BISP6 - Sixth Workshop on Bayesian Inference in Stochastic Processes


Sixth Workshop on


Accademia Cusano, Bressanone/Brixen (BZ), Italy

June, 18-20, 2009


Alexandros Beskos

Diffusion limits for MCMC paths

We examine the complexity of MCMC algorithms in high dimensions. Earlier works in the literature have shown that the step of Random-Walk Metropolis should be scaled as 1/n (n being the dimensionality index); for the (Metropolised) Langevin algorithm the scaling is 1/n^{1/3}. Both results have been theoretically justified in the simplified scenario of iid targets. We consider the so-called `hybrid Monte-Carlo' MCMC algorithm used by physicists in molecular dynamics and other applications.

Bridging the machinery employed above with tools from numerical analysis we show that the MCMC trajectory of the hybrid algorithm converges (when appropriately rescaled) to a hypoelliptic SDE. Such a result provides a complete characterization of the efficiency of the algorithm: we conclude that the hybrid algorithm should be scaled as 1/n^{1/6}, with optimal acceptance probability 0.743.

Paul G. Blackwell, Catlin E. Buck, L. A. Collins, R. Rothlisberger and J. J. Wheatley

Inference for Continuous-Time Processes Observed at Uncertain Times: Counting Layers in Ice Cores

Environmental signals `trapped' in polar ice, such as atmospheric levels of particular chemicals or isotopes, offer valuable information about past climates. These signals can be measured from ice cores, but their interpretation requires knowledge of the relationship between the depth within the core and the actual age of the ice. One approach exploits the fact that some signals have strong annual cycles, and so counting these cycles|`layer counting'|can in principle give a precise date. In practice, layer counting is currently done by eye, based on plotting the signal level against depth, with no formal measure of uncertainty. In this paper, we describe a statistical approach which models the underlying signal as a sine wave of varying level and amplitude. Measurements from the ice core are modelled as a sig nal of this form plus errors coming from an autocorrelated continuous-time (Ornstein-Uhlenbeck) process. The time-depth relationship is modelled as a piecewise linear function. Bayesian analysis of this model, using reversible jump Markov chain Monte Carlo, allows inference about the age-depth rela- tionship, and specifically the number of years covered by a particular section of ice core, including quantitative assessment of the uncertainty involved.

Catherine Calder

Kernel-Based Models for Space-Time Data

Kernel-based models offer a flexible and descriptive framework for studying space-time processes. Nonstationary and anisotropic covariance structures can be readily accommodated by allowing kernel parameters to vary over space and time. In addition, dimension reduction strategies make model fitting computationally feasible for large datasets. We explore various properties of this class of statistical models, as well as the implications of dimension reduction strategies on these properties. In addition, we illustrate the use of one version of these models in a study of carbonaceous aerosols over mainland Southeast Asia. Finally, efficient MCMC strategies are introduced to further facilitate model fitting and comparison.

Fernanda D'Ippoliti and Sara Pasquali

Bayesian Inference for SVCCJ model

We present a Bayesian approach to parameter estimation in a stochastic volatility model with contemporaneous and correlated jumps (SVCCJ) starting from time series of financial data.

In the present work, we develop a Bayesian estimation methodology for the SVCCJ model formulated in D'Ippoliti et al. (2008) that can be applied to other stochastic volatility jump-diffusion processes. The asset pricing model presented in D'Ippoliti et al. (2008) describes the evolution of the underlying asset by a SDE with jumps and two diffusion terms; the first term with constant volatility and the second one of Heston type. The dynamics of the volatility follows a square-root process with jumps.

The choice of the value of parameters is a critical point in the specification of a model: in financial setting, a good estimation of the parameters of the model is an essential requirement for a good valuation of asset prices as well. In financial market, prices are recorded at discrete times. It follows that the problem of parameter estimation from time series can be brought back to parametric inference for discretely observed diffusion processes with jumps. While values of the underlying are available on the market, volatility values are not given. Hence, we consider volatility values as missing data. Moreover, if the number of observed underlying prices is small, it is necessary to generate latent data between two consecutive observations of the underlying in order to obtain reliable estimates for parameters. Volatility values and latent observations for the underlying are considered as missing data and an MCMC method is applied to simulate them.

