Abstract
Particle Metropolis–Hastings (PMH) allows for Bayesian parameter inference in nonlinear state space models by combining Markov chain Monte Carlo (MCMC) and particle filtering. The latter is used to estimate the intractable likelihood. In its original formulation, PMH makes use of a marginal MCMC proposal for the parameters, typically a Gaussian random walk. However, this can lead to a poor exploration of the parameter space and an inefficient use of the generated particles. We propose a number of alternative versions of PMH that incorporate gradient and Hessian information about the posterior into the proposal. This information is more or less obtained as a byproduct of the likelihood estimation. Indeed, we show how to estimate the required information using a fixed-lag particle smoother, with a computational cost growing linearly in the number of particles. We conclude that the proposed methods can: (i) decrease the length of the burn-in phase, (ii) increase the mixing of the Markov chain at the stationary phase, and (iii) make the proposal distribution scale invariant which simplifies tuning.
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The data is obtained from the Earthquake Data Base System of the U.S. Geological Survey, which can be accessed at http://earthquake.usgs.gov/earthquakes/eqarchives/.
References
Andrieu, C., Roberts, G.O.: The pseudo-marginal approach for efficient Monte Carlo computations. Ann. Stat. 37(2), 697–725 (2009)
Andrieu, C., Thoms, J.: A tutorial on adaptive MCMC. Stat. Comput. 18(4), 343–373 (2008)
Andrieu, C., Vihola, M.: Convergence properties of pseudo-marginal Markov chain Monte Carlo algorithms. Pre-print arXiv:1012.1484v1 (2011)
Andrieu, C., Doucet, A., Holenstein, R.: Particle Markov chain Monte Carlo methods. J. R. Stat. Soc. 72(3), 269–342 (2010)
Beaumont, M.A.: Estimation of population growth or decline in genetically monitored populations. Genetics 164(3), 1139–1160 (2003)
Cappé, O., Moulines, E., Rydén, T.: Inference in Hidden Markov Models. Springer, Berlin (2005)
Carpenter, J., Clifford, P., Fearnhead, P.: Improved particle filter for nonlinear problems. IEE Proc. Radar Sonar Navig. 146(1), 2–7 (1999)
Dahlin, J.: Sequential Monte Carlo for inference in nonlinear state space models. Licentiate’s thesis no. 1652, Linköping University (2014)
Dahlin, J., Lindsten, F., Schön, T.B.: Particle Metropolis Hastings using Langevin dynamics. In: Proceedings of the 38th International Conference on Acoustics, Speech, and Signal Processing (ICASSP), Vancouver (2013)
Dahlin, J., Lindsten, F., Schön, T.B.: Second-order particle MCMC for Bayesian parameter inference. In: Proceedings of the 19th IFAC World Congress, Cape Town (2014)
Del Moral, P.: Feynman-Kac Formulae—Genealogical and Interacting Particle Systems with Applications. Probability and its applications. Springer, Berlin (2004)
Del Moral, P., Doucet, A., Singh, S.: Forward smoothing using sequential Monte Carlo. Pre-print arXiv:1012.5390v1 (2010)
Diaconis, P., Holmes, S., Neal, R.: Analysis of a nonreversible Markov chain sampler. Ann. Appl. Probab. 10(3), 685–1064 (2000)
Doucet, A., Johansen, A.: A tutorial on particle filtering and smoothing: fifteen years later. In: Crisan, D., Rozovsky, B. (eds.) The Oxford Handbook of Nonlinear Filtering. Oxford University Press, Oxford (2011)
Doucet, A., Jacob, P., Johansen, A.M.: Discussion on Riemann manifold Langevin and Hamiltonian Monte Carlo methods. J. R. Stat. Soc. 73(2), 162 (2011)
Doucet, A., Pitt, M.K., Deligiannidis, G., Kohn, R.: Efficient implementation of Markov chain Monte Carlo when using an unbiased likelihood estimator. arXiv.org, Pre-print arXiv:1210.1871v3 (2012)
Doucet, A., Jacob, P.E., Rubenthaler, S.: Derivative-free estimation of the score vector and observed information matrix with application to state-space models. Pre-print arXiv:1304.5768v2 (2013)
