Using the Bayesian Framework to Combine Simulations and Physical Observations for Statistical Inference

Higdon, David; Heitmann, Katrin; Lawrence, Earl; Habib, Salman

doi:10.1002/9780470685853.ch5

Cited by 4 publications

(3 citation statements)

References 17 publications

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“…The statistical model did not account for systematic model bias. Our approach is consistent with the mathematical formulation of solving large scale inverse problems using computer models and observed data [see, e.g., the cosmic microwave background application in Higdon et al (2011)]. With only one observed geomagnetic storm, model bias is confounded with the residual process; with multiple storms we could potentially include a full bias term across space and time.…”

Section: Results and Sequential Designmentioning

confidence: 73%

Parameter tuning for a multi-fidelity dynamical model of the magnetosphere

Kleiber¹,

Sain²,

Heaton³

et al. 2013

Ann. Appl. Stat.

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Geomagnetic storms play a critical role in space weather physics with the potential for far reaching economic impacts including power grid outages, air traffic rerouting, satellite damage and GPS disruption. The LFM-MIX is a state-of-the-art coupled magnetosphericionospheric model capable of simulating geomagnetic storms. Imbedded in this model are physical equations for turning the magnetohydrodynamic state parameters into energy and flux of electrons entering the ionosphere, involving a set of input parameters. The exact values of these input parameters in the model are unknown, and we seek to quantify the uncertainty about these parameters when model output is compared to observations. The model is available at different fidelities: a lower fidelity which is faster to run, and a higher fidelity but more computationally intense version. Model output and observational data are large spatiotemporal systems; the traditional design and analysis of computer experiments is unable to cope with such large data sets that involve multiple fidelities of model output. We develop an approach to this inverse problem for large spatiotemporal data sets that incorporates two different versions of the physical model. After an initial design, we propose a sequential design based on expected improvement. For the LFM-MIX, the additional run suggested by expected improvement diminishes posterior uncertainty by ruling out a posterior mode and shrinking the width of the posterior distribution. We also illustrate our approach using the Lorenz '96 system of equations for a simplified atmosphere, using known input parameters. For the Lorenz '96 system, after performing sequential runs based on expected improvement, the posterior mode converges to the true value and the posterior variability is reduced.

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Section: Results and Sequential Designmentioning

confidence: 73%

Parameter tuning for a multi-fidelity dynamical model of the magnetosphere

Kleiber¹,

Sain²,

Heaton³

et al. 2013

Ann. Appl. Stat.

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“…n is assumed normally distributed about each mode with a single variance r 2 (i.e., n is the same for every mode) and represents the combined uncertainty in the model and the measurements. [38] While it is unlikely that the noise is actually the same for every mode, Section III-E-3 discusses a posteriori methods by which this and other assumptions are deemed acceptable. The formulation of Eq.…”

Section: Building a Statistical Rus Modelmentioning

confidence: 99%

“…[5] means that the resonance modes produced by either the forward model or by measurement are themselves random variables. [16,38] Thus, a statement of the conditional probability of measuring a set of modes M from a specimen with a fixed set of parameters h may be written in terms of the forward model f as PðMjhÞ $ Nðf ðhÞ; rÞ; ½6…”

Section: Building a Statistical Rus Modelmentioning

confidence: 99%

Elastic Properties of Novel Co- and CoNi-Based Superalloys Determined through Bayesian Inference and Resonant Ultrasound Spectroscopy

Goodlet

Mills

Bales

et al. 2018

Metall Mater Trans A

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Bayesian inference is employed to precisely evaluate single crystal elastic properties of novel c À c 0 Co-and CoNi-based superalloys from simple and non-destructive resonant ultrasound spectroscopy (RUS) measurements. Nine alloys from three Co-, CoNi-, and Ni-based alloy classes were evaluated in the fully aged condition, with one alloy per class also evaluated in the solution heat-treated condition. Comparisons are made between the elastic properties of the three alloy classes and among the alloys of a single class, with the following trends observed. A monotonic rise in the c 44 (shear) elastic constant by a total of 12 pct is observed between the three alloy classes as Co is substituted for Ni. Elastic anisotropy (A) is also increased, with a large majority of the nearly 13 pct increase occurring after Co becomes the dominant constituent. Together the five CoNi alloys, with Co:Ni ratios from 1:1 to 1.5:1, exhibited remarkably similar properties with an average A 1.8 pct greater than the Ni-based alloy CMSX-4. Custom code demonstrating a substantial advance over previously reported methods for RUS inversion is also reported here for the first time. CmdStan-RUS is built upon the open-source probabilistic programing language of Stan and formulates the inverse problem using Bayesian methods. Bayesian posterior distributions are efficiently computed with Hamiltonian Monte Carlo (HMC), while initial parameterization is randomly generated from weakly informative prior distributions. Remarkably robust convergence behavior is demonstrated across multiple independent HMC chains in spite of initial parameterization often very far from actual parameter values. Experimental procedures are substantially simplified by allowing any arbitrary misorientation between the specimen and crystal axes, as elastic properties and misorientation are estimated simultaneously.

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Fast inversion of a flexible regression model for multivariate pollen counts data

Salter‐Townshend

Haslett

2012

Environmetrics

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We introduce a rich class of models π(Y | C,θ) for multivariate constrained, zero‐inflated counts data. We use new methodology for fast Bayesian inference on unobserved θ given matched training data {}()falsemml-underliney̲iMathClass-punc,falsemml-underlinec̲iMathClass-punc;iMathClass-rel=1MathClass-punc,MathClass-op…MathClass-punc,7742 and propose a new algorithm for fast inversion to π(C | Y). Such models arise in palaeoclimate reconstruction, where Y represents vector of counts of a proxy such as pollen and C represents climate. Subsequently, falsemml-underliney̲f is a vector of counts in a sample reflecting ancient pollen rain and thus ancient climate (the palaeoclimate). The methodology applies in principle to palaeoclimate reconstruction from other proxy types (e.g. chironomids, diatoms, testate amoebae). Furthermore, the generic issue—the statistical inversion of a multivariate relationship—is found in many areas of application (e.g. clustering, supervised classification, medical imaging, oil shale modelling). The contribution of the paper is the application of a Nested‐Dirichlet–Multinomial model in conjunction with models for zero inflation and non‐parametric response surface modelling. We present the models and the tools to perform fast Bayesian inference on the forward problem and inversion, by using integrated nested Laplace approximations. We demonstrate the increase in precision associated with including more climate proxies and the increase in error associated with doing this naïvely. Copyright © 2012 John Wiley & Sons, Ltd.

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Using the Bayesian Framework to Combine Simulations and Physical Observations for Statistical Inference

Cited by 4 publications

References 17 publications

Parameter tuning for a multi-fidelity dynamical model of the magnetosphere

Parameter tuning for a multi-fidelity dynamical model of the magnetosphere

Elastic Properties of Novel Co- and CoNi-Based Superalloys Determined through Bayesian Inference and Resonant Ultrasound Spectroscopy

Fast inversion of a flexible regression model for multivariate pollen counts data

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