The Bayesian method used to estimate parameters can be summarized as follows: (i) draw a value for each parameter from its prior distribution, (ii) generate missing data given the current value of the parameters, (iii) sample a value for each parameter from its posterior distribution. One iteration of the Markov chain is completed when missing data and parameters are updated. Applying recursively the procedure, we obtain a sequence of values for each parameter giving an approximation of its posterior distribution.

In recent years, the problem of parameter estimation in a Bayesian setting for stochastic volatility models has been widely studied. Among others, Eraker (2001) and Durham and Gallant (2002) consider both volatility and underlying prices as latent data. To the best of our knowledge, the issue of parameter estimation in stochastic volatility with jumps model has not been addressed in this way. Asgharian and Bengtsson (2006), Eraker and al. (2003), and Nakajima and Omori (in press) analyze the problem of parameter estimation using a large number of observations and only volatility values are missing data.

Starting from the MCMC algorithms proposed by Eraker et al. (2003) and Durham and Gallant (2002) we extend the procedure to stochastic volatility models with jumps. Moreover, some numerical results are provided to illustrate the procedure.


1.Asgharian H., and Bengtsson C., 2006, Jump Spillover in International Equity Markets, Journal of Financial Econometrics, Vol. 4, No. 2, 167-203.
2.D'Ippoliti F., Moretto E., Pasquali S., and Trivellato S., 2008, The effect of stochastic volatility and jumps on derivative pricing, submitted.
3.Durham G. B., and Gallant A. R., 2002, Numerical Techniques for Maximum Likelihood Estimation of Continuous-time Diffusion Processes, The Journal of Business and Economic Statistics, Vol. 20, 297-316.
4.Eraker B., 2001, MCMC Analysis of Diffusion Models with Application to Finance Journal of Business and Economic Statistics, Vol. 19-2 (April), 177-191.
5.Eraker B., Johannes M., and Polson N., 2003, The Impact of Jumps in Volatilty and Returns, The Journal of Finance, Vol. LVIII, No. 3, June, 1269-1300.
6.Nakajima J., and Omori Y., 2007, Leverage, heavy-tails and correlated jumps in a stochastic volatility models, in press.

Stefano Favaro, Alessandra Guglielmi and Stephen G. Walker

Some developments of the Feigin-Tweedie Markov chain

We define and investigate a new class of measure-valued Markov chains having as unique invariant measure the law of a Dirichlet process. This class of Markov chains includes as a particular case the well known Markov chain introduced by Feigin and Tweedie (1989), which has been widely used in order to characterize linear functionals of the Dirichlet process and to provide approximation procedures for estimating the law of the mean of a Dirichlet process.

Tom Fricker and Jeremy Oakley

Emulators for Multiple Output Computer Models

An emulator is a statistical surrogate for an expensive computer model, used to obtain a fast probabilistic prediction of the output. Emulators can be constructed by considering the input-output relationship as an unknown function and modelling the uncertainty using a Gaussian process prior on the function.

If the computer model produces multiple outputs, the emulator must capture two types of correlation: correlation over the input space, and correlation between different outputs. We show that the usual mathematically-convenient approach of treating the two types as separable can result in misspecified input-space correlation functions for some or all of the outputs. We propose an emulator with a nonseparable covariance, based on the linear model of coregionalization (LMC) taken from the geostatistical literature. By allowing different outputs to have different correlation functions, the LMC emulator can provide better estimates of prediction uncertainty across the outputs. The advantages of the LMC over a separable structure are demonstrated in the emulation of a simple climate model.

Sylvia Frühwirth-Schnatter and Helga Wagner

Stochastic Model Specification Search for State Space Models

State space models are widely used in time series analysis to deal with processes which gradually change over time. Model specification, however, is a difficult task as one has to decide first, which components to include into the model, and second, whether these components are fixed or stochastic.

