Variational Bayesian identification and prediction of stochastic nonlinear dynamic causal models. (English) Zbl 1229.62027
Summary: We describe a general variational Bayesian approach for approximate inference on nonlinear stochastic dynamic models. This scheme extends established approximate inference on hidden-states to cover: (i) nonlinear evolution and observation functions, (ii) unknown parameters and (precision) hyperparameters and (iii) model comparison and prediction under uncertainty. Model identification or inversion entails the estimation of the marginal likelihood or evidence of a model. This difficult integration problem can be finessed by optimising a free-energy bound on the evidence using results from the variational calculus. This yields a deterministic update scheme that optimises an approximation to the posterior density on the unknown model variables. We derive such a variational Bayesian scheme in the context of nonlinear stochastic dynamic hierarchical models, for both model identification and time-series prediction. The computational complexity of the scheme is comparable to that of an extended Kalman filter, which is critical when inverting high dimensional models or long time-series. Using Monte-Carlo simulations, we assess the estimation efficiency of this variational Bayesian approach using three stochastic variants of chaotic dynamic systems. We also demonstrate the model comparison capabilities of the method, its self-consistency and its predictive power.
MSC:
| 62F15 | Bayesian inference |
| 62M10 | Time series, auto-correlation, regression, etc. in statistics (GARCH) |
| 62M20 | Inference from stochastic processes and prediction |
| 37D45 | Strange attractors, chaotic dynamics of systems with hyperbolic behavior |
| 65C05 | Monte Carlo methods |
Keywords:
approximate inference; model comparison; EM; Laplace approximation; free-energy; SDE; nonlinear state-space models; DCM; Kalman filter; rauch smootherSoftware:
BayesDAReferences:
| [1] | Friston, K. J.; Harrison, L.; Penny, W., Dynamic causal modelling, Neuroimage, 19, 1273-1302 (2003) |
| [2] | Kiebel, S. J.; Garrido, M. I.; Friston, K. J., Dynamic causal modelling of evoked responses: The role of intrinsic connections, Neuroimage, 36, 332-345 (2007) |
| [3] | Judd, K.; Smith, L. A., Indistinguishable states II: The imperfect model scenario, Physica D, 196, 224-242 (2004) · Zbl 1081.62074 |
| [4] | Saarinen, A.; LInne, M. L.; Yli-Harja, O., Stochastic differential equation model for cerebellar granule cell excitability, Plos Comput. Bio., 4 (2008) |
| [5] | Herrmann, C. S., Human EEG responses to 1-100 Hz flicker: Resonance phenomena in visual cortex and their potential correlation to cognitive phenomena, Exp. Brain Res., 137, 149-160 (1988) |
| [6] | Jimenez, J. C.; Ozaki, T., An approximate innovation method for the estimation of diffusion processes from discrete data, J. Time Ser. Anal., 76, 77-97 (2006) · Zbl 1115.62100 |
| [7] | Friston, K. J.; Trujillo, N. J.; Daunizeau, J., DEM: A variational treatment of dynamical systems, Neuroimage, 41, 849-885 (2008) |
| [8] | A. Joly-Dave, The fronts and Atlantic storm-track experiment (FASTEX): Scientific objectives and experimental design, Bull. Am. Soc. Meteorol, Mto-France, Toulous, France, 1997. http://citeseer.ist.psu.edu/496255.html; A. Joly-Dave, The fronts and Atlantic storm-track experiment (FASTEX): Scientific objectives and experimental design, Bull. Am. Soc. Meteorol, Mto-France, Toulous, France, 1997. http://citeseer.ist.psu.edu/496255.html |
| [9] | Wikle, C. K.; Berliner, L. M., A Bayesian tutorial for data assimilation, Physica D, 230, 1-16 (2007) · Zbl 1113.62032 |
| [10] | Briers, M.; Doucet, A.; Maskell, S., Smoothing algorithm for state-space models, IEEE Trans. Signal Process. (2004) |
| [11] | Kushner, H. J., (Probability Methods for Approximations in Stochastic Control and for Elliptic Equations. Probability Methods for Approximations in Stochastic Control and for Elliptic Equations, Mathematics in Science and Engineering, vol. 129 (1977), Accademic Press: Accademic Press New York) · Zbl 0547.93076 |
| [12] | Pardoux, E., Filtrage non-lineaire et equations aux derivees partielles stochastiques associees, Ecole d’ete de probabilites de Saint-Flour XIX - 1989, (Lectures Notes in Mathematics, vol. 1464 (1991), Springer-Verlag) · Zbl 0732.60050 |
