Abstract
We show how to speed up sequential Monte Carlo (SMC) for Bayesian inference in large data problems by data subsampling. SMC sequentially updates a cloud of particles through a sequence of distributions, beginning with a distribution that is easy to sample from such as the prior and ending with the posterior distribution. Each update of the particle cloud consists of three steps: reweighting, resampling, and moving. In the move step, each particle is moved using a Markov kernel; this is typically the most computationally expensive part, particularly when the dataset is large. It is crucial to have an efficient move step to ensure particle diversity. Our article makes two important contributions. First, in order to speed up the SMC computation, we use an approximately unbiased and efficient annealed likelihood estimator based on data subsampling. The subsampling approach is more memory efficient than the corresponding full data SMC, which is an advantage for parallel computation. Second, we use a Metropolis within Gibbs kernel with two conditional updates. A Hamiltonian Monte Carlo update makes distant moves for the model parameters, and a block pseudo-marginal proposal is used for the particles corresponding to the auxiliary variables for the data subsampling. We demonstrate both the usefulness and limitations of the methodology for estimating four generalized linear models and a generalized additive model with large datasets.
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References
Baldi, P., Sadowski, P., Whiteson, D.: Searching for exotic particle in high energy physics with deep learning. Nat. Commun. 5, 1–9 (2014)
Bardenet, R., Doucet, A., Holmes, C.: On Markov chain Monte Carlo methods for tall data. J. Mach. Learn. Res. 18(1), 1515–1557 (2017)
Beskos, A., Jasra, A., Kantas, N., Thiery, A.: On the convergence of adaptive sequential Monte Carlo methods. Ann. Appl. Probab. 26(2), 1111–1146 (2016)
Betancourt, M.: A conceptual introduction to Hamiltonian Monte Carlo. ArXiv preprint arXiv:1701.02434 (2017)
Brooks, S., Gelman, A., Jones, G., Meng, X.-L.: Handbook of Markov chain Monte Carlo. CRC Press, Boca Raton (2011)
Buchholz, A., Chopin, N., Jacob, P.E.: Adaptive tuning of Hamiltonian Monte Carlo within sequential Monte Carlo. ArXiv preprint arXiv:1808.07730 (2018)
Ceperley, D., Dewing, M.: The penalty method for random walks with uncertain energies. J. Chem. Phys. 110(20), 9812–9820 (1999)
Chib, S., Jeliazkov, I.: Marginal likelihood from the Metropolis–Hastings output. J. Am. Stat. Assoc. 96(453), 270–281 (2001)
Chopin, N.: A sequential particle filter method for static models. Biometrika 89(3), 539–552 (2002)
Dang, K.-D., Quiroz, M., Kohn, R., Tran, M.-N., Villani, M.: Hamiltonian Monte Carlo with energy conserving subsampling. J. Mach. Learn. Res. 20(100), 1–31 (2019)
Daviet, R.: Inference with Hamiltonian sequential Monte Carlo simulators. http://www.remidaviet.com/files/HSMC-paper.pdf (2016)
Del Moral, P., Doucet, A., Jasra, A.: Sequential Monte Carlo samplers. J. Roy. Stat. Soc. B 68(3), 411–436 (2006)
Del Moral, P., Doucet, A., Jasra, A.: An adaptive Sequential Monte Carlo for approximate Bayesian computation. Stat. Comput. 22(5), 1009–1020 (2012)
Deligiannidis, G., Doucet, A., Pitt, M.K.: The correlated pseudomarginal method. J. R. Stat. Soc. Ser. B Stat. Methodol. 80(5), 839–870 (2018)
Doucet, A., De Freitas, N., Gordon, N.: An introduction to sequential Monte Carlo methods. In: Sequential Monte Carlo Methods in Practice, pp. 3–14. Springer (2001)
Duan, J.C., Fulop, A.: Density-tempered marginalised sequential Monte Carlo samplers. J. Bus. Econ. Stat. 33(2), 192–202 (2015)
Duane, S., Kennedy, A.D., Pendleton, B.J., Roweth, D.: Hybrid Monte Carlo. Phys. Lett. B 195(2), 216–222 (1987)
Fearnhead, P., Taylor, B.M.: An adaptive sequential Monte Carlo sampler. Bayesian Anal. 8(2), 411–438 (2013)