Using a Bayesian approach, one could determine the posterior probabilities of each model separately, which requires estimation of the marginal likelihood for each model by some numerical method. A modern approach to Bayesian model selection is to apply some model space MCMC methods by sampling jointly model indicators and parameters as is done, e.g., in the stochastic variable selection approach for regression models.

In this talk we discuss model space MCMC for state space models. To this aim, we rewrite the state space model in a non-centered version and extend the stochastic variable selection approach to state space models. This allows to choose appropriate components and to decide, if these components are deterministic or stochastic. Details will be provided for time-varying parameter models and unobserved component time series models. The method is extended to non-Gaussian state space models for binary, multinomial or count data, where we make use of auxiliary mixture sampling.

Robert B. Gramacy

Particle Learning for Trees with Applications to Sequential Design and Optimization

We devise a sequential Monte Carlo method, via particle learning (PL), for on-line sampling from the posterior distribution of a sequential process over trees. The mixing problems which typically plague MCMC approaches to similar Bayesian CART models are circumvented by an automatic mixing of global (re-sample) and local (propagate) rules -- the cornerstones of PL. We consider regression trees with both constant and linear mean functions at the leaves, and thereby index a thrifty but powerful family of nonparametric regression models. We further exploit the on-line nature of inference with extensions for sequential design (a.k.a. active learning) under tree models for both (improved) surface prediction and global optimization applications. In both cases, we demonstrate how our proposed algorithms provide better results compared to higher-powered methods but use a fraction of the computational cost.

Peter Green and Fay Hosking

Markov modelling in genome-wide association studies`

We propose a Bayesian modelling approach to the analysis of genome-wide association studies based on single nucleotide polymorphism (SNP) data. Our latent seed model combines various aspects of k-means clustering, hidden Markov models (HMMs) and logistic regression into a fully Bayesian model. It is fitted using Markov chain Monte Carlo methods, with Metropolis-Hastings update steps. The approach is flexible, both in allowing different types of genetic models, and because it can be easily extended while remaining computationally feasible due to the use of fast algorithms for HMMs. It allows for inference primarily on the location of the causal locus and also on other parameters of interest. The latent seed model is used here to analyse three data sets, using both synthetic and real disease phenotypes with real SNP data, and shows promising results. Our method is able to correctly identify the causal locus in examples where single SNP analysis is both successful and unsuccessful at identifying the causal SNP.

Jim Griffin

Semiparametric Models for Financial Data

Financial data such as asset prices or stock indices have well-studied features such as volatility clustering and heavy tails, which have been traditionally modelled using stochastic volatility. However, it is difficult to specify the dependence of the volatility. Initially, stationary model were proposed. However, more recently there has been interest in non-stationary models using long memory or Markov switching processes. In this talk, we will use the framework of infinite superpositions of shot-noise processes (also known as OU processes) to give flexible specifications of the dependence structure and make inference using nonparametric methods. This talk will cover modelling and computational issues with applications to daily and high frequency data.

Timothy J. Heaton, Paul G. Blackwell and Caitlin E. Buck

Reconstructing a Wiener process from observations at imprecise times: Bayesian radiocarbon calibration

For accurate radiocarbon dating, it is necessary to identify fluctuations in the level of radioactive carbon 14C present in the atmosphere through time. The processes underlying these variations are not understood and so a data-based calibration curve is required.

In this talk we present a novel MCMC approach to the production of the internationally agreed curve and the individual challenges involved. Our methodology models the calibration data as noisy observations of a Wiener process and updates sample paths through use of a Metropolis-within-Gibbs algorithm.

Implementation of this algorithm is complicated by certain specific features of the data used, namely that many data points:

  • relate to the mean of the Wiener process over a period of time rather than at a specific point,
  • have calendar dates found using methods (e.g. Uranium-Thorium) which are themselves uncertain,
  • have ordering constraints and correlations in their calendar date uncertainty - for example data are sampled along the same core or have floating calendar dates matched to another sample for which the calendar age is more accurately known.
  • We give an overview of these issues and discuss their implications for the resulting sampler.

