Sequential Bayesian Inference

PKF

Probabilistic Kalman Filters
The Probabilistic Kalman Filters (PKF) are tools for approximate probabilistic learning in
nonlinear statespace 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 sigmapoints, 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 sigmapoints 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 derivativefree Onestep Unscented Filters (OUF) using the unscented transform
[Julier and Ullman, 1997] and also GaussHermite 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 onestep unscented filter compares favorably to Recurrent Neural Networks
and Extended Kalman filters on modeling nonstationary highfrequency financial timeseries and stochastic
volatility modeling [Nikolaev,N. and Smirnov,E., 2007b].

MCF

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 nonGaussian 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 nonrecursively 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 nonGaussian 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].

References
Nikolaev,N. and Smirnov,E. (2007,a). Stochastic Volatility Inference with Monte Carlo Filters, Wilmott Magazine,
John Wiley and Sons, July, pp.7281. (www.wilmott.com)
Nikolaev,N. and Smirnov, E. (2007,b). A OneStep Unscented Particle Filter for Nonlinear Dynamical Systems, In: Proc.
Int. Conf. on Artificial Neural Networks, ICANN2007, LNCS 4668, SpringerVerlag, Berlin, pp.747756.
Doucet,A., de Freitas,N. and Gordon,N. (Eds). (2001). Sequential Monte Carlo Methods in Practice, SpringerVerlag, New York.
Gordon,N.J., Salmond,D.J. and Smith,A.F.M. (1993). Novel Approach to Nonlinear/nonGaussian Bayesian State Estimation,
Proceedings IEEF, vol.140, pp.107113.
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.182193.
Kotecha,J.H. and Djuric,P.M. (2003a). Gaussian Particle Filtering, IEEE Trans. on Signal Processing, vol.51, N:10,
pp.25922601.
Kotecha,J.H. and Djuric,P.M. (2003b). Gaussian Sum Particle Filtering, IEEE Trans. on Signal Processing, vol.51, N:10,
pp.26022612.
Norgaard,M., Poulsen,N.K. and Ravn,O. (2000). New Developments in State Estimation for Nonlinear Systems, Automatica, vol.36,
pp.16271638.
van der Merwe,R. and Wan,E. (2001). The SquareRoot 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.34613464.
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.143152.
Relevant Sequential Bayesian Inference Sites:
n.nikolaev@gold.ac.uk