ASPI :
Applications of interacting particle systems to statistics
Post-doctoral position opened in 2006 / 2007
Stability of the optimal Bayesian filter as a random dynamical system
Location : IRISA / INRIA Rennes
Duration : one year, starting September 2006 (or later)
Application deadline : March 31, 2006
Team :
ASPI (Applications
of interacting particle systems to statistics)
Contact :
François Le Gland
(tél. : +33 (0)2 99 84 73 62,
e-mail : legland@irisa.fr)
Background :
hidden Markov models, Bayesian filtering, dynamical systems
Subject :
In full generality, filtering is about estimating the hidden state of
a system from noisy and partial observations, usually in a Markov
context. When optimality is considered in the mean-square error sense,
the problem reduces to computing the conditional probability
distribution of the hidden state given past observations, known as
the Bayesian optimal filter. This is a random element
in the space of probability distributions, characterized as the
solution of a random evolution equation driven by the noisy observations,
known as the filtering equation, which depends on
- the Markov transition probabilities,
- and the likelihood functions.
Even if these ingredients of the filtering equation are known exactly,
its initial condition, which is the probability distribution of the
hidden initial state, is usually unknown in practice, and the filtering
equation is initialized with an arbitrary probability distribution.
It is therefore important to study stability properties of the filtering
equation with respect to its initial condition, which has several
important practical implications on
- numerical approximations with approximation error that holds
uniformly in time,
- stability properties of the filtering equation with respect to
model parameters,
- large-time asymptotics in statistics of general hidden Markov
models, etc.
This longstanding issue in Bayesian filtering has received a positive
answer in works by Atar-Zeitouni, Del Moral-Guionnet and Le Gland-Oudjane,
under rather strong mixing assumptions which essentially can
hold when the state space is compact, and has proved surprisingly
difficult to answer in the general noncompact case, even though
several attempts have been made in the recent years by Budhiraja-Ocone,
Chigansky-Liptser, Le Gland-Oudjane and Oudjane-Rubenthaler, yielding
to a positive answer in some special cases.
This calls for new ideas, and the objective of this post-doc
project is
- to explore promising connections
between the stability problem in Bayesian filtering and the theory of
random dynamical systems, which offers a unified approach to study
systems subject to noise,
- and to study the concentration of the measure property for the
Bayesian filter, so as to derive upper bounds for the exponential rate
of forgetting under various asymptotics, such as small (or large)
noise intensities.
Bibliography :
- R. Atar, O. Zeitouni,
Exponential stability for nonlinear filtering,
Annales de l'Institut Henri Poincaré (Probabilités
et Statistiques),
33, 6, pp. 697-725, 1997.
- A. Budhiraja, D.L. Ocone,
Exponential stability in discrete-time filtering
for non-ergodic signals,
Stochastic Processes and their Applications,
82, 2, pp. 245-257, Aug. 1999.
- P. Chigansky, R.S. Liptser,
Stability of nonlinear filters in nonmixing case,
The Annals of Applied Probability,
14, 4, pp. 2038-2056, Nov. 2004.
- P. Del Moral, A. Guionnet,
On the stability of interacting processes with
applications to filtering and genetic algorithms,
Annales de l'Institut Henri Poincaré (Probabilités
et Statistiques),
37, 2, pp. 155-194, Mar. 2001.
- F. Le Gland, N. Oudjane,
A robustification approach to stability and to
uniform particle approximation of nonlinear filters : the example
of pseudo-mixing signals,
Stochastic Processes and their Applications,
106, 2, pp. 279-316, Aug. 2003.
- F. Le Gland, N. Oudjane,
Stability and uniform approximation of nonlinear filters
using the Hilbert metric, and application to particle filters,
The Annals of Applied Probability,
14, 1, pp. 144-187, Feb. 2004.
- N. Oudjane, S. Rubenthaler,
Stability and uniform particle approximation of
nonlinear filters in case of non ergodic signal,
Stochastic Analysis and Applications,
23, 3, pp. 421-448, May 2005.
