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Research Papers

# Bayesian Annealed Sequential Importance Sampling: An Unbiased Version of Transitional Markov Chain Monte Carlo

[+] Author and Article Information
Stephen Wu

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

Panagiotis Angelikopoulos

CSELab,
ETH-Zurich,
Zurich CH-8092, Switzerland

Professor
Department of Mechanical Engineering,
University of Thessaly,
Volos 38334, Greece
e-mail: costasp@mie.uth.gr

Petros Koumoutsakos

Professor
CSELab,
ETH-Zurich,
Zurich CH-8092, Switzerland;
Department of Mechanical Engineering,
Massachusetts Institute of Technology,
Cambridge, MA 02138
e-mail: petros@ethz.ch

1Present address: The Institute of Statistical Mathematics, 10–3 Midori-cho, Tachikawa, Tokyo 190-8562, Japan.

2Corresponding author.

Manuscript received July 7, 2016; final manuscript received March 27, 2017; published online September 7, 2017. Assoc. Editor: Yan Wang.

ASME J. Risk Uncertainty Part B 4(1), 011008 (Sep 07, 2017) (13 pages) Paper No: RISK-16-1101; doi: 10.1115/1.4037450 History: Received July 07, 2016; Revised March 27, 2017

## Abstract

The transitional Markov chain Monte Carlo (TMCMC) is one of the efficient algorithms for performing Markov chain Monte Carlo (MCMC) in the context of Bayesian uncertainty quantification in parallel computing architectures. However, the features that are associated with its efficient sampling are also responsible for its introducing of bias in the sampling. We demonstrate that the Markov chains of each subsample in TMCMC may result in uneven chain lengths that distort the intermediate target distributions and introduce bias accumulation in each stage of the TMCMC algorithm. We remedy this drawback of TMCMC by proposing uniform chain lengths, with or without burn-in, so that the algorithm emphasizes sequential importance sampling (SIS) over MCMC. The proposed Bayesian annealed sequential importance sampling (BASIS) removes the bias of the original TMCMC and at the same time increases its parallel efficiency. We demonstrate the advantages and drawbacks of BASIS in modeling of bridge dynamics using finite elements and a disk-wall collision using discrete element methods.

