# Workshop Programme

## for period 22 - 25 April 2014

### Monte Carlo Inference for Complex Statistical Models

22 - 25 April 2014

Timetable

 Tuesday 22 April 08:30-09:05 Registration 09:05-09:15 Welcome from Christie Marr (INI Deputy Director) 09:15-10:15 Andrieu, C (University of Bristol) tba Sem 1 10:15-10:30 Morning Coffee 10:30-11:05 Chopin, N (Centre de Recherche en Économie et Statistique (CREST)) Sequential Quasi-Monte Carlo Sem 1 Co-author: Mathieu Gerber (Université de Lausanne and CREST) We develop a new class of algorithms, SQMC (Sequential Quasi-Monte Carlo), as a variant of SMC (Sequential Monte Carlo) based on low-discrepancy points. The complexity of SQMC is O(N\log N), where N is the number of simulations at each iteration, and its error rate is smaller than the Monte Carlo rate O(N^{-1/2}). The only requirement to implement SQMC is the ability to write the simulation of particle x^n_t given x^n_{t-1} as a deterministic function of x^n_{t-1} and uniform variates. We show that SQMC is amenable to the same extensions as standard SMC, such as forward smoothing, backward smoothing, unbiased likelihood evaluation, and so on. In particular, SQMC may replace SMC within a PMCMC (particle Markov chain Monte Carlo) algorithm. We establish several convergence results. We provide numerical evidence in several difficult scenarios than SQMC significantly outperforms SMC in terms of approximation error. 11:05-11:40 Moulines, E (Télécom ParisTech) On the uniform ergodicity of the particle Gibbs sampler Sem 1 Co-authors: Randal Douc (Telecom SudParis), Fred Lindsten (Cambridge) The particle Gibbs sampler is a systematic way of using a particle filter within Markov chain Monte Carlo (MCMC). This results in an off-the-shelf Markov kernel on the space of state trajectories, which can be used to simulate from the full joint smoothing distribution for a state space model in an MCMC scheme. We show that the PG Markov kernel is uniformly ergodic under rather general assumptions, that we will carefully review and discuss. In particular, we provide an explicit rate of convergence which reveals that: (i) for fixed number of data points, the convergence rate can be made arbitrarily good by increasing the number of particles, and (ii) under general mixing assumptions, the convergence rate can be kept constant by increasing the number of particles superlinearly with the number of observations. We illustrate the applicability of our result by studying in detail two common state space models with non-compact state spaces. 11:40-12:15 Thiery, AH (National University of Singapore) Speeding-up Pseudo-marginal MCMC using a surrogate model Sem 1 The pseudo-marginal MCMC algorithm is a powerful tool for exploring the posterior distribution when only unbiased stochastic estimates of the target density are available. In many situations, although there is no closed-form expression for this density, computationally cheap deterministic estimates are also available; one can then use a delayed-acceptance strategy for exploiting these cheap approximations. As powerful as they are, the use of such algorithms are difficult in practice: it involves tuning the MCMC proposals and choosing the computational budget that one is willing to invest in the creation of the unbiased estimates while taking into account the quality of the cheap deterministic approximations. In this talk we discuss how high-dimensional asymptotic results can help in the tuning of these delayed-acceptance pseudo-marginal MCMC algorithms. This is joint work with Chris Sherlock and Andrew Golightly. 12:30-13:30 Lunch at Wolfson Court 13:45-14:20 Pitt, M (University of Warwick) Efficient implementation of Markov chain Monte Carlo when using an unbiased likelihood estimator Sem 1 When an unbiased estimator of the likelihood is used within an Metropolis-Hastings scheme, it is necessary to tradeoff the number of samples used to evaluate the likelihood against the computing time. Many samples will result in a scheme which has similar properties to the case where the likelihood is exactly known but will be expensive. Few samples will result in faster estimation but at the expense of slower mixing of the Markov chain. We explore the relationship between the number of samples and the efficiency of the resulting Metropolis-Hastings estimates. Under the assumption that the distribution of the additive noise introduced by the log-likelihood estimator is independent of the point at which this log-likelihood is evaluated and other relatively mild assumptions, we provide guidelines on the number of samples to select for a general Metropolis-Hastings proposal. We illustrate on a complex stochastic volatility model that these assumptions are approximately satisfied experimentally and that the theoretical insights with regards to inefficiency and computational time hold true. Keywords: Bayesian inference; Estimated likelihood; Metropolis-Hastings; Particle filtering. 14:20-14:55 Reich, S (Universität Potsdam) Particle filters for infinite-dimensional systems: combining localization and optimal transportation Sem 1 Co-author: Yuan Cheng (University of Potsdam) Particle filters or sequential Monte Carlo methods are powerful tools for adjusting model state to data. However they suffer from the curse of dimensionality and have not yet found wide-spread application in the context of spatio-temporal evolution models. On the other hand, the ensemble Kalman filter with its simple Gaussian approximation has successfully been applied to such models using the concept of localization. Localization allows one to account for a spatial decay of correlation in a filter algorithm. In my talk, I will propose novel particle filter implementations which are suitable for localization and, as the ensemble Kalman filter, fit into the broad class of linear transform filters. In case of a particle filter this transformation will be determined by ideas from optimal transportation while in case of the ensemble Kalman filter one essentially relies on the linear Kalman update formulas. This common framework also allows for a mixture of particle and ensemble Kalman filters. Numerical results will be provided for the Lorenz-96 model which is a crude model for nonlinear advection. 