On the quantification and efficient propagation of imprecise probabilities resulting from small datasets

Zhang, Jiaxin; Shields, Michael D.

doi:10.1016/j.ymssp.2017.04.042

Cited by 76 publications

(57 citation statements)

References 40 publications

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“…Given scarcity of data, it is often impossible to identify a unique best model so that we need to quantify modelform uncertainty and retain multiple candidate models and their associated probabilities -a method referred as to multimodel inference [40]. The previous work of the authors [23] presented an information-theoretic approach for multimodel inference to quantify and propagate these model-form uncertainties. This work seeks to generalize this in a fully Bayesian framework.…”

Section: Bayesian Multimodel Inference and Model-form Uncertaintymentioning

confidence: 99%

“…(6) may be thought of as an implicit approximation to evidence p(d|M j ) under a noninformative parameter prior (or Jeffreys parameter prior ) even though it does not explicitly depend on a parameter prior. The information-theoretic multimodel selection (introduced in [40] and employed in the authors' previous work [23]) can be shown as a special case of the Bayesian evidence-based multimodel selection used herein. Akaike [33] showed that the maximized log-likelihood is a biased estimator of the K-L information and that the bias is approximately equal to K j .…”

Section: Bayesian Evidence Calculationmentioning

confidence: 99%

“…Because both the BIC and AIC are asymptotic quantities that require large dataset size, they are of limited practical use here. Although a small data correction of the AIC, denoted AICc, has been derived [36,42] and used in our previous work [23], this again implies a certain prior form and our objective here is to investigate the effect of the prior. Consequently, we must rely on other estimators for Eq.…”

Section: Bayesian Evidence Calculationmentioning

confidence: 99%

“…, m), each of which has an associated joint parameter pdf p(θ j |d, M j ). A method for propagation of this complete uncertainty has been previously proposed by the authors [23] and is reviewed in the following section.…”

Section: Bayesian Model Parameter Estimationmentioning

confidence: 99%

“…Sections 2 and 3 provide the basic theory for multimodel Bayesian uncertainty quantification for model-form (Section 2) and parameter uncertainties (Section 3). Section 4 reviews a new Monte Carlo-based method for multimodel uncertainty propagation proposed in a recent work by the authors [23]. Some minor modifications are included.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

The effect of prior probabilities on quantification and propagation of imprecise probabilities resulting from small datasets

Zhang

Shields

2018

Computer Methods in Applied Mechanics and Engineering

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This paper outlines a methodology for Bayesian multimodel uncertainty quantification (UQ) and propagation and presents an investigation into the effect of prior probabilities on the resulting uncertainties. The UQ methodology is adapted from the information-theoretic method previously presented by the authors (Zhang and Shields, 2018) to a fully Bayesian construction that enables greater flexibility in quantifying uncertainty in probability model form. Being Bayesian in nature and rooted in UQ from small datasets, prior probabilities in both probability model form and model parameters are shown to have a significant impact on quantified uncertainties and, consequently, on the uncertainties propagated through a physics-based model. These effects are specifically investigated for a simplified plate buckling problem with uncertainties in material properties derived from a small number of experiments using noninformative priors and priors derived from past studies of varying appropriateness. It is illustrated that prior probabilities can have a significant impact on multimodel UQ for small datasets and inappropriate (but seemingly reasonable) priors may even have lingering effects that bias probabilities even for large datasets. When applied to uncertainty propagation, this may result in probability bounds on response quantities that do not include the true probabilities.

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Section: Bayesian Multimodel Inference and Model-form Uncertaintymentioning

confidence: 99%

Section: Bayesian Evidence Calculationmentioning

confidence: 99%

Section: Bayesian Evidence Calculationmentioning

confidence: 99%

Section: Bayesian Model Parameter Estimationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The effect of prior probabilities on quantification and propagation of imprecise probabilities resulting from small datasets

Zhang

Shields

2018

Computer Methods in Applied Mechanics and Engineering

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Modern Monte Carlo methods for efficient uncertainty quantification and propagation: A survey

Zhang

2020

WIREs Computational Stats

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Uncertainty quantification (UQ) includes the characterization, integration, and propagation of uncertainties that result from stochastic variations and a lack of knowledge or data in the natural world. Monte Carlo (MC) method is a sampling‐based approach that has widely used for quantification and propagation of uncertainties. However, the standard MC method is often time‐consuming if the simulation‐based model is computationally intensive. This article gives an overview of modern MC methods to address the existing challenges of the standard MC in the context of UQ. Specifically, multilevel Monte Carlo (MLMC) extending the concept of control variates achieves a significant reduction of the computational cost by performing most evaluations with low accuracy and corresponding low cost, and relatively few evaluations at high accuracy and corresponding high cost. Multifidelity Monte Carlo (MFMC) accelerates the convergence of standard Monte Carlo by generalizing the control variates with different models having varying fidelities and varying computational costs. Multimodel Monte Carlo method (MMMC), having a different setting of MLMC and MFMC, aims to address the issue of UQ and propagation when data for characterizing probability distributions are limited. Multimodel inference combined with importance sampling is proposed for quantifying and efficiently propagating the uncertainties resulting from small data sets. All of these three modern MC methods achieve a significant improvement of computational efficiency for probabilistic UQ, particularly uncertainty propagation. An algorithm summary and the corresponding code implementation are provided for each of the modern MC methods. The extension and application of these methods are discussed in detail. This article is categorized under: Statistical and Graphical Methods of Data Analysis > Monte Carlo Methods Statistical and Graphical Methods of Data Analysis > Sampling

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Computational Modeling and Simulation to Quantify the Effects of Obstructions on the Performance of Ventricular Catheters Used in Hydrocephalus Treatment

TerMaath

Stefanski

Killeffer

2022

Methods in Molecular Biology

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On the quantification and efficient propagation of imprecise probabilities resulting from small datasets

Cited by 76 publications

References 40 publications

The effect of prior probabilities on quantification and propagation of imprecise probabilities resulting from small datasets

The effect of prior probabilities on quantification and propagation of imprecise probabilities resulting from small datasets

Modern Monte Carlo methods for efficient uncertainty quantification and propagation: A survey

Computational Modeling and Simulation to Quantify the Effects of Obstructions on the Performance of Ventricular Catheters Used in Hydrocephalus Treatment

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