Bayesian Analysis

Optimal Bayesian Experimental Design for Models with Intractable Likelihoods Using Indirect Inference Applied to Biological Process Models

Caitríona M. Ryan, Christopher C. Drovandi, and Anthony N. Pettitt

Full-text: Open access


This paper addresses the problem of determining optimal designs for biological process models with intractable likelihoods, with the goal of parameter inference. The Bayesian approach is to choose a design that maximises the mean of a utility, and the utility is a function of the posterior distribution. Therefore, its estimation requires likelihood evaluations. However, many problems in experimental design involve models with intractable likelihoods, that is, likelihoods that are neither analytic nor can be computed in a reasonable amount of time. We propose a novel solution using indirect inference (II), a well established method in the literature, and the Markov chain Monte Carlo (MCMC) algorithm of Müller et al. (2004). Indirect inference employs an auxiliary model with a tractable likelihood in conjunction with the generative model, the assumed true model of interest, which has an intractable likelihood. Our approach is to estimate a map between the parameters of the generative and auxiliary models, using simulations from the generative model. An II posterior distribution is formed to expedite utility estimation. We also present a modification to the utility that allows the Müller algorithm to sample from a substantially sharpened utility surface, with little computational effort. Unlike competing methods, the II approach can handle complex design problems for models with intractable likelihoods on a continuous design space, with possible extension to many observations. The methodology is demonstrated using two stochastic models; a simple tractable death process used to validate the approach, and a motivating stochastic model for the population evolution of macroparasites.

Article information

Bayesian Anal., Volume 11, Number 3 (2016), 857-883.

First available in Project Euclid: 9 October 2015

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

approximate Bayesian computation auxiliary model Bayesian experimental design indirect inference Markov chain Monte Carlo Markov processes


Ryan, Caitríona M.; Drovandi, Christopher C.; Pettitt, Anthony N. Optimal Bayesian Experimental Design for Models with Intractable Likelihoods Using Indirect Inference Applied to Biological Process Models. Bayesian Anal. 11 (2016), no. 3, 857--883. doi:10.1214/15-BA977.

