LIST OF SPEAKERS Pavel Chiganski (pavel.chigansky@gmail.com) Dan Crisan (d.crisan@imperial.ac.uk) Persi Diaconis Douc Randal (douc@barbes.polytechnique.fr) Andreas Eberle (eberle@uni-bonn.de) Kari Heine (kari.heine@tut.fi) Marina Kleptsyna (marina.kleptsyna@univ-lemans.fr) Sylvain Rubenthaler (rubentha@math.unice.fr) Wilhelm Stannat (stannat@mathematik.tu-darmstadt.de) Alexander Veretennikov (veretenn@maths.leeds.ac.uk) PROGRAM Tuesday 29 May 9am-10am S. Rubenthaler, The convergence to equilibirum of neutral genetic models. 10am-11am D. Crisan, Stability of the Discrete Time Filter in Terms of the Tails of Noise Distributions 11am-11.30am Coffee Break 11.30am-12.30apm K. Heine, Uniform approximations of discrete time filter 12.30pm -2pm Lunch 2pm-3pm P. Diaconis, Fastest mixing Markov chain - a survey 3pm-3.30pm Coffee Break 3.30pm-4.30pm A. Eberle, Stability of sequential MCMC methods Wednesday 30 May 9am-10am M. Kleptsyna, On discrete time ergodic filters with wrong initial data. 10am-11am A. Veretennikov, On filtering with wrong initial data, continuous time. 11am-11.30am Coffee Break 11.30am-12.30apm R. Douc, Forgetting of the initial distribution for the filter in Hidden Markov Models 12.30pm -2pm Lunch 2pm-3pm P. Chigansky, Model robustness of finite state nonlinear filtering over the infinite time horizon 3pm-3.30pm Coffee Break 3.30pm-4.30pm W. Stannat, Asymptotic stability of Feynman-Kac propagators LIST OF ABSTRACTS * P. Chigansky Title: Model robustness of finite state nonlinear filtering over the infinite time horizon Abstract: We investigate the robustness of nonlinear filtering for continuous time finite state Markov chains, observed in white noise, with respect to misspecification of the model parameters. It is shown that the distance between the optimal filter and that with incorrect model parameters converges to zero uniformly over the infinite time interval as the misspecified model converges to the true model, provided the signal obeys a mixing condition. The filtering error is controlled through the exponential decay of the derivative of the nonlinear filter with respect to its initial condition. We allow simultaneously for misspecification of the initial condition, of the transition rates of the signal, and of the observation function. The first two cases are treated by relatively elementary means, while the latter case requires the use of Skorokhod integrals and tools of anticipative stochastic calculus. * D. Crisan Title Stability of the Discrete Time Filter in Terms of the Tails of Noise Distributions Abstract The stability of the discrete time filter has been a field of active research throughout the recent years. By stability we mean that the effect of the possibly erroneous initial distribution in the filter eventually vanishes as time increases. One of the motivations for the interest in the stability is its close relation to the convergence of various numerical filter approximation schemes, e.g.~particle filters. In this paper, the main result states easily verifiable conditions that are sufficient for the filter stability. Essentially, the conditions state that the filter is stable, if the observation noise is sufficiently light tailed compared to the randomness in the signal process. Compactness of the state space or ergodicity of the signal are not required. * P. Diaconis Title : Fastest mixing Markov chain - a survey Abstract : In a series of papers with Boyd and Xiao, we study the problem of putting weights on the edges of a graph so the associated Markov chain has a given stationary distribution AND the biggest positive spectral gap. Among other things, we discovered that standard heuristics (e.g. Metropolis) can be far from optimal. The tools involved, semi-definite programming, will be introduced in a friendly fashion. We have just begun to extend to continuous spaces. * R. Douc TITLE: Forgetting of the initial distribution for the filter in Hidden Markov Models ABSTRACT: The forgetting of the initial distribution for discrete Hidden Markov Models (HMM) is addressed: a new set of conditions is proposed, to establish the forgetting property of the filter, at a polynomial and geometric rate. Both a pathwise-type convergence of the total variation distance of the filter started from two different initial distributions, and a convergence in expectation are considered. The results are illustrated using different HMM of interest: the dynamic tobit model, the non-linear state space model and the stochastic volatility model. * A. Eberle Title Stability of sequential MCMC methods Abstract Sequential Monte Carlo Samplers are a class of stochastic algorithms for Monte Carlo integral estimation w.r.t. probability distributions, which combine elements of Markov chain Monte Carlo methods and importance sampling/resampling schemes. We develop a stability analysis by functional inequalities for a nonlinear flow of probability measures describing the limit behavior of the algorithms as the number of particles tends to infinity. Stability results are derived both under global and local assumptions on the generator of the underlying Metropolis dynamics. This allows us to prove that the combined methods sometimes have good asymptotic stability properties in multimodal setups where traditional MCMC methods mix extremely slowly. For example, this holds for the mean field Ising model at all temperatures. * K. Heine Title: Uniform approximations of discrete time filter Abstract:Throughout recent years, various particle filter algorithms have been proposed for the approximation of the generally intractable stochastic discrete time filter. Although there are convergence results for finite time intervals, a stronger form of convergence, namely, uniform convergence, is required for bounding the approximation error on an infinite time interval. In this talk, we provide easily verifiable sufficient conditions for the uniform convergence of a certain class of particle filter algorithms. * M. Kleptsyna Title On discrete time ergodic filters with wrong initial data. Abstract: For a class of non-uniformly ergodic Markov chains $(X_n)$, under observations $(Y_n)$ subject to an IID noise, it is shown that a wrong initial data is forgotten in the mean total variation topology. * S. Rubenthaler Title The convergence to equilibirum of neutral genetic models Abstract This work is concerned with the long time behavior of neutral genetic population models, with fixed population size. We design an explicit, finite, exact, genealogical tree based representation of stationary populations that holds both for finite and infinite types (or alleles) models. We then analyze the decays to the equilibrium of finite populations in terms of the convergence to stationarity of their first common ancestor. We estimate the Lyapunov exponent of the distribution flows with respect to the total variation norm. We give bounds on these exponents only depending on the stability with respect to mutation of a single individual; they are inversely proportional to the population size parameter * W. Stannat Title Asymptotic stability of Feynman-Kac propagators Abstract We present results on the asymptotic stability of non-homogeneous measure-valued evolution equations using a variational approach. The results are applied in particular to the pathwise filter equation for the optimal filter of a signal observed with independent additive noise. In this example the variational approach leads to a new interpretation of the rate of stability. * A. Veretennikov Title: On filtering with wrong initial data, continuous time