Stability of noisy Metropolis–Hastings

Medina-Aguayo, Felipe J.; Lee, Anthony; Roberts, Gareth O.

doi:10.1007/s11222-015-9604-3

Cited by 32 publications

(37 citation statements)

References 35 publications

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“…6. We use a noisy pseudomarginal MCMC method, whose fast mixing is helpful for these initial experiments (Medina-Aguayo et al 2015). These posteriors are significantly improved, exhibiting greater mutual agreement and obvious increasing concentration with improving solver quality.…”

Section: Bayesian Posterior Inference Problemsmentioning

confidence: 99%

Statistical analysis of differential equations: introducing probability measures on numerical solutions

et al. 2016

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In this paper, we present a formal quantification of uncertainty induced by numerical solutions of ordinary and partial differential equation models. Numerical solutions of differential equations contain inherent uncertainties due to the finite-dimensional approximation of an unknown and implicitly defined function. When statistically analysing models based on differential equations describing physical, or other naturally occurring, phenomena, it can be important to explicitly account for the uncertainty introduced by the numerical method. Doing so enables objective determination of this source of uncertainty, relative to other uncertainties, such as those caused by data contaminated with noise or model error induced by missing physical or inadequate descriptors. As ever larger scale mathematical models are being used in the sciences, often sacrificing complete resolution of the differential equation on the grids used, formally accounting for the uncertainty in the numerical method is becoming increasingly more important. This paper provides the formal means to incorporate this uncer- School of Mathematics, University of Edinburgh, Edinburgh, Scotland tainty in a statistical model and its subsequent analysis. We show that a wide variety of existing solvers can be randomised, inducing a probability measure over the solutions of such differential equations. These measures exhibit contraction to a Dirac measure around the true unknown solution, where the rates of convergence are consistent with the underlying deterministic numerical method. Furthermore, we employ the method of modified equations to demonstrate enhanced rates of convergence to stochastic perturbations of the original deterministic problem. Ordinary differential equations and elliptic partial differential equations are used to illustrate the approach to quantify uncertainty in both the statistical analysis of the forward and inverse problems.

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Section: Bayesian Posterior Inference Problemsmentioning

confidence: 99%

Statistical analysis of differential equations: introducing probability measures on numerical solutions

et al. 2016

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“…Doucet et al (2015) observe that this limiting distribution is a good fit for the noise, even for modest values of T and N . The log-normal CLT is very useful for theoretical analysis of PMCMC algorithms as shown, for example, by Doucet et al (2015) and Medina-Aguayo et al (2016). We assume log-normality ofp(y|θ) in our GP model.…”

Section: Pseudo-marginal Mcmcmentioning

confidence: 99%

“…However, the drawback is that the limiting distribution of MCWM is only the posterior p(θ|y) in the limit as N → ∞. Medina-Aguayo et al (2016) develop sufficient conditions for the geometric ergodicity and hence the existence of an invariant distribution of MCWM for large enough N . The conditions are that the idealised chain (the chain that uses exact likelihood evaluations) is geometrically ergodic, the weights W are uniformly integrable and the weights satisfy uniform exponential bounds on their densities close to 0.…”

Section: Pseudo-marginal Mcmcmentioning

confidence: 99%

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Accelerating pseudo-marginal MCMC using Gaussian processes

Drovandi

Moores

Boys

2018

Computational Statistics & Data Analysis

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The grouped independence Metropolis-Hastings (GIMH) and Markov chain within Metropolis (MCWM) algorithms are pseudo-marginal methods used to perform Bayesian inference in latent variable models. These methods replace intractable likelihood calculations with unbiased estimates within Markov chain Monte Carlo algorithms. The GIMH method has the posterior of interest as its limiting distribution, but suffers from poor mixing if it is too computationally intensive to obtain high-precision likelihood estimates. The MCWM algorithm has better mixing properties, but tends to give conservative approximations of the posterior and is still expensive. A new method is developed to accelerate the GIMH method by using a Gaussian process (GP) approximation to the log-likelihood and train this GP using a short pilot run of the MCWM algorithm. This new method called GP-GIMH is illustrated on simulated data from a stochastic volatility and a gene network model. The new approach produces reasonable posterior approximations in these examples with at least an order of magnitude improvement in computing time. Code to implement the method for the gene network example can be found at http://www.runmycode.org/companion/view/2663.

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“…∼ r y . The resulting MCwM algorithm has been studied before in [AR09,MLR16]. Let us note here that this MCwM approach should not be confused with the pseudo-marginal method, see [AR09].…”

Section: Introductionmentioning

confidence: 99%

Perturbation bounds for Monte Carlo within Metropolis via restricted approximations

Medina-Aguayo

Rudolf²,

Schweizer

2020

Stochastic Processes and their Applications

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The Monte Carlo within Metropolis (MCwM) algorithm, interpreted as a perturbed Metropolis-Hastings (MH) algorithm, provides an approach for approximate sampling when the target distribution is intractable. Assuming the unperturbed Markov chain is geometrically ergodic, we show explicit estimates of the difference between the nth step distributions of the perturbed MCwM and the unperturbed MH chains. These bounds are based on novel perturbation results for Markov chains which are of interest beyond the MCwM setting. To apply the bounds, we need to control the difference between the transition probabilities of the two chains and to verify stability of the perturbed chain.

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Stability of noisy Metropolis–Hastings

Cited by 32 publications

References 35 publications

Statistical analysis of differential equations: introducing probability measures on numerical solutions

Statistical analysis of differential equations: introducing probability measures on numerical solutions

Accelerating pseudo-marginal MCMC using Gaussian processes

Perturbation bounds for Monte Carlo within Metropolis via restricted approximations

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