| [13] | F.E. Daum, J. Huang, The curse of dimensionality for particle filters, in: Proc. of IEEE Conf. on Aerospace, Big Sky, MT, 2003; F.E. Daum, J. Huang, The curse of dimensionality for particle filters, in: Proc. of IEEE Conf. on Aerospace, Big Sky, MT, 2003 |
| [14] | Julier, S.; Uhlmann, J.; Durrant-Whyte, H. F., A new method for the nonlinear transformation of means and covariances in filters and estimators, IEEE Trans. Automat. Control. (2000) · Zbl 0973.93053 |
| [15] | G.L. Eyink, A variational formulation of optimal nonlinear estimation. ArXiv:physics/0011049; G.L. Eyink, A variational formulation of optimal nonlinear estimation. ArXiv:physics/0011049 |
| [16] | Budhiraja, A.; Chen, L.; Lee, C., A survey of numerical methods for nonlinear filtering problems, Physica D, 230, 27-36 (2007) · Zbl 1114.65301 |
| [17] | Arulampalam, M. S.; Maskell, M.; Gordon, N.; Clapp, T., A tutorial on particle filters for online nonlinear/non-Gaussian Bayesian tracking, IEEE Trans. Signal Process., 50, 2 (2002), (special issue) |
| [18] | Doucet, A.; Tadic, V., Parameter estimation in general state-space models using particle methods, Ann. Inst. Stat. Math., 55, 409-422 (2003) · Zbl 1047.62084 |
| [19] | Wan, E.; Nelson, A., Dual extended Kalman filter methods, (Haykin, S., Filtering and Neural Networks (2001), Wiley: Wiley New York), 123-173, (Chapter 5) |
| [20] | Yedidia, J. S., An Idiosyncratic Journey Beyond Mean Field Theory (2000), MIT Press |
| [21] | M. Beal, Variational algorithms for approximate Bayesian inference, University of London Ph.D. Thesis, 2003; M. Beal, Variational algorithms for approximate Bayesian inference, University of London Ph.D. Thesis, 2003 |
| [22] | M. Beal, Z. Ghahramani, The variational Kalman smoother. Technical Report, University College London, 2001. http://citeseer.ist.psu.edu/ghahramani01variational.html; M. Beal, Z. Ghahramani, The variational Kalman smoother. Technical Report, University College London, 2001. http://citeseer.ist.psu.edu/ghahramani01variational.html |
| [23] | Wang, B.; Titterington, D. M., Convergence and asymptotic normality of variational Bayesian approximations for exponential family models with missing values, ACM Internat. Conf. Proc. Series, 70, 577-584 (2004) |
| [24] | Roweis, S. T.; Ghahramani, Z., An EM algorithm for identification of nonlinear dynamical systems, (Haykin, S., Kalman Filtering and Neural Networks (2001)) |
| [25] | Valpola, H.; Karhunen, J., An unsupervised learning method for nonlinear dynamic state-space models, Neural Comput., 14, 1, 2547-2692 (2002) · Zbl 1057.68094 |
| [26] | C. Archambeau, D. Cornford, M. Opper, J. Shawe-Taylor, Gaussian process approximations of stochastic differential equations, in: JMLR: Workshop and Conferences Proceedings, vol. 1, 2007, pp. 1-16; C. Archambeau, D. Cornford, M. Opper, J. Shawe-Taylor, Gaussian process approximations of stochastic differential equations, in: JMLR: Workshop and Conferences Proceedings, vol. 1, 2007, pp. 1-16 |
| [27] | Wang, B.; Titterington, D. M., Lack of consistency of mean-field and variational Bayes approximations for state-space models, Neural Process. Lett., 20, 151-170 (2004) |
| [28] | Friston, K. J.; Mattout, J.; Trujillo-Barreto, N.; Ashburner, J.; Penny, W., Variational free-energy and the Laplace approximation, Neuroimage, 34, 220-234 (2007) |
| [29] | Gray, R. M., Entropy and Information Theory (1990), Springer-Verlag · Zbl 0722.94001 |
| [30] | Tanaka, T., A theory of mean field approximation, (Kearns, M. S.; Solla, S. A.; Cohn, D. A., Advances in Neural Information Processing Systems (2001)) · Zbl 1189.94012 |
| [31] | Tanaka, T., Information geometry of mean field approximation, Neural Comput., 12, 1951-1968 (2000) · Zbl 0918.41024 |
| [32] | G.E. Hinton, D. Van Camp, Keeping neural networks simple by minimizing the description length of the weights, in: Proc. of COLT-93, 1993, pp. 5-13; G.E. Hinton, D. Van Camp, Keeping neural networks simple by minimizing the description length of the weights, in: Proc. of COLT-93, 1993, pp. 5-13 |
| [33] | Carlin, B. P.; Louis, T. A., Bayes and empirical Bayes methods for data analysis, (Text in Statistical Science (2000), Chapman and Hall/CRC) · Zbl 0871.62012 |
