Dimension-independent likelihood-informed MCMC

Cui, Tiangang; Law, Kody J. H.; Marzouk, Youssef

doi:10.1016/j.jcp.2015.10.008

Cited by 185 publications

(259 citation statements)

References 44 publications

(152 reference statements)

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“…This perspective links the marginal sampler to some recent work in the inverse problem literature that improves on pCN, such as Law () and Cui et al . (), which we review below.…”

Section: Gradient‐based Samplers For Latent Gaussian Modelsmentioning

confidence: 99%

Auxiliary Gradient-Based Sampling Algorithms

Titsias

Papaspiliopoulos

2018

Journal of the Royal Statistical Society Series B: Statistical Methodology

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Summary We introduce a new family of Markov chain Monte Carlo samplers that combine auxiliary variables, Gibbs sampling and Taylor expansions of the target density. Our approach permits the marginalization over the auxiliary variables, yielding marginal samplers, or the augmentation of the auxiliary variables, yielding auxiliary samplers. The well‐known Metropolis‐adjusted Langevin algorithm MALA and preconditioned Crank–Nicolson–Langevin algorithm pCNL are shown to be special cases. We prove that marginal samplers are superior in terms of asymptotic variance and demonstrate cases where they are slower in computing time compared with auxiliary samplers. In the context of latent Gaussian models we propose new auxiliary and marginal samplers whose implementation requires a single tuning parameter, which can be found automatically during the transient phase. Extensive experimentation shows that the increase in efficiency (measured as the effective sample size per unit of computing time) relative to (optimized implementations of) pCNL, elliptical slice sampling and MALA ranges from tenfold in binary classification problems to 25 fold in log‐Gaussian Cox processes to 100 fold in Gaussian process regression, and it is on a par with Riemann manifold Hamiltonian Monte Carlo sampling in an example where that algorithm has the same complexity as the aforementioned algorithms. We explain this remarkable improvement in terms of the way that alternative samplers try to approximate the eigenvalues of the target. We introduce a novel Markov chain Monte Carlo sampling scheme for hyperparameter learning that builds on the auxiliary samplers. The MATLAB code for reproducing the experiments in the paper is publicly available and an on‐line supplement to this paper contains additional experiments and implementation details.

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“…This perspective links the marginal sampler to some recent work in the inverse problem literature that improves on pCN, such as Law () and Cui et al . (), which we review below.…”

Section: Gradient‐based Samplers For Latent Gaussian Modelsmentioning

confidence: 99%

Auxiliary Gradient-Based Sampling Algorithms

Titsias

Papaspiliopoulos

2018

Journal of the Royal Statistical Society Series B: Statistical Methodology

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“…An important task in practical MCMC is identifying good proposal distributions that can minimize autocorrelation. As the forward model used in our case studies does not have adjoint capabilities, advanced MCMC proposals, such as the stochastic Newton and the likelihood‐informed dimension‐independent proposal that rely on the derivatives of the posterior, cannot be used here. We consider the Gaussian random‐walk proposal

q (x, \cdot) = N (x, σ^{2} 𝚺),

where Σ is some covariance matrix and σ is a scalar that dictates the jump size.…”

Section: Bayesian Formulation and Posterior Explorationmentioning

confidence: 99%

A posteriori stochastic correction of reduced models in delayed‐acceptance MCMC, with application to multiphase subsurface inverse problems

Cui

Fox

O’Sullivan

2019

Numerical Meth Engineering

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Summary Sample‐based Bayesian inference provides a route to uncertainty quantification in the geosciences and inverse problems in general but is very computationally demanding in the naïve form, which requires simulating an accurate computer model at each iteration. We present a new approach that constructs a stochastic correction to the error induced by a reduced model, with the correction improving as the algorithm proceeds. This enables sampling from the correct target distribution at reduced computational cost per iteration, as in existing delayed‐acceptance schemes, while avoiding appreciable loss of statistical efficiency that necessarily occurs when using a reduced model. Use of the stochastic correction significantly reduces the computational cost of estimating quantities of interest within desired uncertainty bounds. In contrast, existing schemes that use a reduced model directly as a surrogate do not actually improve computational efficiency in our target applications. We build on recent simplified conditions for adaptive Markov chain Monte Carlo algorithms to give practical approximation schemes and algorithms with guaranteed convergence. The efficacy of this new approach is demonstrated in two computational examples, including calibration of a large‐scale numerical model of a real geothermal reservoir, that show good computational and statistical efficiencies on both synthetic and measured data sets.

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“…Put in another way, the conditioning lemma is exact for the pushforward of the approximate map, π = T η. Nevertheless, if the conditional density π Θ|D=d can be evaluated up to a normalizing constant, then one can quantify the error in the approximation of the conditional using (24). Under these conditions, if one is not satisfied with this error, then any of the approximate maps T d constructed here could be useful as a proposal mechanism for importance sampling or MCMC, to generate (asymptotically) exact samples from the conditional of interest.…”

Section: Inverse Transport: Map From Samplesmentioning

confidence: 99%

“…Accordingly, enormous efforts have been devoted to the design of improved MCMC and SMC samplers-schemes that generate more nearly independent or unweighted samples. While these efforts are too diverse to summarize easily, they often rest on the design of improved (and structure-exploiting) proposal mechanisms within the algorithms [37,3,26,22,62,53,34,24].…”

Section: Introductionmentioning

confidence: 99%

Sampling via Measure Transport: An Introduction

Marzouk¹,

Moselhy²,

Parno³

et al. 2016

Handbook of Uncertainty Quantification

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We present the fundamentals of a measure transport approach to sampling. The idea is to construct a deterministic coupling-i.e., a transport map-between a complex "target" probability measure of interest and a simpler reference measure. Given a transport map, one can generate arbitrarily many independent and unweighted samples from the target simply by pushing forward reference samples through the map. If the map is endowed with a triangular structure, one can also easily generate samples from conditionals of the target measure. We consider two different and complementary scenarios: first, when only evaluations of the unnormalized target density are available, and second, when the target distribution is known only through a finite collection of samples. We show that in both settings the desired transports can be characterized as the solutions of variational problems. We then address practical issues associated with the optimization-based construction of transports: choosing finite-dimensional parameterizations of the map, enforcing monotonicity, quantifying the error of approximate transports, and refining approximate transports by enriching the corresponding approximation spaces. Approximate transports can also be used to "Gaussianize" complex distributions and thus precondition conventional asymptotically exact sampling schemes. We place the measure transport approach in broader context, describing connections with other optimization-based samplers, with inference and density estimation schemes using optimal transport, and with alternative transformation-based approaches to simulation. We also sketch current work aimed at the construction of transport maps in high dimensions, exploiting essential features of the target distribution (e.g., conditional independence, low-rank structure). The approaches and algorithms presented here have direct applications to Bayesian computation and to broader problems of stochastic simulation.

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Dimension-independent likelihood-informed MCMC

Cited by 185 publications

References 44 publications

Auxiliary Gradient-Based Sampling Algorithms

Auxiliary Gradient-Based Sampling Algorithms

A posteriori stochastic correction of reduced models in delayed‐acceptance MCMC, with application to multiphase subsurface inverse problems

Sampling via Measure Transport: An Introduction

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