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## References

Gordon, N. J. , Salmond, D. J. , and Smith, A. F. M. , 1993, “ Novel Approach to Nonlinear/Non-Gaussian Bayesian State Estimation,” IEE Proc. F Radar Signal Process., 140(2), pp. 107–113.
Kitagawa, G. , 1996, “ Monte Carlo Filter and Smoother for Non-Gaussian Nonlinear State Space Models,” J. Comput. Graphical Stat., 5(1), pp. 1–25.
Del Moral, P. , 1996, “ Nonlinear Filtering: Interacting Particle Solution,” Markov Process. Relat. Fields, 2(4), pp. 555–580.
Liu, J. , and Chen, R. , 1998, “ Sequential Monte Carlo Methods for Dynamic Systems,” J. Am. Stat. Assoc., 93(443), pp. 1032–1044.
Neal, R. , 1998, “ Annealed Importance Sampling,” University of Toronto, Toronto, ON, Canada, Technical Report No. 9805.
Beck, J. , and Au, S. , 2002, “ Bayesian Updating of Structural Models and Reliability Using Markov Chain Monte Carlo Simulation,” ASCE J. Eng. Mech., 128(4), pp. 380–391.
Chopin, N. , 2002, “ A Sequential Particle Filter Method for Static Models,” Biometrika, 89(3), pp. 539–551.
Ching, J. , and Chen, Y. , 2007, “ Transitional Markov Chain Monte Carlo Method for Bayesian Model Updating, Model Class Selection, and Model Averaging,” ASCE J. Eng. Mech., 133(7), pp. 816–832.
Ching, J. , and Wang, J. , 2016, “ Application of the Transitional Markov Chain Monte Carlo Algorithm to Probabilistic Site Characterization,” Eng. Geol., 203, pp. 151–167.
Kirkpatrick, S. , Gelatt, C. , and Vecchi, M. , 1983, “ Optimization by Simulated Annealing,” Science, 220(4598), pp. 671–680. [PubMed]
Del Moral, P. , Doucet, A. , and Jasra, A. , 2006, “ Sequential Monte Carlo Samplers,” J. R. Stat. Soc. B, 68(3), pp. 411–436.
Angelikopoulos, P. , Papadimitriou, C. , and Koumoutsakos, P. , 2012, “ Bayesian Uncertainty Quantification and Propagation in Molecular Dynamics Simulations: A High Performance Computing Framework,” J. Chem. Phys., 137(14), p. 144103. [PubMed]
Zhang, Y. , and Yang, W. , 2014, “ A Comparative Study of the Stochastic Simulation Methods Applied in Structural Health Monitoring,” Eng. Comput., 31(7), pp. 1484–1513.
Ortiz, G. , Alvarez, D. , and Bedoya-Ruiz, D. , 2015, “ Identification of BOUC-WEN Type Models Using the Transitional Markov Chain Monte Carlo Method,” Comput. Struct., 146, pp. 252–269.
He, S. , and Ng, C. , 2017, “ Guided Wave-Based Identification of Multiple Cracks in Beams Using a Bayesian Approach,” Mech. Syst. Signal Process., 84(A), pp. 324–345.
Beck, J. , 2010, “ Bayesian System Identification Based on Probability Logic,” Struct. Control Health Monit., 17(7), pp. 825–847.
Hadjidoukas, P. , Angelikopoulos, P. , Papadimitriou, C. , and Koumoutsakos, P. , 2015, “ Π4U: A High Performance Computing Framework for Bayesian Uncertainty Quantification of Complex Models,” J. Comput. Phys., 284, pp. 1–21.
Glynn, P. W. , and Heidelberger, P. , 1991, “ Analysis of Initial Transient Deletion for Replicated Steady-State Simulations,” Oper. Res. Lett., 10(8), pp. 437–443.
Minson, S. , Simons, M. , and Beck, J. , 2013, “ Bayesian Inversion for Finite Fault Earthquake Source Models i-Theory and Algorithm,” Geophys. J. Int., 194(3), pp. 1701–1726.
Betz, W. , Papaioannou, I. , and Straub, D. , 2016, “ Transitional Markov Chain Monte Carlo: Observations and Improvements,” ASCE J. Eng. Mech., 142(5), p. 04016016.
Rosenthal, J. , 2000, “ Parallel Computing and Monte Carlo Algorithms,” Far East J. Theor. Stat., 4(2), pp. 207–236.
Hadjidoukas, P. E. , Angelikopoulos, P. , Rossinelli, D. , Alexeev, D. , Papadimitriou, C. , and Koumoutsakos, P. , 2014, “ Bayesian Uncertainty Quantification and Propagation for Discrete Element Simulations of Granular Materials,” Comput. Methods Appl. Mech. Eng., 282, pp. 218–238.
Papadimitriou, C. , and Papadioti, D.-C. , 2013, “ Component Mode Synthesis Techniques for Finite Element Model Updating,” Comput. Struct., 126, pp. 15–28.
Kruggel-Emden, H. , Wirtz, S. , and Scherer, V. , 2008, “ A Study on Tangential Force Laws Applicable to the Discrete Element Method (DEM) for Materials With Viscoelastic or Plastic Behavior,” Chem. Eng. Sci., 63(6), pp. 1523–1541.
Tsuji, Y. , Tanaka, T. , and Ishida, T. , 1992, “ Lagrangian Numerical Simulation of Plug Flow of Cohesionless Particles in a Horizontal Pipe,” Powder Technol., 71(3), pp. 239–250.

## Figures

Fig. 1

Summary of the TMCMC algorithm

Fig. 2

Mean estimates of running a single stage TMCMC on a 3D standard Gaussian likelihood with varying maximum chain length from 1 to . The black lines represent results directly after resampling and before MCMC (solid line—mean value; dashed lines—one standard deviation bound). Only the third dimension (D-3) is shown. The other two dimensions (D-2) have the exact same behavior.