| [34] | C. Robert, L’analyse statistique Bayesienne, Ed. Economica, 1992; C. Robert, L’analyse statistique Bayesienne, Ed. Economica, 1992 · Zbl 0827.62006 |
| [35] | Kloeden, P. E.; Platen, E., Numerical Solution of Stochastic Differential Equations, Stochastic Modeling and Applied Probability (1999), Springer |
| [36] | Ozaki, T., A bridge between nonlinear time series models and nonlinear stochastic dynamical systems: A local linearization approach, Statistica Sinica, 2, 113-135 (1992) · Zbl 0820.62075 |
| [37] | Kleibergen, F.; Van Dijk, H. K., Non-stationarity in GARCH models: A Bayesian analysis, J. Appl. Econom., 8, S41-S61 (1993) |
| [38] | Meyer, R.; Fournier, D. A.; Berg, A., Stochastic volatility: Bayesian computation using automatic differentiation and the extended Kalman filter, Econom. J., 6, 408-420 (2003) · Zbl 1065.91533 |
| [39] | Sornette, D.; Pisarenko, V. F., Properties of a simple bilinear stochastic model: Estimation and predictability, Physica D, 237, 429-445 (2008) · Zbl 1133.91504 |
| [40] | Tropper, M. M., Ergodic and quasideterministic properties of finite-dimensional stochastic systems, J. Stat. Phys., 17, 491-509 (1977) · Zbl 1255.60174 |
| [41] | Björck, A., Numerical Methods for Least Squares Problems (1996), SIAM: SIAM Philadelphia · Zbl 0847.65023 |
| [42] | Lacour, C., Nonparametric estimation of the stationary density and the transition density of a Markov chain, Stoch. Process. Appl., 118, 232-260 (2008) · Zbl 1129.62028 |
| [43] | Angeli, D.; Ferrell, J. E.; Sontag, E. D., Detection of multistability, bifurcations, and hysteresis in a large class of biological positive-feedback systems, Proc. Nat. Atl. Sci., 101, 1822-1827 (2004) |
| [44] | Lorenz, E. N., Deterministic nonperiodic flow, J. Atmospheric Sci., 20, 130-141 (1963) · Zbl 1417.37129 |
| [45] | H. Keller, Attractors and bifurcations of the stochastic Lorenz system, Technical Report No. 389, Universitat Bremen, 1996. citeseer.ist.psu.edu/keller96attractors.html; H. Keller, Attractors and bifurcations of the stochastic Lorenz system, Technical Report No. 389, Universitat Bremen, 1996. citeseer.ist.psu.edu/keller96attractors.html |
| [46] | Fitzhugh, R., Impulses and physiological states in theoretical models of nerve membranes, Biophys. J., 1, 445-466 (1961) |
| [47] | J.S. Nagumo, S. Arimoto, S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proc. IRE 1962 50, pp. 2061-2070; J.S. Nagumo, S. Arimoto, S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proc. IRE 1962 50, pp. 2061-2070 |
| [48] | Gill, R. D.; Levit, B. Y., Applications of the van trees inequality: a Bayesian Cramer-Rao bound, Bernouilli, 1, 59-79 (1995) · Zbl 0830.62035 |
| [49] | Slotine, J.; Li, W., Applied Nonlinear Control (1991), Prentice-Hall, Inc: Prentice-Hall, Inc New Jersey · Zbl 0753.93036 |
| [50] | Gardner, W. A.; Napolitano, A.; Paura, L., Cyclostationarity: Half a century of research, Sig. Process., 86, 639-697 (2006) · Zbl 1163.94338 |
| [51] | Gelman, A.; Carlin, J. B.; Stern, H. S.; Rubin, D. B., Bayesian Data Analysis (2004), Chapman & Hall/CRC editions · Zbl 1039.62018 |
| [52] | Hanson, F. B.; Ryan, D., Mean and quasideterministic equivalence for linear stochastic dynamics, Math. Biosci., 93, 1-14 (1988) · Zbl 0679.92018 |
| [53] | Friston, K. J.; Ashburner, J.; Kiebel, S. J.; Nichols, T.; Penny, W. D., Statistical Parametric Mapping, The Analysis of Functional Brain Images (2006), Academic Press, Elsevier Ltd., ISBN: 10: 0-12-372560-7 |
| [54] | Petrelis, F.; Aumaitre, S.; Mallick, K., Escape from a potential well, stochastic resonance and zero-frequency component of the noise, Europhys. Lett., 79, 40004 (2007) |
| [55] | Ghahramani, Z.; Hinton, G. A., Variational learning for switching state-space models, Neural Comput., 12, 831-864 (2000) |
| [56] | Ito, H. M., Ergodicity of randomly perturbed Lorenz model, J. Stat. Phys., 35, 151-158 (1984) · Zbl 0589.60055 |
| [57] | Turbiner, A., Anharmonic oscillator and double-well potential: Approximating eigenfunctions, Lett. Math. Phys., 74, 169-180 (2005) · Zbl 1092.34049 |
| [58] | Crisan, D.; Lyons, T., A particle approximation of the solution of the Kushner-Stratonovitch equation, Probab. Theory Related Fields, 115, 549-578 (1999) · Zbl 0951.93068 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
