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Title: Accelerating Asymptotically Exact MCMC for Computationally Intensive Models via Local Approximations

Abstract

We construct a new framework for accelerating Markov chain Monte Carlo in posterior sampling problems where standard methods are limited by the computational cost of the likelihood, or of numerical models embedded therein. Our approach introduces local approximations of these models into the Metropolis–Hastings kernel, borrowing ideas from deterministic approximation theory, optimization, and experimental design. Previous efforts at integrating approximate models into inference typically sacrifice either the sampler’s exactness or efficiency; our work seeks to address these limitations by exploiting useful convergence characteristics of local approximations. We prove the ergodicity of our approximate Markov chain, showing that it samples asymptotically from the exact posterior distribution of interest. We describe variations of the algorithm that employ either local polynomial approximations or local Gaussian process regressors. Our theoretical results reinforce the key observation underlying this article: when the likelihood has some local regularity, the number of model evaluations per Markov chain Monte Carlo (MCMC) step can be greatly reduced without biasing the Monte Carlo average. Numerical experiments demonstrate multiple order-of-magnitude reductions in the number of forward model evaluations used in representative ordinary differential equation (ODE) and partial differential equation (PDE) inference problems, with both synthetic and real data. Supplementary materials for thismore » article are available online.« less

Authors:
 [1];  [1];  [2];  [3]
  1. Massachusetts Inst. of Technology (MIT), Cambridge, MA (United States). Dept. of Aeronautics and Astronautics
  2. Harvard Univ., Cambridge, MA (United States). Dept. of Statistics
  3. Univ. of Ottawa, Ottawa (Canada). Dept. of Mathematics and Statistics
Publication Date:
Research Org.:
Massachusetts Inst. of Technology (MIT), Cambridge, MA (United States)
Sponsoring Org.:
USDOE Office of Science (SC)
OSTI Identifier:
1535376
Grant/Contract Number:  
SC0007099
Resource Type:
Accepted Manuscript
Journal Name:
Journal of the American Statistical Association
Additional Journal Information:
Journal Volume: 111; Journal Issue: 516; Journal ID: ISSN 0162-1459
Publisher:
Taylor & Francis
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; Mathematics

Citation Formats

Conrad, Patrick R., Marzouk, Youssef M., Pillai, Natesh S., and Smith, Aaron. Accelerating Asymptotically Exact MCMC for Computationally Intensive Models via Local Approximations. United States: N. p., 2015. Web. doi:10.1080/01621459.2015.1096787.
Conrad, Patrick R., Marzouk, Youssef M., Pillai, Natesh S., & Smith, Aaron. Accelerating Asymptotically Exact MCMC for Computationally Intensive Models via Local Approximations. United States. doi:10.1080/01621459.2015.1096787.
Conrad, Patrick R., Marzouk, Youssef M., Pillai, Natesh S., and Smith, Aaron. Wed . "Accelerating Asymptotically Exact MCMC for Computationally Intensive Models via Local Approximations". United States. doi:10.1080/01621459.2015.1096787. https://www.osti.gov/servlets/purl/1535376.
@article{osti_1535376,
title = {Accelerating Asymptotically Exact MCMC for Computationally Intensive Models via Local Approximations},
author = {Conrad, Patrick R. and Marzouk, Youssef M. and Pillai, Natesh S. and Smith, Aaron},
abstractNote = {We construct a new framework for accelerating Markov chain Monte Carlo in posterior sampling problems where standard methods are limited by the computational cost of the likelihood, or of numerical models embedded therein. Our approach introduces local approximations of these models into the Metropolis–Hastings kernel, borrowing ideas from deterministic approximation theory, optimization, and experimental design. Previous efforts at integrating approximate models into inference typically sacrifice either the sampler’s exactness or efficiency; our work seeks to address these limitations by exploiting useful convergence characteristics of local approximations. We prove the ergodicity of our approximate Markov chain, showing that it samples asymptotically from the exact posterior distribution of interest. We describe variations of the algorithm that employ either local polynomial approximations or local Gaussian process regressors. Our theoretical results reinforce the key observation underlying this article: when the likelihood has some local regularity, the number of model evaluations per Markov chain Monte Carlo (MCMC) step can be greatly reduced without biasing the Monte Carlo average. Numerical experiments demonstrate multiple order-of-magnitude reductions in the number of forward model evaluations used in representative ordinary differential equation (ODE) and partial differential equation (PDE) inference problems, with both synthetic and real data. Supplementary materials for this article are available online.},
doi = {10.1080/01621459.2015.1096787},
journal = {Journal of the American Statistical Association},
number = 516,
volume = 111,
place = {United States},
year = {2015},
month = {10}
}