    Amy H. Herring and David A. Savitz

    Infinite latent class transition models for longitudinal predictors

    In many applications, it is of interest to relate a longitudinal predictor to a response variable. For example, one may obtain repeated ultrasound measurements of fetal growth during pregnancy, and repeated growth measurements in infancy, with interest focused on the relationship between early growth and outcomes measured at birth and later in childhood development. To characterize the growth trajectory, we propose a dynamic latent class transition model, which generalizes infinite hidden Markov models to allow a more flexible dependence structure. To build a flexible joint model for prediction, we allow the response distribution to change nonparametrically according to the time-varying state indicators, while favoring a sparse structure. Efficient methods are developed for posterior computation relying on slice sampling, and the methods are applied to data from an epidemiologic study.

    Chaitanya Joshi and Simon Wilson

    A New Method to Approximate Bayesian Inference on Diffusion Process Parameters

    Since, in real life most of the diffusion processes are observed only at discrete time intervals not small enough, both Likelihood based and Bayesian methods of inference become non-trivial. To overcome this problem Bayesian inference is centred around introducing m latent data points between every pair of observations. However it was shown that as mincreases, one can make very precise inference about the diffusion coefficient of the process via the quadratic variation. This dependence results in slow mixing of the naive MCMC schemes and it worsens linearly as the amount of data augmentation increases. Various different approaches have been proposed to get around this problem. Some of them involve transforming the SDE, while most others present innovative MCMC schemes.

    We propose a new method to approximate Bayesian inference on the diffusion process parameters. Our method is simple, computationally efficient, does not involve any transformations, and is not based on the MCMC approach. Principle features of this new method are Gaussian approximation proposed by Durham and Gallant (2002) and a grid search to explore parameter space. In this paper we first introduce our new method and then compare its performance with recently proposed MCMC based schemes on several diffusion processes.

    Qiming Lv, Paul G. Backwell, Catlin Buck, M. Charles, S. Colledge and G. Jones

    Network-based spatio-temporal modelling of the first arrival of prehistoric agriculture in Europe

    Current archaeological wisdom views the spread of Neolithic agriculture as a leap-frogging migration across a non-uniform landscape. To obtain holistic insight into the rate and pattern of such spread, we model it as a spatio-temporal process on an irregular network based on a Delaunay t riangulation. The nodes in the network represent sites of statistical and geographic importance, augmented by points chosen to ensure that the network meets a minimum angle constraint. Edges indicate major corridors or geographical boundaries, plus additional edges to complete the triangulation. Inference about arrival times of the process combines the spatial structure of the network with observed radiocarbon dates of cereal grains found at some sites (using an existing Bayesian radiocarbon calibration framework). This novel approach breaks down arrival times into travel times along edges, thus explic itly disentangling the spatial dependence of the arrival time among neighboring sites and allowing geographic information and environmental conditions to enter as priors. Preliminary tests on European data show that our model can reduce uncertainties on arrival times for nodes where data are available and make predictions on the others. It therefore allows coherent analysis of patterns and processes rather than isolated investigation of individual sites.

    Anandamayee Majumdar, Debashis Paul and Dianne Bautista

    A Bayesian Approach to Modeling Multivariate Nonstationary Spatial Processes using Generalized Convolution method

    We propose a flexible class of nonstationary stochastic models for multivariate spatial data. The method is based on a general version of convolutions of spatially varying covariance kernels and produces mathematically valid covariance structures. This method generalizes the convolution approach suggested by Majumdar and Gelfand (2007) to extend multivariate spatial covariance functions to the nonstationary case. A Bayesian method for estimation of the parameters in the covariance model based on a Gibbs sampler is proposed, and applied to simulated data. Flexibility or robustness of the model is examined as well, using simulations from several kinds of model-departures. Model comparison is performed with the coregionalization model of Wackernagel (2003) which uses a stationary bivariate model. Based on posterior prediction results, the performance of our model is seen to be considerably better. Our model is then applied to real data where the prediction coverage using this methods results in significantly higher.

