Nikolay Nikolaev, Goldsmiths College, University of London

Sequential Bayesian Inference


Probabilistic Kalman Filters

The Probabilistic Kalman Filters (PKF) are tools for approximate probabilistic learning in nonlinear state-space models. PKF apply the Bayes' theorem sequentially in time to infer averaged statistics, that is the mean and variance, of the state posterior distribution from the provided data. Such are the Unscented Kalman filter (UKF) [van de Merwe and Wan, 2001] and the Central Difference Filter (CDF) [Norgaard et al., 2000]. They achieve higher accuracy on modelling nonlinear dynamical systems than standard extended Kalman filters because during the time step they calculate the predictive state density by deterministic sampling (thus avoiding linearisation of the observation model using derivatives), and after that they perform the measurement update of the posterior distribution with the standard equations. Their improved performance is due to avoiding the computation of analytical derivatives like Jacobians or Hessians.

UKF rely on a contemporary technique known as the unscented transform [Julier and Ullman, 1997] to pick state samples, called sigma-points, from carefully selected location points. The spread of the location points is determined in such a way so as to obtain a density estimate with the same statistical properties as the true, unknown state distribution. The usefulness of drawing sigma-points is in that they enable to achieve higher accuracy of approximation of the state mean and variance, which is up to the second order for any kind of model nonlinearity.

Recent research develops fully derivative-free One-step Unscented Filters (OUF) using the unscented transform [Julier and Ullman, 1997] and also Gauss-Hermite Quadratures [Zoeter et al., 2004]. A distinguishing characteristic of the OUF is that during both the time and measurement update steps of the training process it samples the state deterministically using either quadratures or the unscented transform. The OUF is more reliable than the UKF because it performs sampling to compute also the measurement update, which allows to achieve more efficient approximation to the posterior.

Empirical investigations show that the one-step unscented filter compares favorably to Recurrent Neural Networks and Extended Kalman filters on modeling nonstationary high-frequency financial time-series and stochastic volatility modeling [Nikolaev,N. and Smirnov,E., 2007b].


Monte Carlo (Particle) Filters

Monte Carlo (Particle) Filters use stochastic sampling to approximate the integrals of the distributions of interest arising in sequential inference, instead of evaluating them by means of linearization with the derivatives of the system equations. They are alternative to deterministic sampling approaches, and are especially useful for accurate modelling of analytically intractable nonstandard distributions (without a predominant mode) encountered when nonlinear observation models are considered. The Monte Carlo methods [Doucet et al., 2001] compute reliably both the predictive and filtering distributions by averaging over populations of samples. These methods are less sensitive to the specificities in the data and fit well the distributions even when the data are contaminated by non-Gaussian noise.

Particle filters (PF) [Gordon et al., 1993], [Doucet et al., 2001] are such a Monte Carlo method for calculating the expectation of a probability density function by sampling. According to it the probability density function is represented by a population of random samples (particles) each with a corresponding weight, and its representation is obtained by summation over the weighted samples. This approach allows to approximate arbitrary well any general distribution by increasing the number of samples.

Current research investigates Gaussian particle filters (GPF) [Kotecha and Djuric, 2003a] which use mixtures of Gaussians to represent the distributions of interest, and calculate non-recursively the weights to avoid degeneration of the learning process. The mixtures are computed by particle filtering techniques, while only their mean and variances are propageted sequentially in time. Adopting mixture distributions helps to produce more accurate results because they provide potential to model successively even non-Gaussian densities. This claim is based on theorems from classical filtering theory which state that the prediction and filtering densities in dynamical models can be described arbitrarily closely by a sufficient number of mixtures from normal distributions. A further improvement suggested bt Gaussian Sum Particle Filters (GSPF) [Kotecha and Djuric, 2003b] is to use also a mixture density to tackle the state noise. This mixture for the state noise brings further flexibility and helps to capture better the unknown nature of the state noise.

The conducted recent research found out that GSPF performs better than standard particle filters and auxiliary particle filters on on modelling stochastic volatilities [Nikolaev and Smirnov, 2007a].


Nikolaev,N. and Smirnov,E. (2007,a). Stochastic Volatility Inference with Monte Carlo Filters, Wilmott Magazine, John Wiley and Sons, July, pp.72-81. (

Nikolaev,N. and Smirnov, E. (2007,b). A One-Step Unscented Particle Filter for Nonlinear Dynamical Systems, In: Proc. Int. Conf. on Artificial Neural Networks, ICANN-2007, LNCS 4668, Springer-Verlag, Berlin, pp.747-756.

Doucet,A., de Freitas,N. and Gordon,N. (Eds). (2001). Sequential Monte Carlo Methods in Practice, Springer-Verlag, New York.

Gordon,N.J., Salmond,D.J. and Smith,A.F.M. (1993). Novel Approach to Nonlinear/non-Gaussian Bayesian State Estimation, Proceedings IEE-F, vol.140, pp.107-113.

Julier,S.J., and Uhlmann,J.K. (1997). A New Extension of the Kalman Filter to Nonlinear Systems, In: Proc. SPIE Int. Soc. Opt. Eng., vol.3068, Orlando, FL, pp.182-193.

Kotecha,J.H. and Djuric,P.M. (2003a). Gaussian Particle Filtering, IEEE Trans. on Signal Processing, vol.51, N:10, pp.2592-2601.

Kotecha,J.H. and Djuric,P.M. (2003b). Gaussian Sum Particle Filtering, IEEE Trans. on Signal Processing, vol.51, N:10, pp.2602-2612.

Norgaard,M., Poulsen,N.K. and Ravn,O. (2000). New Developments in State Estimation for Nonlinear Systems, Automatica, vol.36, pp.1627-1638.

van der Merwe,R. and Wan,E. (2001). The Square-Root Unscented Kalman Filter for State and Parameter Estimation, In: Proc. of the Int. Conf. on Acoustics, Speech, and Signal Processing (ICASSP), (Salt Lake City, Utah), IEEE Press, pp.3461-3464.

Zoeter,O., Ypma,A. and Heskes,T. (2004). Improved Unscented Kalman Smoothing for Stock Volatility Estimation, In: Proc. of the 14th IEEE Signal Processing Society Workshop, A.Barros, J.Principe, J.Larsen, T.Adali, and S.Douglas (Eds.). IEEE Press, NJ, pp.143-152.

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