Brazilian Journal of Probability and Statistics

Barker’s algorithm for Bayesian inference with intractable likelihoods

Flávio B. Gonçalves, Krzysztof Łatuszyński, and Gareth O. Roberts

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Abstract

In this expository paper, we abstract and describe a simple MCMC scheme for sampling from intractable target densities. The approach has been introduced in Gonçalves, Łatuszyński and Roberts (2017a) in the specific context of jump-diffusions, and is based on the Barker’s algorithm paired with a simple Bernoulli factory type scheme, the so called 2-coin algorithm. In many settings, it is an alternative to standard Metropolis–Hastings pseudo-marginal method for simulating from intractable target densities. Although Barker’s is well known to be slightly less efficient than Metropolis–Hastings, the key advantage of our approach is that it allows to implement the “marginal Barker’s” instead of the extended state space pseudo-marginal Metropolis–Hastings, owing to the special form of the accept/reject probability. We shall illustrate our methodology in the context of Bayesian inference for discretely observed Wright–Fisher family of diffusions.

Article information

Source
Braz. J. Probab. Stat., Volume 31, Number 4 (2017), 732-745.

Dates
Received: December 2016
Accepted: August 2017
First available in Project Euclid: 15 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.bjps/1513328765

Digital Object Identifier
doi:10.1214/17-BJPS374

Mathematical Reviews number (MathSciNet)
MR3738176

Zentralblatt MATH identifier
1385.65013

Keywords
Intractable likelihood Bayesian inference Barker’s algorithm Bernoulli factory 2-coin algorithm stochastic differential equations Wright–Fisher diffusion

Citation

Gonçalves, Flávio B.; Łatuszyński, Krzysztof; Roberts, Gareth O. Barker’s algorithm for Bayesian inference with intractable likelihoods. Braz. J. Probab. Stat. 31 (2017), no. 4, 732--745. doi:10.1214/17-BJPS374. https://projecteuclid.org/euclid.bjps/1513328765


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References

  • Andrieu, C. and Roberts, G. O. (2009). The pseudo-marginal approach for efficient Monte Carlo computations. The Annals of Statistics 37, 697–725.
  • Andrieu, C. and Vihola, M. (2015). Convergence properties of pseudo-marginal Markov chain Monte Carlo algorithms. The Annals of Applied Probability 25, 1030–1077.
  • Asmussen, S., Glynn, P. and Thorisson, H. (1992). Stationarity detection in the initial transient problem. ACM Transactions on Modeling and Computer Simulation 2, 130–157.
  • Barker, A. A. (1965). Monte Carlo calculations of the radial distribution functions for a protonelectron plasma. Australian Journal of Physics 18, 119–133.
  • Beaumont, M. A. (2003). Estimation of population growth or decline in genetically monitored populations. Genetics 164, 1139–1160.
  • Beskos, A., Papaspiliopoulos, O. and Roberts, G. O. (2006a). Retrospective exact simulation of diffusion sample paths with applications. Bernoulli 12, 1077–1098.
  • Beskos, A., Papaspiliopoulos, O. and Roberts, G. O. (2008). A new factorisation of diffusion measure and sample path reconstruction. Methodology and Computing in Applied Probability 10, 85–104.
  • Beskos, A., Papaspiliopoulos, O., Roberts, G. O. and Fearnhead, P. (2006b). Exact and computationally efficient likelihood-based inference for discretely observed diffusion processes (with discussion). Journal of the Royal Statistical Society, Series B 68, 333–382.
  • Flegal, J. M. and Herbei, R. (2012). Exact sampling for intractable probability distributions via a Bernoulli factory. Electronic Journal of Statistics 6, 10–37.
  • Gonçalves, F. B., Łatuszyński, K. G. and Roberts, G. O. (2017a). Exact Monte Carlo likelihood-based inference for jump-diffusion processes. Available at arXiv:1707.00332.
  • Gonçalves, F. B., Roberts, G. O. and Łatuszyński, K. G. (2017b). Exact Bayesian inference for diffusion driven Cox processes. In preparation.
  • Herbei, R. and Berliner, L. M. (2014). Estimating ocean circulation: An MCMC approach with approximated likelihoods via the Bernoulli factory. Journal of the American Statistical Association 109, 944–954.
  • Huber, M. (2015). Optimal linear Bernoulli factories for small mean problems. Available at arXiv:1507.00843.
  • Huber, M. (2016). Nearly optimal Bernoulli factories for linear functions. Combinatorics, Probability & Computing 25, 577–591.
  • Jacob, P. E. and Thiery, A. H. (2015). On nonnegative unbiased estimators. The Annals of Statistics 43, 769–784.
  • Jenkins, P. A. and Spanó, D. (2016). Exact simulation of the Wright–Fisher diffusion. The Annals of Applied Probability 27, 1478–1509.
  • Keane, M. and O’Brien, G. (1994). A Bernoulli factory. ACM Transactions on Modeling and Computer Simulation 4, 213–219.
  • Kloeden, P. and Platen, E. (1995). Numerical Solution of Stochastic Differential Equations. New York: Springer.
  • Łatuszyński, K., Kosmidis, I., Papaspiliopoulos, O. and Roberts, G. (2011). Simulating events of unknown probabilities via reverse time martingales. Random Structures & Algorithms 38, 441–452.
  • Łatuszyński, K., Palczewski, J. and Roberts, G. (2017). Exact inference for a Markov switching diffusion model with discretely observed data. In preparation.
  • Łatuszyński, K. and Roberts, G. O. (2013). CLTs and asymptotic variance of time-sampled Markov chains. Methodology and Computing in Applied Probability 15, 237–247.
  • Mira, A. (2001). Ordering and improving the performance of Monte Carlo Markov chains. Statistical Science 16, 340–350.
  • Mira, A. and Geyer, C. (1999). Ordering Monte Carlo Markov chains. Technical report, School of Statistics, Univ. Minnesota.
  • Nacu, Ş. and Peres, Y. (2005). Fast simulation of new coins from old. The Annals of Applied Probability 15, 93–115.
  • Papaspiliopoulos, O. and Roberts, G. O. (2008). Retrospective Markov chain Monte Carlo methods for Dirichlet process hierarchical models. Biometrika 95, 169–186.
  • Peskun, P. H. (1973). Optimum Monte Carlo sampling using Markov chains. Biometrika 60, 607–612.
  • Roberts, G. and Rosenthal, J. (1997). Geometric ergodicity and hybrid Markov chains. Electronic Communications in Probability 2, 13–25.
  • Schraiber, J. G., Griffiths, R. C. and Evans, S. N. (2013). Analysis and rejection sampling of Wright–Fisher diffusion bridges. Theoretical Population Biology 89, 64–74.
  • Sermaidis, G., Papaspiliopoulos, O., Roberts, G. O., Beskos, A. and Fearnhead, P. (2013). Markov chain Monte Carlo for exact inference for diffusions. Scandinavian Journal of Statistics 40, 294–321.
  • Von Neumann, J. (1951). Various techniques used in connection with random digits. In Monte Carlo Method, Vol. 12. National Bureau of Standards.