14:55-15:30 Stuart, A (University of Warwick) The Filtering Distribution For Partially Observed Chaotic Dynamical Systems Sem 1 Co-author: Daniel Sanz (University of Warwick) Many physical systems can be successfully modelled by a deterministic dynamical system for which, however, the initial conditions may contain uncertainty. In the presence of chaos this can lead to undesirable growth of uncertainty over time. However, when noisy observations of the system are present these may be used to compensate for the uncertainty in the initial state. This scenario is naturally modelled by viewing the initial state as given by a probability distribution, and to then condition this probability distribution on the noisy observations, thereby reducing uncertainty. Filtering refers to the situation where the conditional distribution on the system state is updated sequentially, at the time of each observation. In this talk we investigate the asymptotic behaviour of this filtering distribution for large time. We focus on a class of dissipative systems that includes the Lorenz '63 and '96 models, and the Navier-Stokes equations on a 2D torus. We first st udy the behaviour of a variant on the 3DVAR filter, creating a unified analysis which subsumes the existing work in [1,2] which, itself, builds on [3]. The optimality property of the true filtering distribution is then used, when combined with this modified 3DVAR analysis, to provide general conditions on the observation of our wide class of chaotic dissipative systems which ensure that the filtering distributions concentrate around the true state of the underlying system in the long-time asymptotic regime. [1] C.E.A. Brett, K.F. Lam, K.J.H. Law, D.S. McCormick, M.R. Scott and A.M. Stuart, Accuracy and stability of filters for dissipative PDEs.'' Physica D 245(2013). [2] K.J.H. Law, A. Shukla and A.M. Stuart, Analysis of the 3DVAR Filter for the Partially Observed Lorenz '63 Model.'' Discrete and Continuous Dynamical Systems A, 34(2014). [3] K. Hayden, E. Olsen and E.S. Titi, Discrete data assimilation in the Lorenz and 2D Navier-Stokes equations.'' Physica D 240(2011). 15:30-15:50 Afternoon Tea 15:50-16:25 Lee, A (University of Warwick) Locally adaptive Monte Carlo methods Sem 1 Co-authors: Christophe Andrieu (University of Bristol), Arnaud Doucet (University of Oxford) In various situations of interest, natural implementations of Monte Carlo algorithms such as Markov chain Monte Carlo and sequential Monte Carlo can perform poorly due to uneven performance in different parts of the space in which they operate. For example, in Markov chain Monte Carlo a Markov kernel may behave increasingly poorly in the tails of the target distribution of interest and in sequential Monte Carlo the quality of associated estimates may plummet if too few particles are used at a particular time. We overview a particular strategy, local adaptation, that seeks to overcome some of these phenomena in practice. 16:25-17:00 Whiteley, NP (University of Bristol) Particle filtering subject to interaction constraints Sem 1 Co-authors: Kari Heine (Bristol), Taylan Cemgil (Bogazici), Anthony Lee (Warwick) The potential benefits of parallel or distributed implementation motivates study of the interaction structure of particle filters. Can we do away with resampling, or at least re-structure it in such a way as to be more naturally suited to non-serial implementation? What role does resampling really play in endowing these algorithms with attractive properties? This talk will introduce some new algorithms in this context and discuss their theoretical properties. Related Links: http://arxiv.org/abs/1309.2918 17:00-18:00 Welcome Wine Reception
 Friday 25 April 09:50-10:25 Stumpf, M (Imperial College London) Approximate Bayesian Inference for Stochastic Processes Sem 1 Co-authors: Paul Kirk (Imperial College London), Angelique Ale (Imperial College London), Ann Babtie (Imperial College London), Sarah Filippi (Imperial College London), Eszter Lakatos (Imperial College London), Daniel Silk (Imperial College London), Thomas Thorn (University of Edinburgh) We consider approximate Bayesian computation (ABC) approaches to model the dynamics and evolution of molecular networks. Initially conceived to cope with problems with intractable likelihoods, ABC has gained popularity over the past decade. But there are still considerable problems in applying ABC to real-world problems, some of which are shared with exact Bayesian inference, but some are due to the nature of ABC. Here we will present some recent advances that allow us to apply ABC sequential Monte Carlo (SMC) to real biological problems. The rate of convergence of ABC-SMC depends crucially on the schedule of thresholds, ?t, t=1,2,…,T, and the perturbation kernels used to generate proposals from the previous population of parameters. We show how both of these can be tuned individually, and jointly. Careful calibration of the ABC-SMC approach can result in a 10-fold reduction in the computational burden (or more). I will also provide an overview of an alternative approach where, rather than approximating the likelihood in an ABC framework, we provide approximations to the master equation that describes the evolution of the stochastic system, that go beyond the conventional linear noise approximation (LNA). This allows us to tackle systems with interesting dynamics", that are typically beyond the scope of the LNA, and we will show how to use such approaches in exact Bayesian inference procedures (including nested sampling and SMC). 