Particle Markov Chain Monte Carlo methods are used to carry out inference in nonlinear and non-Gaussian state space models, where the posterior density of the states is approximated using particles. Current approaches usually perform Bayesian inference using either a particle marginal Metropolis–Hastings (PMMH) algorithm or a particle Gibbs (PG) sampler. This paper shows how the two ways of generating variables mentioned above can be combined in a flexible manner to give sampling schemes that converge to a desired target distribution. The advantage of our approach is that the sampling scheme can be tailored to obtain good results for different applications. For example, when some parameters and the states are highly correlated, such parameters can be generated using PMMH, while all other parameters are generated using PG because it is easier to obtain good proposals for the parameters within the PG framework. We derive some convergence properties of our sampling scheme and also investigate its performance empirically by applying it to univariate and multivariate stochastic volatility models and comparing it to other PMCMC methods proposed in the literature.
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Andrieu, C., Roberts, G.O.: The pseudo-marginal approach for efficient Monte Carlo computations. Ann. Stat. 37(2), 697–725 (2009)
Andrieu, C., Doucet, A., Holenstein, R.: Particle Markov chain Monte Carlo methods. J. R. Stat. Soc. Ser. B 72(3), 269–342 (2010)
Brix, A.F., Lunde, A., Wei, W.: A general Schwartz model for energy spot price—estimation using a particle MCMC method. Energy Econ. 72, 560–582 (2018)
Carter, C., Kohn, R.: Markov chain Monte Carlo in conditionally Gaussian state space models. Biometrika 83(3), 589–601 (1996)
Chib, S., Pitt, M.K., Shephard, N.: Likelihood based inference for diffusion driven models. Working Paper (2004)
Chib, S., Nardari, F., Shephard, N.: Analysis of high dimensional multivariate stochastic volatility models. J. Econom. 134(2), 341–371 (2006)
Dahlin, J., Lindsten, F., Schön, T.: Particle Metropolis–Hastings using gradient and Hessian information. Stat. Comput. 25(1), 81–92 (2015)
Deligiannidis, G., Doucet, A., Pitt, M.K.: The correlated pseudo-marginal method. J. R. Stat. Soc. Ser. B 80(5), 839–870 (2018)
Douc, R., Cappé, O.: Comparison of resampling schemes for particle filtering. In: Proceedings of the 4th International Symposium on Image and Signal Processing and Analysis, 2005. ISPA 2005, pp. 64–69. IEEE (2005)
Doucet, A., Godsill, S., Andrieu, C.: On sequential Monte Carlo sampling methods for Bayesian filtering. Stat. Comput. 10(3), 197–208 (2000)
Durbin, J., Koopman, S.: Time Series Analysis of State Space Methods, second edn. Oxford University Press, Oxford (2012)
Fearnhead, P., Meligkotsidou, L.: Augmentation schemes for particle MCMC. Stat. Comput. 26(6), 1293–1306 (2016)
Gerlach, R., Carter, C., Kohn, R.: Efficient Bayesian inference for dynamic mixture models. J. Am. Stat. Assoc. 95(451), 819–828 (2000)
Godsill, S., Doucet, A., West, M.: Monte Carlo smoothing for nonlinear time series. J. Am. Stat. Assoc. 99(465), 156–168 (2004)
Guo, D., Wang, X., Chen, R.: New sequential Monte Carlo methods for nonlinear dynamic systems. Stat. Comput. 15(2), 135–147 (2005)
Ignatieva, K., Rodrigues, P., Seeger, N.: Empirical analysis of affine versus nonaffine variance specifications in jump-diffusion models for equity indices. J. Bus. Econ. Stat. 33(1), 68–75 (2015)
Kastner, G., Fruhwirth-Schnatter, S., Lopes, H.F.: Efficient Bayesian inference for multivariate factor stochastic volatility models. J. Comput. Graph. Stat. 26(4), 905–917 (2017)
Kim, S., Shephard, N., Chib, S.: Stochastic volatility: likelihood inference and comparison with ARCH models. Rev. Econ. Stud. 65(3), 361–393 (1998)
Kitagawa, G.: Monte Carlo filter and smoother for non-Gaussian nonlinear state space models. J. Comput. Graph. Stat. 5(1), 1–25 (1996)
Kleppe, T.S., Yu, J., Skaug, H.: Estimating the GARCH diffusion: simulated maximum likelihood in continuous time. SMU Economics and Statistics Working Paper Series, p. 13 (2010)
Lindsten, F., B. Schön, T.: On the use of backward simulation in particle Markov chain Monte Carlo methods. arxiv:1110.2873 (2012a)
Lindsten, F., Schön, T.B.: On the use of backward simulation in the particle Gibbs sampler. In: Proceedings of the 37th International Conference on Acoustics, Speech, and Signal Processing, pp. 3845–3848. ICASSP (2012b)
Lindsten, F., Schon, T.B.: Backward simulation methods for Monte Carlo statistical inference. Found. Trends Mach. Learn. 6(1), 1–143 (2013)
Lindsten, F., Jordan, M.I., Schön, T.B.: Particle Gibbs with ancestor sampling. J. Mach. Learn. Res. 15, 2145–2184 (2014)
Lindsten, F., Bunch, P., Singh, S.S., Schön, T.B.: Particle ancestor sampling for near-degenerate or intractable state transition models. arxiv:1505.0635v1 (2015)
Nemeth, C., Fearnhead, P., Mihaylova, L.: Particle approximations of the score and observed information matrix for parameter estimation in state-space models with linear computational cost. J. Comput. Graph. Stat. 25(4), 1138–1157 (2016a)
Nemeth, C., Sherlock, C., Fearnhead, P.: Particle Metropolis-adjusted Langevin algorithms. Biometrika 103(3), 701–717 (2016b)
Olsson, J., Ryden, T.: Rao-Blackwellization of particle Markov chain Monte Carlo methods using forward filtering backward sampling. IEEE Trans. Signal Process. 59(10), 4606–4619 (2011)
Pitt, M.K., Silva, RdS, 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)
Roberts, G.O., Rosenthal, J.S.: Examples of adaptive MCMC. J. Comput. Graph. Stat. 18(2), 349–367 (2009)
Stein, E., Stein, J.: Stock price distributions with stochastic volatility: an analytic approach. Rev. Financ. Stud. 4, 727–752 (1991)
Stramer, O., Bognar, M.: Bayesian inference for irreducible diffusion processes using the pseudo-marginal approach. Bayesian Anal. 6(2), 231–258 (2011)
Van Der Merwe, R., Doucet, A., De Freitas, N., Wan, E.: The unscented particle filter. Advances in neural information processing systems, pp. 584–590 (2001)
Wu, X., Zhou, G., Wang, S.: Estimation of market prices of risks in the G.A.R.C.H. diffusion model. Economic Research-Ekonomska Istraživanja 31(1), 15–36 (2018)
The work of the authors was partially supported by an ARC Research Council Grant DP120104014. The work of Robert Kohn and David Gunawan was also partially supported by the ARC Center of Excellence Grant CE140100049.
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Mendes, E.F., Carter, C.K., Gunawan, D. et al. A flexible particle Markov chain Monte Carlo method. Stat Comput (2020). https://doi.org/10.1007/s11222-019-09916-7
- Diffusion equation
- Factor stochastic volatility model
- Particle Gibbs sampler