    Rui Paulo

    Derivatives of computer models

    We will explore several possibilities of incorporating derivative information in problems of calibration and validation of computer models. A key ingredient in the analysis of computer models is the specification of a statistical model relating the output of the computer model and the reality it aims at reproducing, which involves a discrepancy function. This idea was originally proposed by Kennedy and O'Hagan (2001). We propose an alternative formulation of this discrepancy function which involves the use of derivative information of the computer model. This formulation is motivated by a perturbation and linearization idea. Simulated examples are used to assess the advantages and disadvantages of incorporating derivative information in this class of problems.

    Pepa Ramirez Cobo

    Bayesian inference for the Markovian Arrival process

    In Lucantoni (1990) the term Batch Markovian Arrival process (BMAP) is first used to describe the Versatile Markovian point process introduced by Neuts (1979). In the BMAP arrivals are allowed to occur in batches where different types of arrivals can have different batch size distribution. The MAP is a special case of BMAP, where all batch sizes are equal to one. The idea of a BMAP is to keep the tractability of the Poisson arrival process but significantly generalizes it to allow the inclusion of dependent interarrival times, non-exponential interarrival-time distributions, and correlated batch sizes. This makes the BMAP an effective and powerful Internet data traffic model: it is able to capture dependence and correlation, one of the main features in Internet-related data.

    In this work we develop Bayesian inference for the 2-states MAP, where times of arrivals depend on the current state of a hidden underlying Markov Process. In practice, only the times when arrivals occur are observed, and neither the values of the states at these time points or the transient changes are available. We base our approach on a Metropolis-Hastings scheme, where the likelihood function is derived from what we define as the Effective Markovian Arrival process. We will illustrate our methodology with simulated and a real teletraffic data set. We will also delineate some ideas concerning the identifiability of the MAP.

    Gareth O. Roberts, Yvo Pokern, Omiros Papaspiliopoulos and Andrew Stuart

    Bayesian non-parametric analysis of diffusions

    This presentation will describe recent progress on Bayesian non-parametric analysis of diffusion drift functions given continuous data on a finite time interval. It turns out that Gaussian processes can be used as conjugate priors, and we describe methodology for characterising posterior mean and covariance structure in terms of solutions to differential equations with coefficients given as functions of the observed diffusion local time.

    Abel Rodriguez and David B. Dunson

    Nonparametric inference in spatio-temporal processes using Probit stick breaking processes

    Constructing flexible models for stochastic processes is an important challenge in a number of settings. We describe a novel class of Bayesian nonparametric priors based on stick-breaking constructions where the weights of the process are constructed as probit transformations of normal random variables. We show that these priors are extremely flexible, allowing us to generate a great variety of models while preserving computational simplicity. Particular emphasis is placed on the construction of rich temporal and spatial processes, which are applied to two problems in finance and ecology.

    Esther Salazar, Hedibert F. Lopes and Dani Gamerman

    Spatial dynamic factor analysis

    A new class of space-time models derived from standard dynamic factor models is proposed. The temporal dependence is modeled by latent factors while the spatial dependence is modeled by the factor loadings. Factor analytic arguments are used to help identify temporal components that summarize most of the spatial variation of a given region. The temporal evolution of the factors is described in a number of forms to account for different aspects of time variation such as trend and seasonality. The spatial dependence is incorporated into the factor loadings by a combination of deterministic and stochastic elements thus giving them more flexibility and generalizing previous approaches. The new structure implies nonseparable space-time variation to observables, despite its conditionally independent nature, while reducing the overall dimensionality, and hence complexity, of the problem. The number of factors is treated as another unknown parameter and fully Bayesian inference is performed via a reversible jump Markov Chain Monte Carlo algorithm. The new class of models is tested against one synthetic dataset and applied to real data obtained from the Clean Air Status and Trends Network (CASTNet). The factor model decompositionis shown to capture important aspects of spatial and temporal behavior of the data.

