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

Zhang, Jiaxin; Shields, Michael D.

doi:10.1016/j.cma.2018.01.045

Cited by 34 publications

(17 citation statements)

References 61 publications

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“…However, the evidence is very critical in Bayesian multimodel inference and needs to be calculated with caution. We refer to (Zhang & Shields, 2018a) for a detail discussion of the evidence calculation.…”

Section: Modern MC Methods For Uqmentioning

confidence: 99%

“…Along with the MMMC framework, Zhang and Shields further investigated the effect of prior probability on quantification and propagation of imprecise probabilities resulting from small data sets (Zhang & Shields, 2018a). It is demonstrated that prior probabilities play a critical role in Bayesian multimodel UQ framework for small data sets, and inappropriate priors may lead to biased probabilities as well as inaccurate estimators even for large data sets.…”

Section: Modern MC Methods For Uqmentioning

confidence: 99%

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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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Section: Modern MC Methods For Uqmentioning

confidence: 99%

Section: Modern MC Methods For Uqmentioning

confidence: 99%

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

Zhang

2020

WIREs Computational Stats

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“…Zhang et al [19] studied the effect of prior on prediction and estimation in Bayesian inference. ey think that model parameters are shown to have a significant impact on quantified uncertainties.…”

Section: Posterior Distribution Of Dirichlet Conjugate Priormentioning

confidence: 99%

“…In the actual attack, the damage grade of the target cannot be determined. Assuming that the threshold value Λ of damage probability is 0.8, and then the cumulative value of ammunition required for different damage grades can be obtained; that is, the optimal ammunition demand φ Γ * r for reaching each damage grade can be obtained by referring to equation (19). e grade of zero damage means that the target equipment's combat performance is nearly unaffected, and no ammunition is consumed.…”

Section: Example Analysismentioning

confidence: 99%

Bayesian Estimation of Ammunition Demand Based on Multinomial Distribution

Shi

et al. 2021

Discrete Dynamics in Nature and Society

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In view of the small sample size of combat ammunition trial data and the difficulty of forecasting the demand for combat ammunition, a Bayesian inference method based on multinomial distribution is proposed. Firstly, considering the different damage grades of ammunition hitting targets, the damage results are approximated as multinomial distribution, and a Bayesian inference model of ammunition demand based on multinomial distribution is established, which provides a theoretical basis for forecasting the ammunition demand of multigrade damage under the condition of small samples. Secondly, the conjugate Dirichlet distribution of multinomial distribution is selected as a prior distribution, and Dempster–Shafer evidence theory (D-S theory) is introduced to fuse multisource previous information. Bayesian inference is made through the Markov chain Monte Carlo method based on Gibbs sampling, and ammunition demand at different damage grades is obtained by referring to cumulative damage probability. The study result shows that the Bayesian inference method based on multinomial distribution is highly maneuverable and can be used to predict ammunition demand of different damage grades under the condition of small samples.

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“…An effective sampling method would contribute to describing the design space more accurately and finding the global optimal solution more potentially. As one of the most common methods, Latin hypercube sampling (LHS) has been widely used for design of experiments, 51 Monte Carlo simulations, 52,53 and uncertainty quantification, [54][55][56] employed in nearly every field of computational engineering and science. Stein 57 showed that LHS can filter the variance associated with the additive components (main effects) of a transformation, which states that the main effects and low-order interactions are perhaps to govern the most general transformations and it leads LHS to reduce variance significantly.…”

Section: Introductionmentioning

confidence: 99%

Buckling surrogate-based optimization framework for hierarchical stiffened composite shells by enhanced variance reduction method

Tian

Zhang

et al. 2019

Journal of Reinforced Plastics and Composites

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The surrogate-based optimization of hierarchical stiffened composite shells against buckling is a typical multimodal and multivariables optimization problem. To improve the computational efficiency and global optimizing ability of the surrogate-based optimization of hierarchical stiffened composite shells, an enhanced variance reduction method based on Latinized partially stratified sampling and multifidelity analysis methods is proposed in this paper and then integrated into the surrogate-based optimization framework. In the offline step of the optimization framework, candidate pairing strategies of design variables are generated by Latinized partially stratified sampling and compared by performing priori optimizations based on the low-fidelity analysis method, and the optimal pairing strategy is therefore determined. On the basis of the optimal pairing strategy, the surrogate-based optimization is carried out using the highfidelity analysis method in the online step. With less computational cost in the offline step, the proposed enhanced variance reduction method overcomes the limitation of Latinized partially stratified sampling that the optimal pairing strategy is not obvious in complex problems. Then, extensive optimization examples are carried out to verify the efficiency and effectiveness of the proposed optimization framework. Given an approximate computational cost, the optimal buckling result of the proposed framework using enhanced variance reduction method increases by 18.2% than that of the traditional framework based on Latin hypercube sampling. In particular, the advantage of enhanced variance reduction method in the space-filling ability is highlighted in comparison to Latin hypercube sampling. When achieving an approximate global optimal solution, the proposed framework reduces the total computational cost by 76.3% than the traditional framework. Finally, the numerical implementation of asymptotic homogenization method is used herein for the accurate prediction of effective stiffness coefficients of the initial design and optimal results. Through comparison, it is concluded that the high axial stiffness and bending stiffness are the main mechanism for the high loadcarrying capacity of optimal results.

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

Cited by 34 publications

References 61 publications

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

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

Bayesian Estimation of Ammunition Demand Based on Multinomial Distribution

Buckling surrogate-based optimization framework for hierarchical stiffened composite shells by enhanced variance reduction method

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