Fig. 3

Standard deviation estimates of running a single stage TMCMC on a 3D standard Gaussian likelihood with varying maximum chain length from 1 to . The black lines represent results directly after resampling and before MCMC (solid line—mean value; dashed lines—one standard deviation bound). Only the third dimension (D-3) is shown. The other two dimensions (D-2) have the exact same behavior.

Fig. 4

Correlation estimates of running a single stage TMCMC on a 3D standard Gaussian likelihood with varying maximum chain length from 1 to . The black lines represent results directly after resampling and before MCMC (solid line—mean value; dashed lines—one standard deviation bound). Only the correlation between the second and the third dimension (D-2 and D-3) is shown. The other two pairs have the exact same behavior.

Fig. 5

Acceptance rate of all MCMC runs when running a single stage TMCMC on a 3D standard Gaussian likelihood with varying maximum chain length from 1 to

Fig. 6

Results of running TMCMC with varying lmax and β2

Fig. 7

Results of running different TMCMC with burn-in period for every MCMC step. Dashed lines represent lmax = 1, and solid lines represent lmax = , i.e., the original TMCMC algorithm.

Fig. 8

Results of running different TMCMC with various dimension Gaussian likelihoods. Dashed lines represent lmax = 1, and solid lines represent lmax = , i.e., the original TMCMC algorithm.

Fig. 9

Results of running different TMCMC with bimodal Gaussian likelihoods. Dashed lines represent lmax = 1 and solid lines represent lmax = , i.e., the original TMCMC algorithm.

Fig. 10

Mean estimates of running different TMCMC with bimodal Gaussian likelihoods (two dimensions: D-1 and D-2). Dashed lines represent lmax = 1, and solid lines represent lmax = , i.e., the original TMCMC algorithm.

Fig. 11

Standard deviation estimates of running different TMCMC with bimodal Gaussian likelihoods (two dimensions: D-1 and D-2). Dashed lines represent lmax = 1, and solid lines represent lmax = , i.e., the original TMCMC algorithm.

Fig. 12

Estimates for correlation and the weights’ ratio between the two Gaussian distributions when running different TMCMC with bimodal Gaussian likelihoods. Dashed lines represent lmax = 1, and solid lines represent lmax = , i.e., the original TMCMC algorithm.

Fig. 13

Results of running different TMCMC with approximately unidentifiable Gaussian likelihoods. Dashed lines represent lmax = 1, and solid lines represent lmax = , i.e., the original TMCMC algorithm.

Fig. 14

Mean estimates of running different TMCMC with approximately unidentifiable Gaussian likelihoods (six dimensions: D-1, …, D-6). Dashed lines represent lmax = 1, and solid lines represent lmax = , i.e., the original TMCMC algorithm.

Fig. 15

Standard deviation estimates of running different TMCMC with approximately unidentifiable Gaussian likelihoods (six dimensions: D-1, …, D-6). Dashed lines represent lmax = 1 and solid lines represent lmax = , i.e., the original TMCMC algorithm.

Fig. 16

Testing stagewise evidence convergence of TMCMC using 3D Gaussian likelihood with uniform prior. No burn-in in all stages.

Fig. 17

Testing stagewise evidence convergence of TMCMC using 3D Gaussian likelihood with uniform prior. Twenty steps of burn-in for each Markov chain in the first two stages only.

Fig. 18

Testing stagewise evidence convergence of TMCMC using 3D Gaussian likelihood with uniform prior. Fifty steps of burn-in for each Markov chain in the first two stages only.

Fig. 19

Testing stagewise evidence convergence of TMCMC using 3D Gaussian likelihood with uniform prior. Twenty steps of burn-in for each Markov chain in all stages.

Fig. 20

Simulation of a disk impacting at an inclined surface [17]

Fig. 21

Posterior samples for one case of the disk-wall collision example. Upper diagonal: projection of the posterior samples in all pairs of 2D parameter space (color scale corresponds to log-likelihood values of the samples). Diagonal: marginal distributions of the model parameters estimated using kernel histograms. Box-plots denote the mean and 95% percentiles. Lower diagonal: projected densities in 2D parameter space constructed via a kernel estimate (color scale corresponds to log-posterior values).

Fig. 22

Subdivided structural components of the Metsovo bridge [17]

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