    Yuan Shen, Dan Cornford, Cedric Archambeau and Manfred Opper

    A Hybrid Approach to Bayesian Smoothing of Partially Observed Multivariate Diffusions with High Efficiency and Accuracy

    There is increasing interest in Bayesian inference in diffusion processes, with ongoing efforts to develop computationally efficient algorithms. We present two extensions to a recently developed Variational Gaussian Process Approximation (VGPA) method.

    Aiming at high-dimensional systems, we adopt a model reduction approach to VGPA. By adopting a basis function expansion, both the time-dependent control parameters of the approximate process and its moment equations are projected onto a lower-dimensional subspace. This allows us both to reduce the computational complexity and to eliminate the time discretisation used in the previous algorithm. We show an efficient smoother and compare this with existing methods.

    We go on to show how we combine our algorithms with Markov Chain Monte Carlo methods. The optimised approximate process from VGPA is used to devise an adaptive mixture of an independence and a random walk sampler, together with a blocking strategy. We show this results in a sampling scheme with better mixing than a state-of-art Hybrid Monte Carlo (HMC) path sampler.

    Comparing with VGPA and HMC, the efficiency and accuracy of our approaches are validated on the stochastic Lorenz system. We also report its application to the 20-dimensional diffusion approximation of the turbulence like Kumamoto-Shivashinsky system.

    Nozer D. Singpurwalla

    The Lattice QCD Problem of Particle Physics and its Bayesian Analysis

    In this talk I will start with an overview presentation of particle physics, a topic that has recently attracted the attention of many physicists, and which has spawned several Noble Prizes. This will be followed by the introduction of a mathematical problem that this topic brings forth. The problem is at the heart of the very nature of what constitutes matter. Standard statistical approaches to solving this problem have failed to produce satisfactory answers, causing the physicists to call for a use of Bayesian methods. I have been able to do this by first looking at the anatomy of the underlying equations which characterize the Lattice QCD, and noticing a pattern. I then endow the parameters of these equations with suitable proper elicited priors. Statistically, the problem boils down to estimating an infinite number of parameters using a finite number of equations. The role of MCMC has been essential to obtain results that match those given by the theoretical physicists. The computing work has been done by Josh Landon.

    Mark F.J. Steel and Thais C.O. Fonseca

    Non-Gaussian Bayesian spatiotemporal modeling

    The aim of this work is to construct non-Gaussian and nonseparable covariance functions for processes that vary continuously in space and time. Stochastic modelling of phenomena over space and time is important in many areas of application. Choosing an appropriate model can be difficult as one must take care to use valid covariance structures. We start from a general and flexible way of constructing valid nonseparable covariance functions through mixing over separable Gaussian covariance functions. We then generalize the resulting models by allowing for individual outliers as well as regions with larger variances. We induce this through scale mixing with separate positive-valued processes. Smooth mixing processes are applied to the underlying correlated Gaussian processes in space and in time, thus leading to regions in space and time with increased spread. We also apply a separate uncorrelated mixing process to the nugget effect to generate individual outliers. We examine properties of these models and consider posterior and predictive Bayesian inference. We implement the latter through a Markov chain Monte Carlo sampler and apply our modelling approach to simulated and real data (temperature data from the Basque country and wind speed data from Brazil).

    Marc A. Suchard

    Spatial-Temporal Processes in Phylogenetics

    Kingman's coalescent process opens the door for estimation of population genetics model parameters from molecular sequences. One paramount parameter of interest is the effective population size. Temporal variation of this quantity characterizes the demographic history of a population. Since researchers are rarely able to choose a priori a deterministic model describing effective population size dynamics for data at hand, non-parametric curve fitting methods based on multiple change-point (MCP) models have been developed. We propose an alternative to change-point modeling that exploits Gaussian Markov random fields to achieve temporal smoothing of the effective population size in a Bayesian framework. The main advantage of our approach is that, in contrast to MCP models, the explicit temporal smoothing does not require strong prior decisions. To approximate the posterior distribution of the population dynamics, we use efficient, fast mixing MCMC algorithms designed for highly structured Gaussian models. In a simulation study, we demonstrate that the proposed temporal smoothing method, named Bayesian skyride, successfully recovers "true" population size trajectories in all simulation scenarios and competes well with the MCP approaches without evoking strong prior assumptions. We apply our Bayesian skyride method to two real data sets. We analyze sequences of hepatitis C virus contemporaneously sampled in Egypt, reproducing all key known aspects of the viral population dynamics. Next, we estimate the demographic histories of human influenza A hemagglutinin sequences, serially sampled throughout three flu seasons. Finally, We extend models of temporal variation to include spatial diffusion. Such spatial processes enable formal Bayesian hypothesis testing about the relative contributions of geography and human transport in infectious disease evolution. We demonstrate how to successfully embed estimation of spatial diffusion in an already highly structured stochastic model through fast computation using graphics processing units. These stream processors yield over 100-fold run-time improvement for important aspects of the spatial-temporal model.