Giordani, P., Jacobson, T., Von Schedvin, E., Villani, M.: Taking the twists into account: predicting firm bankruptcy risk with splines of financial ratios. J. Financ. Quant. Anal. 49(4), 1071–1099 (2014)
Guldas, H., Cemgil, A.T., Whiteley, N., Heine, K.: A practical introduction to butterfly and adaptive resampling in sequential Monte Carlo. IFAC-PapersOnLine 48(28), 787–792 (2015)
Heine, K., Whiteley, N., Cemgil, A.T.: Parallelizing particle filters with butterfly interactions. Scand. J. Stat. 47, 361–396 (2019)
Jasra, A., Stephens, D.A., Doucet, A., Tsagaris, T.: Inference for Lévy-driven stochastic volatility models via adaptive Sequential Monte Carlo. Scand. J. Stat. 38(1), 1–22 (2011)
Jeffreys, H.: The Theory of Probability. OUP, Oxford (1961)
Johnson, A.A., Jones, G.L., Neath, R.C.: Component-wise Markov chain Monte Carlo: uniform and geometric ergodicity under mixing and composition. Stat. Sci. 28(3), 360–375 (2013)
Kass, R.E., Raftery, A.E.: Bayes factors. J. Am. Stat. Assoc. 90(430), 773–795 (1995)
Lee, A., Yau, C., Giles, M.B., Doucet, A., Holmes, C.C.: On the utility of graphics cards to perform massively parallel simulation of advanced Monte Carlo methods. J. Comput. Graph. Stat. 19(4), 769–789 (2010)
Liu, J.S.: Monte Carlo Strategies in Scientific Computing. Springer, New York (2001)
Murray, L.M., Lee, A., Jacob, P.E.: Parallel resampling in the particle filter. J. Comput. Graph. Stat. 25(3), 789–805 (2016)
Neal, R.: Annealed importance sampling. Stat. Comput. 11, 125–139 (2001)
Neal, R.M.: MCMC using Hamiltonian dynamics. Handbook of Markov chain Monte Carlo (2011)
Quiroz, M., Villani, M.: Dynamic mixture-of-experts models for longitudinal and discrete-time survival data. https://github.com/mattiasvillani/Papers/raw/master/DynamicMixture.pdf (2013)
Quiroz, M., Tran, M.-N., Villani, M., Kohn, R., Dang, K.-D.: The block-Poisson estimator for optimally tuned exact subsampling MCMC. ArXiv preprint arXiv:1603.08232v5 (2018a)
Quiroz, M., Villani, M., Kohn, R., Tran, M.-N., Dang, K.-D.: Subsampling MCMC: an introduction for the survey statistician. Sankhya A 80, 33–69 (2018b)
Quiroz, M., Kohn, R., Villani, M., Tran, M.N.: Speeding up MCMC by efficient data subsampling. J. Am. Stat. Assoc. 114, 831–843 (2019)
Roberts, G.O., Stramer, O.: Langevin diffusions and Metropolis-Hastings algorithms. Methodol. Comput. Appl. Probab. 4(4), 337–357 (2002)
Roberts, G.O., Gelman, A., Gilks, W.R.: Weak convergence and optimal scaling of random walk Metropolis-Hastings. Ann. Appl. Probab. 7(1), 110–120 (1997)
Sim, A., Filippi, S., Stumpf, M.P.: Information geometry and sequential Monte Carlo. ArXiv preprint arXiv:1212.0764 (2012)
South, L.F., Pettitt, A.N., Friel, N., Drovandi, C.C.: Efficient use of derivative information within SMC methods for static Bayesian models. https://eprints.qut.edu.au/108150/ (2017)
South, L.F., Pettitt, A.N., Drovandi, C.C., et al.: Sequential Monte Carlo samplers with independent Markov chain Monte Carlo proposals. Bayesian Anal. 14(3), 753–776 (2019)
Tran, M.N., Kohn, R., Quiroz, M., Villani, M.: The block-pseudo marginal sampler. Arxiv preprint arXiv:1603.02485v5 (2017)
Wang, L., Wang, S., Bouchard-Côté, A.: An annealed sequential Monte Carlo method for Bayesian phylogenetics. Syst. Biol. 69(1), 155–183 (2020)
Acknowledgements
We thank the Associate Editor and two reviewers for helping to improve both the content and the presentation of the article. Khue-Dung Dang, David Gunawan, Matias Quiroz and Robert Kohn were partially supported by Australian Research Council Center of Excellence grant CE140100049.
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Gunawan, D., Dang, KD., Quiroz, M. et al. Subsampling sequential Monte Carlo for static Bayesian models. Stat Comput 30, 1741–1758 (2020). https://doi.org/10.1007/s11222-020-09969-z
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DOI: https://doi.org/10.1007/s11222-020-09969-z