Abstract
We consider Bayesian online static parameter estimation for state-space models. This is a very important problem, but is very computationally challenging as the state-of-the art methods that are exact, often have a computational cost that grows with the time parameter; perhaps the most successful algorithm is that of SM C2 (Chopin et al., J R Stat Soc B 75: 397–426 2013). We present a version of the SM C2 algorithm which has computational cost that does not grow with the time parameter. In addition, under assumptions, the algorithm is shown to provide consistent estimates of expectations w.r.t. the posterior. However, the cost to achieve this consistency can be exponential in the dimension of the parameter space; if this exponential cost is avoided, typically the algorithm is biased. The bias is investigated from a theoretical perspective and, under assumptions, we find that the bias does not accumulate as the time parameter grows. The algorithm is implemented on several Bayesian statistical models.
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References
Andrieu C, Doucet A, Holenstein R (2010) Particle Markov chain Monte Carlo methods (with discussion). J R Statist Soc Ser B 72:269–342
Andrieu C, Doucet A, Tadić V (2009) On-line parameter estimation in general state-space models using pseudo-likelihood. Unpublished Technical Report
Beskos A, Jasra A, Kantas N, Thiery A (2016) On the convergence of adaptive sequential Monte Carlo methods. Ann Appl Probab 26:1111–1146
Borwanker J, Kallianpur G, Prakasa Rao BLS (1971) The Bernstein-Von Mises theorem for Markov processes. Ann Math Stat 42:1241–1253
Cappé O, Ryden T, Moulines É (2005) Inference in hidden Markov models. Springer, New York
Centanni S, Minozzo M (2006) A Monte Carlo approach to filtering for a class of marked doubly stochastic Poisson processes. J Amer Stat Assoc 101:1582–1597
Cérou F, Del Moral P, Guyader A (2011) A non-asymptotic variance theorem for un-normalized Feynman-Kac particle models. Ann Inst Henri Poincare 47:629–649
Chan H P, Heng C W, Jasra A (2016) Theory of parallel particle filters for hidden Markov models. Adv Appl Probab 48:69–87
Chopin N, Jacob P, Papaspiliopoulos O (2013) SM C2: a sequential Monte Carlo algorithm with particle Markov chain Monte Carlo updates. J R Stat Soc B 75:397–426
Crisan D, Miguez J (2014) Particle-kernel estimation of the filter density in state-space models. Bernoulli 20:1879–1929
Crisan D, Miguez J (2014) Nested particle filters for online parameter estimation in discrete-time state-space Markov models. ar**v preprint
Deligiannidis G, Doucet A, Pitt M K (2015) The correlated pseudo-marginal method. ar**v preprint
Del Moral P (2004) Feynman-Kac formulae: genealogical and interacting particle systems with applications. Springer, New York
Del Moral P (2013) Mean field simulation for Monte Carlo integration. Chapman & Hall, London
Del Moral P, Doucet A, Jasra A (2006) Sequential Monte Carlo samplers. J R Stat Soc B 68:411–436
Douc R, Moulines E, Olsson J, Van Handel R (2011) Consistency of the maximum likelihood estimator for general hidden Markov models. Ann Stat 39:474–513
Doucet A, Johansen A (2011) A tutorial on particle filtering and smoothing: fifteen years later. In: Crisan et B, Rozovsky D (eds) Handbook of nonlinear filtering. Oxford University Press, Oxford
Fearnhead P (2002) MCMC, sufficient statistics and particle filters. J Comp Graph Stat 11:848–862
Gilks W R, Berzuini C (2001) Following a moving target - Monte Carlo inference for dynamic Bayesian models. J R Stat Soc B 63:127–146
Jacob P, Murray L, Rubenthaler S (2015) Path storage in the particle filter. Stat Comp 25:487–496
Kantas N, Doucet A, Singh S S, Maciejowski J M, Chopin N (2015) On particle methods for parameter estimation in state-space models. Stat Sci 30:328–351
Polson NG, Stroud JR, Müller P (2008) Practical filtering with sequential parameter learning. J R Stat Soc B 70:413–428
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Moral, P.D., Jasra, A. & Zhou, Y. Biased Online Parameter Inference for State-Space Models. Methodol Comput Appl Probab 19, 727–749 (2017). https://doi.org/10.1007/s11009-016-9511-x
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DOI: https://doi.org/10.1007/s11009-016-9511-x