Duane, S., Kennedy, A.D., Pendleton, B.J., Roweth, D.: Hybrid Monte Carlo. Phys. Lett. B 195(2), 216–222 (1987)
Everitt, R.G.: Bayesian parameter estimation for latent Markov random fields and social networks. J. Comput. Gr. Stat. 21(4), 940–960 (2012)
Flury, T., Shephard, N.: Bayesian inference based only on simulated likelihood: particle filter analysis of dynamic economic models. Econom. Theory 27(5), 933–956 (2011)
Girolami, M., Calderhead, B.: Riemann manifold Langevin and Hamiltonian Monte Carlo methods. J. R. Stat. Soc. 73(2), 1–37 (2011)
Golightly, A., Wilkinson, D.J.: Bayesian parameter inference for stochastic biochemical network models using particle Markov chain Monte Carlo. Interface Focus 1(6), 807–820 (2011)
Gordon, N.J., Salmond, D.J., Smith, A.F.M.: Novel approach to nonlinear/non-Gaussian Bayesian state estimation. IEEE Proc. Radar Signal Process. 140(2), 107–113 (1993)
Kitagawa, G., Sato, S.: Monte Carlo smoothing and self-organising state-space model. In: Doucet, A., de Fretias, N., Gordon, N. (eds.) Sequential Monte Carlo methods in practice, pp. 177–195. Springer, Berlin (2001)
Langrock, R.: Some applications of nonlinear and non-Gaussian state-space modelling by means of hidden Markov models. J. Appl. Stat. 38(12), 2955–2970 (2011)
Neal, R.M.: MCMC using Hamiltonian dynamics. In: Brooks, S., Gelman, A., Jones, G., Meng, X.L. (eds.) Handbook of Markov Chain Monte Carlo. Chapman & Hall, London (2010)
Nemeth, C., Fearnhead, P.: Particle Metropolis adjusted Langevin algorithms for state-space models. Pre-print arXiv:1402.0694v1 (2014)
Nocedal, J., Wright, S.: Numerical Optimization, 2nd edn. Springer, Berlin (2006)
Olsson, J., Cappé, O., Douc, R., Moulines, E.: Sequential Monte Carlo smoothing with application to parameter estimation in nonlinear state space models. Bernoulli 14(1), 155–179 (2008)
Peters, G.W., Hosack, G.R., Hayes, K.R.: Ecological non-linear state space model selection via adaptive particle Markov chain Monte Carlo. Pre-print arXiv:1005.2238v1 (2010)
Pitt, M.K., Shephard, N.: Filtering via simulation: auxiliary particle filters. J. Am. Stat. Assoc. 94(446), 590–599 (1999)
Pitt, M.K., Silva, R.S., Giordani, P., Kohn, R.: On some properties of Markov chain Monte Carlo simulation methods based on the particle filter. J. Econom. 171(2), 134–151 (2012)
Poyiadjis, G., Doucet, A., Singh, S.S.: Particle approximations of the score and observed information matrix in state space models with application to parameter estimation. Biometrika 98(1), 65–80 (2011)
Rauch, H.E., Tung, F., Striebel, C.T.: Maximum likelihood estimates of linear dynamic systems. AIAA J. 3(8), 1445–1450 (1965)
Robert, C.P., Casella, G.: Monte Carlo Statistical Methods, 2nd edn. Springer, Berlin (2004)
Roberts, G.O., Rosenthal, J.S.: Optimal scaling of discrete approximations to Langevin diffusions. J. R. Stat. Soc. Ser. B (Stat. Methodol.) 60(1), 255–268 (1998)
Roberts, G.O., Stramer, O.: Langevin diffusions and Metropolis–Hastings algorithms. Methodol. Comput. Appl. Probab. 4(4), 337–357 (2003)
Roberts, G.O., Gelman, A., Gilks, W.R.: Weak convergence and optimal scaling of random walk Metropolis algorithms. Ann. Appl. Probab. 7(1), 110–120 (1997)
Sherlock, C., Thiery, A.H., Roberts, G.O., Rosenthal, J.S. On the efficency of pseudo-marginal random walk Metropolis algorithms. Pre-print arXiv:1309.7209v1 (2013)
Acknowledgments
This work was supported by: Learning of complex dynamical systems (Contract number: 637-2014-466) and Probabilistic modeling of dynamical systems (Contract number: 621-2013-5524) and CADICS, a Linnaeus Center, all funded by the Swedish Research Council.
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Dahlin, J., Lindsten, F. & Schön, T.B. Particle Metropolis–Hastings using gradient and Hessian information. Stat Comput 25, 81–92 (2015). https://doi.org/10.1007/s11222-014-9510-0
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DOI: https://doi.org/10.1007/s11222-014-9510-0