Research Papers

Hierarchical Stochastic Model in Bayesian Inference for Engineering Applications: Theoretical Implications and Efficient Approximation

[+] Author and Article Information
Stephen Wu

CSELab ETH-Zurich,
CH-8092, Switzerland
e-mail: stewu@ism.ac.jp

Panagiotis Angelikopoulos

CSELab ETH-Zurich,
CH-8092, Switzerland

James L. Beck

Department of Mechanical and
Civil Engineering,
California Institute of Technology,
Pasadena, CA 91125
e-mail: jimbeck@caltech.edu

Petros Koumoutsakos

CSELab ETH-Zurich,
CH-8092, Switzerland
e-mail: petros@ethz.ch

1Pressent address: The Institute of Statistical Mathematics, Tokyo 190-8562, Japan.

2Pressent address: D.E. Shaw Research LLC, NY, NY 10036.

Manuscript received December 21, 2017; final manuscript received June 6, 2018; published online August 14, 2018. Assoc. Editor: Siu-Kui Au.

ASME J. Risk Uncertainty Part B 5(1), 011006 (Aug 14, 2018) (12 pages) Paper No: RISK-17-1104; doi: 10.1115/1.4040571 History: Received December 21, 2017; Revised June 06, 2018

Hierarchical Bayesian models (HBMs) have been increasingly used for various engineering applications. We classify two types of HBM found in the literature as hierarchical prior model (HPM) and hierarchical stochastic model (HSM). Then, we focus on studying the theoretical implications of the HSM. Using examples of polynomial functions, we show that the HSM is capable of separating different types of uncertainties in a system and quantifying uncertainty of reduced order models under the Bayesian model class selection framework. To tackle the huge computational cost for analyzing HSM, we propose an efficient approximation scheme based on importance sampling (IS) and empirical interpolation method (EIM). We illustrate our method using two engineering examples—a molecular dynamics simulation for Krypton and a pharmacokinetic/pharmacodynamics (PKPD) model for cancer drug.

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Grahic Jump Location
Fig. 2

Three different types of contaminated data sets with x between 0 and 1. The black dash lines denote the actual function without any error.

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Fig. 3

Three different types of contaminated data sets with x between 0.4 and 1. The black dash lines denote the actual function without any error.

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Fig. 1

Graphical representations of the two HBMs: (a) hierachical prior model and (b) hierarchical stochastic model

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Fig. 6

Posterior robust prediction for different models in different cases (sample size = 100). The dash lines denote the mean prediction and the gray area encloses 90% of the total probability density for the predicted value. Crosses are the data points. The four columns correspond to those in Table 4.

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Fig. 4

Different groupings of data sets. Different colors and marker combinations represent different data sets. The dashed lines denote the actual function without any error: (a) actual grouping, (b) grouping across x, (c) over grouping, and (d) random grouping.

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Fig. 5

Eight data sets from y = x2 (P2) and eight data sets from y = x3 (P3). The noises are additive Gaussian error with standard deviation σ̂y=0.1.

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Fig. 7

Graphical representations of the two alternative HSMs: (a) independent additive error parameters and (b) common additive error parameters

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Fig. 9

Posterior results for the Krypton MD simulation study: (a) posterior of ϵLJ and σLJ, (b) posterior of σy, and (c) posterior-robust prediction

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Fig. 10

Posterior distributions for parameters of the basic PK model. Upper diagonal: projection of the posterior samples for all pairs of 2D parameter space (colors indicate log-likelihood values of the samples). Diagonal: marginal distributions of the model parameters estimated using kernel histograms. Box-plots denote the means and the 5 and 95 percentiles. Lower diagonal: projected densities in 2D parameter space constructed via a kernel estimate (coloring according to log-posterior values): (a) classical Bayesian and (b) basic HSM.

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Fig. 11

Posterior robust predictions for the plasma concentrations of aflibercept based on the basic HSM. Circles are the data points. Dark gray and light gray regions are the 50% and 90% quantile range of the posterior distribution, respectively: (a) bound aflibercept and (b) free aflibercept.

Grahic Jump Location
Fig. 8

Results of the EIM approximation for the likelihood function. Left 2 plots: chosen bases of σy (cross), ϵLJ and σLJ (circles) for experiment 6. The grayscale contour shows the actual likelihood values for this experiment. Right plot: maximum error of the EIM estimate for each experiment as a function of the number of basis. The error is normalized by the maximum value of the actual function.



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