Bayesian Analysis

Improving the Efficiency of Fully Bayesian Optimal Design of Experiments Using Randomised Quasi-Monte Carlo

Christopher C. Drovandi and Minh-Ngoc Tran

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Optimal experimental design is an important methodology for most efficiently allocating resources in an experiment to best achieve some goal. Bayesian experimental design considers the potential impact that various choices of the controllable variables have on the posterior distribution of the unknowns. Optimal Bayesian design involves maximising an expected utility function, which is an analytically intractable integral over the prior predictive distribution. These integrals are typically estimated via standard Monte Carlo methods. In this paper, we demonstrate that the use of randomised quasi-Monte Carlo can bring significant reductions to the variance of the estimated expected utility. This variance reduction can then lead to a more efficient optimisation algorithm for maximising the expected utility.

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Bayesian Anal., Volume 13, Number 1 (2018), 139-162.

First available in Project Euclid: 30 December 2016

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approximate Bayesian computation evidence experimental design importance sampling mutual information Laplace approximation quasi-Monte Carlo

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Drovandi, Christopher C.; Tran, Minh-Ngoc. Improving the Efficiency of Fully Bayesian Optimal Design of Experiments Using Randomised Quasi-Monte Carlo. Bayesian Anal. 13 (2018), no. 1, 139--162. doi:10.1214/16-BA1045.

