Paul Fearnhead
Title: Conditional simulation of Continuous-time Markov processes
The transition density of many continuous-time Markov processes is intractable. Durham and Gallant have recently shown how to perform an efficient importance sampler that estimates the transition density for diffusion processes. We consider extending this idea to other continuous-time Markov processes; and look at simulating the conditional sample path for such processes.

Tristan Marshall
Title: Perfect Simulation for parameters of a discretely-observed diffusion process
We are concerned with the following basic problem: We have a series of partial observations along the path of some diffusion process X, where X is a member of some parametric family of diffusion processes; we wish to perform Bayesian inference on the parameter of X given the partial observations. We build on the work of Beskos et al. (2005 RSS B Read Paper), who introduce a novel algorithm to exactly simulate the missing paths of the diffusion. Using this algorithm, they show how to construct a Gibbs sampler whose stationary distribution is exactly the posterior distribution of the parameter given the observations. This is an improvement on existing methods, which relied on approximating the missing paths, and so did not converge to the true posterior. Here we show how we can further obtain exact, independent samples from the stationary distribution of this Gibbs Sampler, at least in certain cases - an example of Perfect Simulation. Hence we can sample exactly from the posterior distribution of the parameter, removing the need to assess convergence of our Markov chain. Hence we can eliminate all sources of error except for Monte Carlo error, albeit at an increased computational cost.

Simon Godsill
Title: Sequential inference for jump diffusion models in tracking applications
In this work we propose models for tracking of manoeuvring objects based on a continuous time jump-diffusion model. The jumps in the model allow for sudden changes to new manoeuvres while the diffusion component allows for continuous random disturbances to the object during motion. The models are formulated as linear Gaussian SDEs, conditioned upon the times of jumps. This allows for very efficient Monte Carlo inference, since the trajectory of the object, conditioned on jump times, may be marginalised exactly using the Kalman filter and the prediction error decomposition. We present particle filter implementations of the models for highly manoeuvrable objects, partially observed in high levels of noise. Broader applications of the models to continuous time ARMA models, driven by jumps and Brownian motion, are also considered.

Andrew Golightly
Title: MCMC Sampling for Partially Observed Diffusion Processes with Applications
Diffusion processes governed by stochastic differential equations (SDEs) are a well established tool for modelling continuous time data from a wide range of areas. Consequently, there has been much interest in the task of how to estimate diffusion parameters from partial and discrete observations. Unfortunately, likelihood based inference can be problematic as closed form transition densities are rarely available. One widely used solution adopts the treatment of Pedersen (1995, Scandinavian Journal of Statistics 22:55-71) and involves the introduction of latent data points between every pair of observations to allow an Euler-Maruyama approximation of the true transition densities to become accurate. In recent literature, Markov chain Monte Carlo (MCMC) methods have been used to sample the posterior distribution of latent data and model parameters; however, naive schemes suffer from a mixing problem, highlighted by Roberts and Stramer (2001, Biometrika 88(4):603-621), that worsens with the degree of augmentation. We will consider some recently developed MCMC schemes that are not adversely affected by the amount of augmentation. The methodology will be illustrated by estimating parameters governing a Lotka-Volterra system using partial and discrete data.

Yvo Pokern
Title: Parameter estimation for partially observed hyperelliptic diffusions
The problem of parameter estimation for partially observed hypoelliptic diffusion processes with high frequency discrete time sampling is considered. Since exact likelihoods for the transition densities are typically not known, approximations are used that are expected to work well in the limit of small inter-sample times and large total observation times. The combination of hypoellipticity and partial observation leads to ill-conditioning requiring a judicious combination of approximate likelihoods for the various parameters to be estimated. We combine these in a deterministic scan Gibbs sampler alternating between missing data in the unobserved solution components, and parameters. Numerical experiments display asymptotic consistency of the method when applied to simulated data. A simple application to molecular dynamics data is presented.

Osnat Stramer
Title: On a General Brownian Bridge Sampler
I will discuss the discretized simulation method to approximate the transition density of a diffusion processes using the importance sampling approach. I will present a general form of the Brownian bridge sampler and provide some theoretical justification for its use. Some of the results will be illustrate with a simulated example.

Bruno Casella
Title: Partial Implicit Langevin Schemes in MCMC Applications
This talk concerns the application of partial implicit discretization methods to improve the performances of the standard Euler-Langevin algorithms (ULA and MALA). When dealing with the Euler-Langevin algorithms we might observe a major undesired e_ect: even if the original Langevin di_usion is ergodic or geometrically ergodic, the Euler scheme (ULA) may not be. When this happens, the resulting Metropolis adjusted chain (MALA) will typically fail to be geometrically ergodic. We will show that in such situations the advantage of using partial implicit discretization methods instead the naive Euler method is twofold. On one side, partial implicit methods are able to recover geometric ergodicity from the original di_usion and, on the other, they can be successfully employed as proposals for geometrically ergodic Metropolis-Hastings Langevin algorithms.

Kasper K. Berthelsen
Title: Bayesian inference when the data distribution has an intractable normalising constants
We consider a Bayesian setup where the data distribution is specified by an unnormalised density with an intractable normalising constant. The intractable normalising constant makes conventional Metropolis-Hastings sampling of the posterior impractical. Combining recent ideas we consider an auxiliary variable approach which eliminates the need to evaluate (ratios of) intractable normalising constants. Here we consider the case when the auxiliary variable is a random vector. We expect the mixing properties of the algorithm to improve as the length of auxiliary random vector is increased. We explore this effect in the context of a specific example. Furthermore, we adopt an adaptive approach for tuning the algorithm including the length of auxiliary random vector. This talk is based on joint work with Gareth O. Roberts.

Andrew Stuart
Title: Nonparametric Estimation of the Drift
First and second order equations of Langevin type arise naturally in many applications such as molecular dynamics. For such problems there appears to be a vast difference between algorithms based on likelihood maximization, as appear in the statistics literature, and algorithms based on fitting to the empirical measure, as are often used by practitioners. By taking a non-parametric view we show that, in fact, the two approaches are closely related. This leads to a general approach to non-parametric estimation for reversible diffusions, and for non-reversible diffusions of a particular form, by applying the maximum likelihood principle non-parametrically. Joint work with Yvo Pokern (Warwick, Statistics) and Eric Vanden Eijnden (Courant).

Claudia Kluppelberg
Title: The continous-time GARCH model
In [1] we introduce a continuous-time GARCH [COGARCH(1,1)] model which, driven by a single L´evy noise process, exhibits the same second order properties as the discrete-time GARCH(1,1) model. Moreover, the COGARCH(1,1) model has heavy tails and clusters in the extremes. The second order structure of the COGARCH(1,1) model allows for some estimation procedure based on the ARMA(1,1) autocorrelation structure of the model and other moments. The model can be fitted to high-frequency data, and the statistical analysis also provides an estimator for the volatility; cf. [3].

Philip O'Neill
Title: Bayesian inference for stochastic epidemic models in structured populations based on final outcome data
We consider the problem of Bayesian inference for infection rates in a continuous-time multi-type stochastic epidemic model in which the population has a given structure, given data on final outcome. This amounts to partially observing the epidemic at the start and end of the outbreak only. For such data, a likelihood is both analytically and numerically intractable. This problem can be overcome by imputation of suitable latent variables. We describe two such approaches based on different representations of the epidemic model. We also consider extentions to the methodology for the situation where the observed data are a fraction of the entire population. The methods are illustrated with data on influenza outbreaks.