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Cited by: 18 works
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Works referenced in this record:

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High-Order Collocation Methods for Differential Equations with Random Inputs
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Adaptive Construction of Surrogates for the Bayesian Solution of Inverse Problems
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Adaptive Smolyak Pseudospectral Approximations
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Approximate Bayesian computational methods
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Stochastic spectral methods for efficient Bayesian solution of inverse problems
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Uncertainty Quantification and Weak Approximation of an Elliptic Inverse Problem
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General state space Markov chains and MCMC algorithms
journal, January 2004


Statistical inverse problems: Discretization, model reduction and inverse crimes
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    Works referencing / citing this record:

    Preconditioning Markov Chain Monte Carlo Simulations Using Coarse-Scale Models
    journal, January 2006

    • Efendiev, Y.; Hou, T.; Luo, W.
    • SIAM Journal on Scientific Computing, Vol. 28, Issue 2
    • DOI: 10.1137/050628568

    Bayesian inverse problems with Monte Carlo forward models
    journal, February 2013

    • Marzouk, Youssef; Langmore, Ian; Bal, Guillaume
    • Inverse Problems and Imaging, Vol. 7, Issue 1
    • DOI: 10.3934/ipi.2013.7.81

    Uncertainty Quantification and Weak Approximation of an Elliptic Inverse Problem
    journal, January 2011

    • Dashti, M.; Stuart, A. M.
    • SIAM Journal on Numerical Analysis, Vol. 49, Issue 6
    • DOI: 10.1137/100814664

    Bayesian calibration of computer models
    journal, August 2001

    • Kennedy, Marc C.; O'Hagan, Anthony
    • Journal of the Royal Statistical Society: Series B (Statistical Methodology), Vol. 63, Issue 3
    • DOI: 10.1111/1467-9868.00294

    Spline-Based Emulators for Radiative Shock Experiments With Measurement Error
    journal, June 2013

    • Chakraborty, Avishek; Mallick, Bani K.; Mcclarren, Ryan G.
    • Journal of the American Statistical Association, Vol. 108, Issue 502
    • DOI: 10.1080/01621459.2013.770688

    Parameter and State Model Reduction for Large-Scale Statistical Inverse Problems
    journal, January 2010

    • Lieberman, Chad; Willcox, Karen; Ghattas, Omar
    • SIAM Journal on Scientific Computing, Vol. 32, Issue 5
    • DOI: 10.1137/090775622

    Approximation of Bayesian Inverse Problems for PDEs
    journal, January 2010

    • Cotter, S. L.; Dashti, M.; Stuart, A. M.
    • SIAM Journal on Numerical Analysis, Vol. 48, Issue 1
    • DOI: 10.1137/090770734

    Construction of a genetic toggle switch in Escherichia coli
    journal, January 2000

    • Gardner, Timothy S.; Cantor, Charles R.; Collins, James J.
    • Nature, Vol. 403, Issue 6767
    • DOI: 10.1038/35002131

    Design and Analysis of Computer Experiments
    journal, November 1989

    • Sacks, Jerome; Welch, William J.; Mitchell, Toby J.
    • Statistical Science, Vol. 4, Issue 4
    • DOI: 10.1214/ss/1177012413

    Bayesian Computation Using Design of Experiments-Based Interpolation Technique
    journal, August 2012