    Matthew Taddy

    Dynamic point process modeling with a DDP

    Bayesian nonparametric methods have been previously employed in flexible modeling for spatial event data that may be viewed as the realization of a Poisson point process. When these spatial events occur over time, it is common to incorporate temporal dependence by viewing the data as a single realization of a point process in both space and time. However, in discrete time settings -- when each time point corresponds to a set of spatially distributed events -- it is more appropriate to model the data as arising from multiple point processes that have an intensity function which is changing in time. This is made possible by employing a novel version of the Dependent Dirichlet process as a prior for time-indexed normalized process intensities. Details of the development of the model and prior measure will be provided, along with an example involving crime event data.

    Elisa Varini and Renata Rotondi

    Bayesian analysis of a state-space model with a nonstationary state process

    We aim to define a model to study sequences of earthquake occurrences in a region by including some features suggested by geophysical and geological findings and by direct observation of the seismic events.

    Since the seismic rate is usually subject to clear temporal variations, we assume that the physical system has different states in agreement with these variations. At least three states can be deduced from the literature corresponding respectively to periods of low, medium and high seismic activity (e.g. seismic quiescence, foreshocks and aftershocks). We consider appropriate to propose a state-space model where the observations come from a point process such that its intensity function has a different expression for each state.

    In a previous preliminary study we defined the state process to be a homogeneous Markov process, that is the transition probabilities from a state to another were constant in time. In the present work, we consider the state process to be a non-homogeneous process such that the transition probabilities at time t depend on the observations up to t as well as the last visited states.

    We perform a Bayesian analysis of some real data sets by exploiting MCMC and particle filtering methods.

    Michail D. Vrettas, Dan Cornford, Yuan Shen and Manfred Opper

    Constrained Variational Bayesian Approximation for Parameter Estimation in Diffusion Processes

    Stochastic differential equations (SDEs) have been used extensively in modelling phenomena that exhibit intrinsic randomness or unknown components. To tackle these problems a range of different methods have been employed based on the Kalman filter and Monte Carlo approaches. An alternative is based on a variational treatment of the inference problem. Recently, a variational Bayesian algorithm has been proposed for inference in diffusion processes that can be used for both state and parameter estimation. In this paper we propose a new approach that is based on a radial basis function (RBF) reparametrisation of this variational approximation algorithm. We focus on estimating the (hyper-) parameters in the drift and diffusion of the stochastic differential equation considered. The new RBF approach is fast and shows great robustness and stability. Here we validate our method on a multidimensional (40 variable Lorentz) dynamical system and compare our results with a state of the art ensemble Kalman filter and a recently reported Markov Chain Monte Carlo based algorithm. We compare the asymptotic behaviour of the algorithms as the observation density increases, and discuss the reasons for the empirically observed differences in performance of the methods.

    Mike West

    Mixtures, Clustering, Spatial & Dynamic Point Processes and Big Data Sets

    Motivated by biological studies involving large data sets from flow cytometry and fluorescent spectroscopy in immunology, we have developed a number of novel Bayesian mixture models for density and intensity estimation, clustering problems and spatio-temporal dynamic modelling. New models and computational methods involve non-Gaussian component mixture modelling with new uses of old nonparametric mixtures, and flexible mixtures for static and dynamic spatial point processes. I will report on some of these developments and immunological "big data" applications.