10:25-10:40 Morning Coffee 10:40-11:15 Schön, TB (Uppsala Universitet) Sequential Monte Carlo methods for graphical models Sem 1 Co-authors: Christian A. Naesseth (Linkoping University), Fredrik Lindsten (University of Cambridge) We develop a sequential Monte Carlo (SMC) algorithm for inference in general probabilistic graphical model. Via a sequential decomposition of the PGM we find a sequence of auxiliary distributions defined on a monotonically increasing sequence of probability spaces. By targeting these auxiliary distributions using purpose built SMC samplers we are able to approximate the full joint distribution defined by the graphical model. Our SMC sampler also provides an unbiased estimate of the partition function (normalization constant) and we show how it can be used within a particle Markov chain Monte Carlo framework. This allows for better approximations of the marginals and for unknown parameters to be estimated. The proposed inference algorithms can deal with an arbitrary graph structure and the domain of the random variables in the graph can be discrete or continuous. Related Links: http://arxiv.org/pdf/1402.0330v1.pdf - Associated paper http://user.it.uu.se/~thosc112/index.html - Speaker (Thomas Schön) home page 11:15-11:50 Murray, L (CSIRO) Sequential Monte Carlo with Highly Informative Observations Sem 1 Co-author: Pierre Del Moral (University of New South Wales) We introduce a sequential Monte Carlo (SMC) method for sampling the state of continuous-time state-space models when observations are highly informative, a situation in which standard SMC methods can perform poorly. The most extreme case is where the observations are exact---of the state itself---and the problem is that of simulating diffusion bridges between given starting and ending states. The basic idea is to introduce a sequence of intermediate weighting and resampling steps between observation times, guiding particles towards the ending state. A few designs that have been useful in practice are given, and demonstrated on some applied problems that feature complex models. 11:50-12:25 Li, K (Uppsala University) Generalised Particle Filters with Gaussian Mixtures Sem 1 Stochastic filtering is defined as the estimation of a partially observed dynamical system. A massive scientific and computational effort has been dedicated to the development of numerical methods for approximating the solution of the filtering problem. Approximating with Gaussian mixtures has been very popular since the 1970s, however the existing work is only based on the success of the numerical implementation and is not theoretically justified. We fill this gap and conduct a rigorous analysis of a new Gaussian mixture approximation to the solution of the filtering problem. In particular, we construct the corresponding approximating algorithm, deduce the L2-convergence rate and prove a central limit type theorem for the approximating system. In addition, we show a numerical example to illustrate some features of this algorithm. This is joint work with Dan Crisan (Imperial College London). References: [1] D. Crisan, K. Li, “A central limit type theorem for Gaussian mixture approximations to the nonlinear filtering problem”, ArXiv1401:6592, (2014). [2] D. Crisan, K. Li, “Generalised particle filters with Gaussian mixtures”, accepted by Stochastic Processes and their Applications, ArXiv1306:0255, (2013). [3] D. Crisan, K. Li, “Generalised particle filters with Gaussian measures”, Proceedings of 19th European Signal Processing Conference, Barcelona, Spain, pp. 659-663, (2011). 12:30-13:30 Lunch at Wolfson Court 14:00-14:45 Sharia, T (Royal Holloway, University of London) Truncated stochastic approximation with moving bounds Sem 1 A wide class of truncated stochastic approximation procedures with moving random bounds will be discussed. This class of procedures has three main characteristics: truncations with random moving bounds, a matrix-valued random step-size sequence, and a dynamically changing random regression function. While we believe that the proposed class of procedures will find its way to a wider range of applications, the main motivation is to accommodate applications to parametric statistical estimation theory. The proposed method allows for incorporation of auxiliary information into the estimation process, and is consistent and asymptotically efficient under certain regularity conditions. Related Links: •http://arxiv.org/pdf/1101.0031v4.pdf - Link to the paper in ArXiv 14:45-15:15 Tadic, V (University of Bristol) Asymptotic Properties of Recursive Maximum Likelihood Estimation in State-Space Models Sem 1 Co-author: Arnaud Doucet (University of Oxford) Recursive maximum likelihood algorithm for state-space models (i.e., for continuous state hidden Markov models) is an iterative estimation method based on particle filter and stochastic gradient search. In this talk, resent results on its asymptotic properties are presented. These results are focused on the asymptotic bias and the asymptotic variance. They also involve diffusion approximation, almost-sure and mean-square convergence of the recursive maximum likelihood algorithm. Some auxiliary (yet, rather interesting) results on the asymptotic properties of the particle filter and the log-likelihood are presented in the talk, too.