit model in a panel data setting with emphasis on sampling a high dimensional correlation matrix and improving the overall efficiency of the data augmentation approach. We reparameterize the correlation matrix in a principled way and then carry out efficient Bayesian inference by using Hamiltonian Monte Carlo sampling. We also propose a novel antithetic variable method to generate samples from the posterior distribution of the random effects and regression coefficients, resulting in significant gains in efficiency. We apply the methodology by analysing stated preference data obtained from Australian general practitioners evaluating alternative contraceptive products. Our analysis suggests that the joint probability of discussing combinations of contraceptive products with a patient shows medical practice variation among the general practitioners, which indicates some resistance even to discuss these products, let alone to recommend them.

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“…Their behaviors are far from well understood [2,5,6,7,12,15]. In this paper we shall consider the Gibbs sampler in the Euclidean space.…”

Section: Introductionmentioning

confidence: 99%

On the Geometrical Convergence of Gibbs Sampler inRd

Hwang

Sheu

1998

Journal of Multivariate Analysis

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The geometrical convergence of the Gibbs sampler for simulating a probability distribution in R d is proved. The distribution has a density which is a bounded perturbation of a log-concave function and satisfies some growth conditions. The analysis is based on a representation of the Gibbs sampler and some powerful results from the theory of Harris recurrent Markov chains. Academic PressAMS 1991 subject classifications: 62E25, 65C05, 82B31.

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Improving Stochastic Relaxation for Gussian Random Fields

Cited by 41 publications

References 10 publications

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

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

Efficient Data Augmentation for Multivariate Probit Models with Panel Data: An Application to General Practitioner Decision Making about Contraceptives

On the Geometrical Convergence of Gibbs Sampler inRd

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