Comment on Article by Chkrebtii, Campbell, Calderhead, and Girolami

Mallick, Bani K.; Yang, Keren; Guha, Nilabja; Efendiev, Yalchin

doi:10.1214/16-ba1038

Cited by 4 publications

(8 citation statements)

References 14 publications

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“…To overcome this issue, Mallick et al (2016) propose a hybrid method that uses a combination of numerical and probabilistic models on different scales. We discuss this strategy in the following section on prior specification.…”

Section: Uncertainty Quantification For Partial Differential Equationsmentioning

confidence: 99%

“…The discussions of Yoo (2016) and Mallick et al (2016) provide an alternative and flexible way of defining the prior process jointly on u and any partial derivatives via basis expansion with Gaussian priors on the coefficients. Although Gaussian processes can also be written in spectral form, the eigenfunctions of the covariance lack the interpretability of B-spline bases, and are difficult to adapt to different resolutions.…”

Section: Prior Specificationmentioning

confidence: 99%

“…The use of numerical output within a UQ algorithm (e.g., Dass, 2016) requires the conversion of numerical error bounds to probability statements, and may result in the loss of the functional structure of the error. Mallick et al (2016) suggest an interesting compromise between discrete approximations and probabilistic methods for multiscale models. They propose to recover large-scale "resolved" dynamics numerically, while the uncertainty in the smaller-scale "unresolved" dynamics, which is typically much greater, can be characterized through Gaussian process prior updating.…”

Section: Relationship To Numerical Solversmentioning

confidence: 99%

“…Dass (2016) and Mallick et al (2016) pose an important question: is posterior coverage guaranteed in the limit? Intuitively, one would suppose that if the uncertainty quantification in the forward model does not underestimate discretization uncertainty, then this should be the case.…”

Section: Calibrationmentioning

confidence: 99%

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Chkrebtii¹,

Campbell²,

Calderhead³

2016

Bayesian Anal.

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Section: Uncertainty Quantification For Partial Differential Equationsmentioning

confidence: 99%

Section: Prior Specificationmentioning

confidence: 99%

Section: Relationship To Numerical Solversmentioning

confidence: 99%

Section: Calibrationmentioning

confidence: 99%

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Rejoinder

Chkrebtii¹,

Campbell²,

Calderhead³

2016

Bayesian Anal.

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“…A detailed comparison of several sampling strategies for LGMs in Filippone et al (2013) showed that the single block updating strategy of Knorr-Held and Rue (2002) has larger effective sample size compared to sufficient augmentation, ancillary augmentation and ancillarity-sufficiency interweaving strategy (Yu and Meng, 2011), and the surrogate method (Murray and Adams, 2010). Another approach to infer LGMs was proposed by Filippone and Girolami (2014), who suggested using a pseudomarginal sampling procedure for the marginal posterior density of the hyperparameters, which relies on the Metropolis-Hastings algorithm and importance sampling. Essentially, samples from the marginal posterior density of the hyperparameters are obtained first, and the latent parameters are sampled from the conditional posterior density of the latent parameters.…”

Section: Introductionmentioning

confidence: 99%

Max-and-Smooth: A Two-Step Approach for Approximate Bayesian Inference in Latent Gaussian Models

et al. 2021

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With modern high-dimensional data, complex statistical models are necessary, requiring computationally feasible inference schemes. We introduce Max-and-Smooth, an approximate Bayesian inference scheme for a flexible class of latent Gaussian models (LGMs) where one or more of the likelihood parameters are modeled by latent additive Gaussian processes. Our proposed inference scheme is a two-step approach. In the first step (Max), the likelihood function is approximated by a Gaussian density with mean and covariance equal to either (a) the maximum likelihood estimate and the inverse observed information, respectively, or (b) the mean and covariance of the normalized likelihood function. In the second step (Smooth), the latent parameters and hyperparameters are inferred and smoothed with the approximated likelihood function. The proposed method ensures that the uncertainty from the first step is correctly propagated to the second step. Because the prior density for the latent parameters is assumed to be Gaussian and the approximated likelihood function is Gaussian, the approximate posterior density of the latent parameters (conditional on the hyperparameters) is also Gaussian, thus facilitating efficient posterior inference in high dimensions. Furthermore, the approximate marginal posterior distribution of the hyperparameters is tractable, and as a result, the hyperparameters can be sampled independently of the latent parameters. We show that the computational cost of Max-and-Smooth is close to being insensitive to the number of independent data replicates, and that it scales well with increased dimension of the latent parameter vector provided that its Gaussian prior density is specified with a sparse precision matrix. In the case of a large number of independent data replicates, sparse precision matrices, and high-dimensional latent vectors, the speedup is substantial in comparison to an MCMC scheme that infers the posterior density from the exact likelihood function. The accuracy of the Gaussian approximation to the likelihood function increases with the number of data replicates per latent model parameter. The proposed inference scheme is demonstrated on one spatially referenced real dataset and on simulated data mimicking spatial, temporal, and spatio-temporal inference problems. Our results show that Max-and-Smooth is accurate and fast.

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Dynamic data-driven Bayesian GMsFEM

Cheung

Guha

2019

Journal of Computational and Applied Mathematics

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In this paper, we propose a Bayesian approach for multiscale problems with the availability of dynamic observational data. Our method selects important degrees of freedom probabilistically in a Generalized multiscale finite element method framework. Due to scale disparity in many multiscale applications, computational models can not resolve all scales. Dominant modes in the Generalized Multiscale Finite Element Method are used as "permanent" basis functions, which we use to compute an inexpensive multiscale solution and the associated uncertainties. Through our Bayesian framework, we can model approximate solutions by selecting the unresolved scales probabilistically. We consider parabolic equations in heterogeneous media. The temporal domain is partitioned into subintervals. Using residual information and given dynamic data, we design appropriate prior distribution for modeling missing subgrid information. The likelihood is designed to minimize the residual in the underlying PDE problem and the mismatch of observational data. Using the resultant posterior distribution, the sampling process identifies important degrees of freedom beyond permanent basis functions. The method adds important degrees of freedom in resolving subgrid information and ensuring the accuracy of the observations.

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Comment on Article by Chkrebtii, Campbell, Calderhead, and Girolami

Cited by 4 publications

References 14 publications

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Max-and-Smooth: A Two-Step Approach for Approximate Bayesian Inference in Latent Gaussian Models

Dynamic data-driven Bayesian GMsFEM

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