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  • Berger, J. (1985). Statistical Decision Theory and Bayesian Analysis. New York: Springer-Verlag.
  • Bernardo, J. M. and Smith, A. (2000). Bayesian Theory. Chichester: Wiley.
  • Bliemer, M. C. J., Rose, J. M., and Hess, S. (2008). “Approximation of Bayesian efficiency in experimental choice designs.” Journal of Choice Modelling, 1(1): 98–126.
  • Box, G. E. and Muller, M. E. (1958). “A note on the generation of random normal deviates.” The Annals of Mathematical Statistics, 29(2): 610–611.
  • Box, G. E. P. and Hill, W. J. (1967). “Discrimination among mechanistic models.” Technometrics, 9: 57–71.
  • Cook, A. R., Gibson, G. J., and Gilligan, C. A. (2008). “Optimal observation times in experimental epidemic processes.” Biometrics, 64(3): 860–868.
  • Dehideniya, M. B., Drovandi, C. C., and McGree, J. M. (2016). “Effcient Bayesian Design for Discriminating Between Models with Intractable Likelihoods in Epidemiology.”
  • Dick, J. and Pillichshammer, F. (2010). Digital Nets and Sequence. Discrepancy Theory and Quasi-Monte Carlo Integration. Cambridge: Cambridge University Press.
  • Drovandi, C. C., McGree, J. M., and Pettitt, A. N. (2013). “Sequential Monte Carlo for Bayesian sequentially designed experiments for discrete data.” Computational Statistics and Data Analysis, 57: 320–335.
  • Drovandi, C. C., McGree, J. M., and Pettitt, A. N. (2014). “A sequential Monte Carlo algorithm to incorporate model uncertainty in Bayesian sequential design.” Journal of Computational and Graphical Statistics, 23: 3–24.
  • Drovandi, C. C. and Pettitt, A. N. (2013). “Bayesian experimental design for models with intractable likelihoods.” Biometrics, 69(4): 937–948.
  • Gerber, M. and Chopin, N. (2015). “Sequential quasi Monte Carlo.” Journal of the Royal Statistical Society: Series B (Statistical Methodology), 77(3): 509–579.
  • Gillespie, D. T. (1977). “Exact stochastic simulation of coupled chemical reactions.” The Journal of Physical Chemistry, 81(25): 2340–2361.
  • Glasserman, P. (2004). Monte Carlo Methods in Financial Engineering. New York: Springer-Verlag.
  • Gotwalt, C. M., Jones, B. A., and Steinberg, D. M. (2009). “Fast computation of designs robust to parameter uncertainty for nonlinear settings.” Technometrics, 51: 88–95.
  • Hainy, M., Müller, W. G., and Wagner, H. (2014). “Likelihood-Free Simulation-Based Optimal Design: An Introduction.” In Melas, V., Mignani, S., Monari, P., and Salmaso, L. (eds.), Topics in Statistical Simulation, 271–278. Springer.
  • Halton, J. H. (1960). “On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals.” Numerische Mathematik, 2: 84–90.
  • Hammersley, J. M. (1960). “Monte Carlo methods for solving multivariable problems.” Annals of the New York Academy of Sciences, 86: 844–874.
  • Huan, X. and Marzouk, Y. M. (2014). “Gradient-based stochastic optimization methods in Bayesian experimental design.” International Journal for Uncertainty Quantification, 4(6): 479–510.
  • Loh, W.-L. (2003). “On the asymptotic distribution of scrambled net quadrature.” Annals of Statistics, 31: 1282–1324.
  • Matousek, J. (1998). “On the L2-discrepancy for anchored boxes.” Journal of Complexity, 14: 527–556.
  • McGree, J. M., Drovandi, C. C., White, G., and Pettitt, A. N. (2016). “A pseudo-marginal sequential Monte Carlo algorithm for random effects models in Bayesian sequential design.” Statistics and Computing, 26(5): 1121–1136.
  • Meyer, R. K. and Nachtsheim, C. J. (1995). “The coordinate-exchange algorithm for constructing exact optimal experimental designs.” Technometrics, 37: 60–69.
  • Moler, C. and van Loan, C. (2003). “Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later.” SIAM Review, 45(1): 3–49.
  • Müller, P., Sansó, B., and De Iorio, M. (2004). “Optimal Bayesian design by inhomogeneous Markov chain simulation.” Journal of the American Statistical Association, 99(467): 788–798.
  • Niederreiter, H. (1992). Random Number Generation and Quasi-Monte Carlo Methods. Philadelphia: Society for Industrial and Applied Mathematics.
  • Overstall, A. M., McGree, J. M., and Drovandi, C. C. (2016). “Fully Bayesian optimal design using the approximate coordinate exchange algorithm and normal-based approximations to posterior quantities.”
  • Overstall, A. M. and Woods, D. C. (2016). “Bayesian Design of Experiments using Approximate Coordinate Exchange.” arXiv:1501.00264.
  • Owen, A. B. (1997). “Scrambled net variance for integrals of smooth functions.” Annals of Statistics, 25: 1541–1562.
  • Robert, C. P. (2007). The Bayesian Choice: From Decision-Theoretic Foundations to Computational Implementation. Springer Science & Business Media.
  • Ryan, E. G., Drovandi, C. C., McGree, J. M., and Pettitt, A. N. (2016). “A review of modern computational algorithms for Bayesian optimal design.” International Statistical Review, 84: 128–154.
  • Ryan, E. G., Drovandi, C. C., and Pettitt, A. N. (2015). “Fully Bayesian experimental design for Pharmacokinetic studies.” Entropy, 17: 1063–1089.
  • Ryan, E. G., Drovandi, C. C., Thompson, M. H., and Pettitt, A. N. (2014). “Towards Bayesian experimental design for nonlinear models that require a large number of sampling times.” Computational Statistics and Data Analysis, 70: 45–60.
  • Tran, M., Nott, D. J., and Kohn, R. (2015). “Variational Bayes with Intractable Likelihood.” arXiv:1503.08621.
  • van der Corput, J. G. (1935). “Verteilungsfunktionen. I. Mitt.” Proceedings. Akadamie van Wetenschappen Amsterdam (in German), 38: 813–821.
  • Weaver, B. P., Williams, B. J., Anderson-Cook, C. M., and Higdon, D. M. (2016). “Computational enhancements to Bayesian design of experiments using Gaussian processes.” Bayesian Analysis, 11(1): 191–213.