    Statistical inverse problems: Discretization, model reduction and inverse crimes
    journal, January 2007

    • Kaipio, Jari; Somersalo, Erkki
    • Journal of Computational and Applied Mathematics, Vol. 198, Issue 2
    • DOI: 10.1016/j.cam.2005.09.027

    Deterministic Sampling of Expensive Posteriors Using Minimum Energy Designs
    journal, March 2019


    Sparse pseudospectral approximation method
    journal, July 2012

    • Constantine, Paul G.; Eldred, Michael S.; Phipps, Eric T.
    • Computer Methods in Applied Mechanics and Engineering, Vol. 229-232
    • DOI: 10.1016/j.cma.2012.03.019

    Approximate Bayesian computational methods
    journal, October 2011

    • Marin, Jean-Michel; Pudlo, Pierre; Robert, Christian P.
    • Statistics and Computing, Vol. 22, Issue 6
    • DOI: 10.1007/s11222-011-9288-2

    Stochastic spectral methods for efficient Bayesian solution of inverse problems
    journal, June 2007

    • Marzouk, Youssef M.; Najm, Habib N.; Rahn, Larry A.
    • Journal of Computational Physics, Vol. 224, Issue 2
    • DOI: 10.1016/j.jcp.2006.10.010

    A Sparse Grid Stochastic Collocation Method for Partial Differential Equations with Random Input Data
    journal, January 2008

    • Nobile, F.; Tempone, R.; Webster, C. G.
    • SIAM Journal on Numerical Analysis, Vol. 46, Issue 5
    • DOI: 10.1137/060663660

    Robust Locally Weighted Regression and Smoothing Scatterplots
    journal, December 1979


    Local Derivative-Free Approximation of Computationally Expensive Posterior Densities
    journal, April 2012

    • Bliznyuk, Nikolay; Ruppert, David; Shoemaker, Christine A.
    • Journal of Computational and Graphical Statistics, Vol. 21, Issue 2
    • DOI: 10.1080/10618600.2012.681255

    General state space Markov chains and MCMC algorithms
    journal, January 2004


    An Adaptive Metropolis Algorithm
    journal, April 2001

    • Haario, Heikki; Saksman, Eero; Tamminen, Johanna
    • Bernoulli, Vol. 7, Issue 2
    • DOI: 10.2307/3318737

    Multivariate Adaptive Regression Splines
    journal, March 1991


    A Stochastic Collocation Approach to Bayesian Inference in Inverse Problems
    journal, January 2009


    A transport-based multifidelity preconditioner for Markov chain Monte Carlo
    journal, November 2019


    Local Gaussian Process Approximation for Large Computer Experiments
    journal, April 2015

    • Gramacy, Robert B.; Apley, Daniel W.
    • Journal of Computational and Graphical Statistics, Vol. 24, Issue 2
    • DOI: 10.1080/10618600.2014.914442

    Adaptive Smolyak Pseudospectral Approximations
    journal, January 2013

    • Conrad, Patrick R.; Marzouk, Youssef M.
    • SIAM Journal on Scientific Computing, Vol. 35, Issue 6
    • DOI: 10.1137/120890715

    Approximating likelihoods for large spatial data sets
    journal, May 2004

    • Stein, Michael L.; Chi, Zhiyi; Welty, Leah J.
    • Journal of the Royal Statistical Society: Series B (Statistical Methodology), Vol. 66, Issue 2
    • DOI: 10.1046/j.1369-7412.2003.05512.x

    Adaptive Construction of Surrogates for the Bayesian Solution of Inverse Problems
    journal, January 2014

    • Li, Jinglai; Marzouk, Youssef M.
    • SIAM Journal on Scientific Computing, Vol. 36, Issue 3
    • DOI: 10.1137/130938189

    High-Order Collocation Methods for Differential Equations with Random Inputs
    journal, January 2005

    • Xiu, Dongbin; Hesthaven, Jan S.
    • SIAM Journal on Scientific Computing, Vol. 27, Issue 3
    • DOI: 10.1137/040615201