Abstract
Approximate Bayesian computation (ABC) has become a popular technique to facilitate Bayesian inference from complex models. In this article we present an ABC approximation designed to perform biased filtering for a Hidden Markov Model when the likelihood function is intractable. We use a sequential Monte Carlo (SMC) algorithm to both fit and sample from our ABC approximation of the target probability density. This approach is shown to, empirically, be more accurate w.r.t. the original filter than competing methods. The theoretical bias of our method is investigated; it is shown that the bias goes to zero at the expense of increased computational effort. Our approach is illustrated on a constrained sequential lasso for portfolio allocation to 15 constituents of the FTSE 100 share index.
Similar content being viewed by others
References
Anagnostopoulos, C., Tasoulis, D., Adams, N.M., Hand, D.J.: Online optimization for variable selection in data streams. In: ECAI 2008, 18th Eur. Conf. Art. Intel., pp. 132–136 (2008)
Anderson, B., Moore, J.: Optimal Filtering. Prentice Hall, Englewood Cliffs (1979)
Balakrishnan, S., Madigan, D.: Algorithms for sparse linear classifiers in the massive data setting. J. Mach. Learn. Res. 9, 313–337 (2008)
Beaumont, M.A., Cornuet, J.M., Marin, J.M., Robert, C.P.: Adaptivity for ABC algorithms: the ABC-PMC scheme. Biometrika 96, 983–990 (2009)
Briers, M., Doucet, A., Maskell, S.: Generalized two-filter smoothing for non-linear and non-Gaussian state-space models. Ann. Inst. Stat. Math. 62, 61–89 (2010)
Campillo, F., Rossi, V.: Convolution particle filter for parameter estimation in general state-space models. IEEE Trans. Aerosp. Electron. Syst. (2009)
Cappé, O., Ryden, T., Moulines, É.: Inference in Hidden Markov Models. Springer, New York (2005)
Carvalho, C.M., West, M.: Dynamic matrix-variate graphical models. Bayesian Anal 2, 69–98 (2007)
Casarin, R., Marin, J.M.: Online data processing: comparison of Bayesian regularized filters. Electron. J. Stat. 3, 239–258 (2009)
Cornebise, J., Peters, G.P.: Comment on Andrieu et al. J. R. Stat. Soc. Ser B (2010, to appear)
Del Moral, P.: Feynman-Kac Formulae: Genealogical and Interacting Particle Systems. Springer, New York (2004)
Del Moral, P., Doucet, A., Jasra, A.: An adaptive sequential Monte Carlo method for approximate Bayesian computation. Technical Report, ISM (2009)
Del Moral, P., Doucet, A., Jasra, A.: On adaptive resampling procedures for sequential Monte Carlo methods. Technical Report, INRIA (2010)
DeMiguel, V., Garlappi, L., Uppal, R.: Optimal versus naive diversification: how inefficient is the 1/N strategy? Rev. Financ. Stud. 22, 1915–1953 (2009a)
DeMiguel, V., Garlappi, L., Nogales, F.J., Uppal, R.: A generalized approach to portfolio optimization: Improving performance by constraining portfolio norms. Manag. Sci. 55, 798–812 (2009b)
Doucet, A., Johansen, A.: A tutorial on particle filtering and smoothing: fifteen years later. In: Crisan, D., Rozovsky, B. (eds.) Oxford Handbook of Nonlinear Filtering. OUP, Oxford (2009)
Doucet, A., Godsill, S., Andrieu, C.: On sequential Monte Carlo sampling methods for Bayesian filtering. Stat. Comput. 10, 197–208 (2000)
Doucet, A., De Freitas, J.F.G., Gordon, N.J.: Sequential Monte Carlo Methods in Practice. Springer, New York (2001)
Doucet, A., Briers, M., Sénécal, S.: Efficient block sampling strategies for sequential Monte Carlo methods. J. Comput. Graph. Stat. 15, 693–711 (2006)
Fabozzi, F., Kolm, P.N., Pachamanova, D.A., Focardi, S.M.: Robust Portfolio Optimization and Management. Wiley, Hoboken (2007)
Helmbold, D.P., Schapire, R.E., Singer, Y., Warmuth, M.K.: On-line portfolio selection using multiplicative updates. Math. Finance 8, 325–347 (1998)
Julier, S., Uhlman, J.: A new extension of the Kalman filter to non-linear systems. In: Proc. Aerosense. (1997)
King, A.A., Ionides, E.L., Pascual, M., Bouma, M.J.: Inapparent infections and cholera dynamics. Nature 454, 877–880 (2008)
Kuhn, D., Parpas, P., Rustem, B., Fonseca, R.: Dynamic mean-variance portfolio analysis under model risk. J. Comput. Finance 12, 91–115 (2009)
Kozubowski, T.J., Panorska, A.K.: Multivariate geometric stable distributions in financial applications. Math. Comput. Mod. 29, 83–92 (1999)
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. Technical Report, University of Oxford (2009)
Li, D., Ng, W.L.: Optimal dynamic portfolio selection: multi-period mean-variance formulation. Math. Finance 10, 387–406 (2000)
Liu, J.S.: Monte Carlo Strategies in Scientific Computing. Springer, New York (2001)
Liu, J., West, M.: Combined parameter and state estimation in simulation-based filtering. In: Doucet, A., de Freitas, N., Gordon, N. (eds.) Sequential Monte Carlo Methods in Practice, pp. 197–223 (2001)
Lo, A.W.: Hedge Funds: An Analytic Perspective. Princeton University Press, Princeton (2008)
Majoram, P., Molitor, J., Plagnol, V., Tavaré, S.: Markov chain Monte Carlo without likelihoods. Proc. Natl. Acad. Sci. 100, 15324–15328 (2003)
Markowitz, H.: Mean-variance analysis in portfolio choice and capital markets. J. Finance 7, 77–91 (1952)
Martin, J.M.: Bayesian methods for time series models. PhD Transfer Thesis, Imperial College London (2010)
Merton, R.C.: On estimating the expected return on a market: an exploratory investigation. J. Financ. Econ. 8, 323–361 (1980)
Pages, G., Pham, H.: Optimal quantization methods for non-linear filtering with discrete-time observations. Bernoulli 11, 893–932 (2005)
Pitt, M.K., Shepherd, N.: Filtering via simulation: auxiliary particle filters. J. Am. Stat. Assoc. 94, 590–599 (1999)
Pritchard, J.K., Seielstad, M.T., Perez-Lezaun, A., Feldman, P.: Population growth of human Y chromosome microsatellites. Mol. Biol. Evol. 16, 1791–1798 (1999)
Smidl, V., Quinn, A.: Variational Bayesian filtering. IEEE Trans. Signal Process. 56, 5020–5030 (2008)
Tibshirani, R.: Regression shrinkage and selection via the lasso. J. R. Stat. Soc. Ser. B 58, 267–288 (1996)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Jasra, A., Singh, S.S., Martin, J.S. et al. Filtering via approximate Bayesian computation. Stat Comput 22, 1223–1237 (2012). https://doi.org/10.1007/s11222-010-9185-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11222-010-9185-0