Export citation


  • Bayarri, M. J., Berger, J. O., Paulo, R., Sacks, J., Cafeo, J. A., Cavendish, J., Lin, C.-H., and Tu, J. (2007). “A framework for validation of computer models.” Technometrics, 49(2): 138–154.
  • Bernardo, J. M. and Smith, A. F. (2000). Bayesian theory. John Wiley & Sons.
  • Chaloner, K. and Larntz, K. (1989). “Optimal Bayesian design applied to logistic regression experiments.” Journal of Statistical Planning and Inference, 21(2): 191–208.
  • Chaloner, K. and Verdinelli, I. (1995). “Bayesian experimental design: A review.” Statistical Science, 10(3): 273–304.
  • Clyde, M. A. (2001). “Experimental design: A Bayesian perspective.” International Encyclopedia of the Social and Behavioral Sciences, 8: 5075–5081.
  • Cook, A. R., Gibson, G. J., and Gilligan, C. A. (2008). “Optimal observation times in experimental epidemic processes.” Biometrics, 64(3): 860–868.
  • Cox, D. R. (1961). “Tests of separate families of hypotheses.” In: Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, volume 1, 105–123.
  • Denham, D. A., Ponnudurai, T., Nelson, G. S., Guy, F., and Rogers, R. (1972). “Studies with Brugia pahangi.” – I. Parasitological observations on primary infections of cats (Felis catus). International Journal for Parasitology, 2(2): 239–247.
  • Drovandi, C. C., McGree, J. M., and Pettitt, A. N. (2013). “Sequential Monte Carlo for Bayesian sequentially designed experiments for discrete data.” Computational Statistics & Data Analysis, 57(1): 320–335.
  • Drovandi, C. C. and Pettitt, A. N. (2011). “Estimation of parameters for macroparasite population evolution using approximate Bayesian computation.” Biometrics, 67(1): 225–233.
  • Drovandi, C. C. and Pettitt, A. N. (2013). “Bayesian experimental design for models with intractable likelihoods.” Biometrics, 69(4): 937–948.
  • Drovandi, C. C., Pettitt, A. N., and Faddy, M. J. (2011). “Approximate Bayesian computation using indirect inference.” Journal of the Royal Statistical Society: Series C (Applied Statistics), 60(3): 317–337.
  • Drovandi, C. C., Pettitt, A. N., and Lee, A. (2015). “Bayesian indirect inference using a parametric auxiliary model.” Statistical Science, 30(1): 72–95.
  • Gallant, A. R. and McCulloch, R. E. (2009). “On the determination of general scientific models with application to asset pricing.” Journal of the American Statistical Association, 104(485): 117–131.
  • Gallant, A. R. and Tauchen, G. (1996). “Which moments to match?” Econometric Theory, 12(04): 657–681.
  • Gillespie, D. T. (1977). “Exact stochastic simulation of coupled chemical reactions.” The Journal of Physical Chemistry, 81(25): 2340–2361.
  • Gourieroux, C., Monfort, A., and Renault, E. (1993). “Indirect inference.” Journal of Applied Econometrics, 8(S1): S85–S118.
  • Hainy, M., Müller, W. G., and Wagner, H. (2013). Likelihood-free simulation-based optimal design. Technical report, Johannes Kepler University of Linz.
  • Heggland, K. and Frigessi, A. (2004). “Estimating functions in indirect inference.” Journal of the Royal Statistical Society: Series B (Statistical Methodology), 66(2): 447–462.
  • Kuo, L., Soyer, R., and Wang, F. (1999). “Optimal design for quantal bioassay via Monte Carlo methods.” Bayesian Statistics VI, 795–802.
  • Lindley, D. V. (1972). Bayesian Statistics, a Review. Capital City Press, Montepelier, Vermont.
  • Michael, E., Grenfell, B. T., Isham, V. S., Denham, D. A., and Bundy, D. A. P. (1998). “Modelling variability in lymphatic filariasis: Macrofilarial dynamics in the Brugia pahangi–cat model.” In: Proceedings of the Royal Society of London. Series B: Biological Sciences, 265(1391): 155–165.
  • Moores, M. T., Drovandi, C. C., Mengersen, K. L., and Robert, C. P. (2015). “Pre-processing for approximate Bayesian computation in image analysis.” Statistics and Computing, 25(1): 23–33.
  • 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.
  • Nelder, J. A. and Mead, R. (1965). “A simplex method for function minimization.” Computer Journal, 7(4): 308–313.
  • Ottesen, E. A. (2006). “Lymphatic filariasis: Treatment, control and elimination.” Advances in Parasitology, 61: 395–441.
  • Raiffa, H. and Schlaifer, R. (1961). Applied Statistical Decision Theory. Division of Research, Graduate School of Business Administration, Harvard University.
  • Reeves, R. W. and Pettitt, A. N. (2005). “A theoretical framework for approximate Bayesian computation.” In: Proceedings of the 20th International Workshop on Statistical Modelling, Sydney, 393–396.
  • Renshaw, E. (1991). Modelling Biological Populations in Space and Time. Cambridge University Press, Cambridge.
  • Riley, S., Donnelly, C. A., and Ferguson, N. M. (2003). “Robust parameter estimation techniques for stochastic within-host macroparasite models.” Journal of Theoretical Biology, 225(4): 419–430.
  • Ryan, E. G., Drovandi, C. C., and Pettitt, A. N. (2015). “Fully Bayesian experimental design for pharmacokinetic studies.” Entropy, 17(3): 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 & Data Analysis, 70: 45–60.
  • Smith, A. A. (1993). “Estimating nonlinear time-series models using simulated vector autoregressions.” Journal of Applied Econometrics, 8(S1): S63–S84.
  • Suswillo, R. R., Denham, D. A., and McGreevy, P. B. (1982). “The number and distribution of Brugia pahangi in cats at different times after a primary infection.” Acta Tropica, 39(2): 151–156.
  • Van Laarhoven, P. J. M. and Aarts, E. H. L. (1987). Simulated annealing: Theory and applications. D. Reidel Publishing Company, Dordrecht.
  • Weaver, B. P., Williams, B. J., Anderson-Cook, C. M., and Higdon, D. M. (2015). “Computational enhancements to Bayesian design of experiments using Gaussian processes.” Bayesian Analysis.
  • Wood, S. N. (2010). “Statistical inference for noisy nonlinear ecological dynamic systems.” Nature, 466(7310): 1102–1104.