Fifth Workshop on
Valencia (Spain), June 14-16, 2007

Abstracts of contributed talks

Dilan Gorur* and Carl Edward Rasmussen: Dirichlet Process Mixtures of Factor Analysers
Mixture of factor analysers (MFA) is a well-known model that combines the dimensionality reduction technique of Factor Analysis (FA) with mixture modeling. The key issue in MFA is deciding on the latent dimension and the number of mixture components to be used. The Bayesian treatment of MFA has been considered by Beal and Ghahramani (2000) using variational approximation and by Fokoué and Titterington (2003) using birth-and –death Markov chain Monte Carlo (MCMC). Here, we present the nonparametric MFA model utilizing a Dirichlet process (DP) prior on the component parameters (that is, the factor loading matrix and the mean vector of each component) and describe an MCMC scheme for inference. The clustering property of the DP provides automatic selection of the number of mixture components. The latent dimensionality of each component is inferred by automatic relevance determination (ARD). Identifying the action potentials of individual neurons from extracellular recordings, known as spike sorting, is a challenging clustering problem. We apply our model for clustering the waveforms recorded from the cortex of a macaque monkey.

Michele Guindani*, Alan Gelfand and Sonia Petrone: Hybrid Dirichlet processes for Bayesian inference with functional data
Inference for functional data is receiving increasing interest in the scientific community. Functional data means that the observations are in principle realizations of a random curve (or a stochastic process), defined on some coordinate space. Examples include regression curves, random surfaces etc. In this work we study Bayesian inference for functional data in presence of random effects and individual heterogenety. Recent Bayesian modeling of functional data has considered mixed effect models, where a residual stationary Gaussian effect is assumed. Arguably, one might prefer the flexibility of a nonstationary, non-Gaussian specification. In a nonparametric setting, this can be accommodated by a mixture model of Gaussian processes where the latent distribution has a "functional" Dirichlet process prior. The mixture model introduces random effects and clustering of the individual curves. However, a limitation of Dirichlet process mixture models is that the latent factor driving curve selection is defined globally, while for functional data it seems more appropriate to allow local effetcs on some portions of the curve. To this aim, we propose a novel class of prior distributions on a random curve, which generalizes the functional Dirichlet process by allowing hybrid species. The proposed hybrid Dirichlet process is used as a prior on the latent distribution in a mixture model of Gaussian processes. This allows to capture hybrid clustering, e.g. parts of a curve belong to different clusters and the curve as a whole could be viewed as a hybrid from a base set of curves. When the number of components of the mixture model if finite, our prior is a generalization of the so called finite-dimensional Dirichlet process. We discuss in some details this case and its limiting behavior as the numebr of components goes to infinity. Applications to simulated and real image data illustrate the procedure.

John Haslett and Andrew Parnell*: Non-parametric Bayesian monotonic regression with applications to radiocarbon dating
We develop a suite of models which can be used to examine radiocarbon-dated sediment cores. Such cores are used as the basis for estimating uncertainty in past events such as climate change (eg Haslett et al., 2006). The task is to reconstruct the sediment history of the core by linking depth to age. The nature of deposition is such that older events occur at lower depths; thus a valid sedimentation history must be monotonic. We incorporate this information via a random sum of gamma increments; a Tweedie distribution. An advantage of this method is that parameters are of random length without the need for reversible jump techniques. The models are easily incorporated into existing radiocarbon dating technology to produce stochastic sedimentation histories which honestly assess uncertainty in the timing of core events, and hence climate change.

Hedibert F. Lopes*, Robert McCulloch and Ruey Tsay: Time-varying Covariances: A Cholesky Decomposition Approach
In this paper, we introduce a new class of time-varying multivariate stochastic volatility models that rely upon the Cholesky decomposition of the vector of time series covariance matrices at each given time period. Our model structure encompasses and/or generalizes several of the existing factor-like stochastic volatility models recently proposed in the financial econometrics literature. We compare our model with competing ones by posterior predictive assessment and portfolio cumulative gains in emerging and developed economy stock markets.

Malcolm J. Price*, Nicky J. Welton and A.E. Ades: Bayesian Inferential Techniques for Modelling the Effect of Treatment on Disease Progression Represented by a Continuous-time Markov Model
A four-state continuous time markov model is used to represent the disease progression of asthma sufferers in randomised controlled trials measuring the relative efficacies of different treatments for asthma. The data, in the form of aggregated discrete time transitions, have a multinomial likelihood. However, the competing risks structure makes it difficult to model different separate treatment effects on different transition probabilities using standard logistic regression models. Instead the transition rates are modelled using log-linear regression, and these rates are converted to transition probabilities using Kolmogorovs forward equations. A range of plausible treatment models is developed in which the relative treatment effects act on different combinations of forward, backward transitions, or both. This specification of a structure for the summary treatment effects allows the results to be generalised across studies making it possible to perform meta-analysis. Bayesian inferential techniques with uninformative priors are used and the parameters are estimated using MCMC simulation in WinBUGS and WBDiff is used to solve the forward equations. Model comparison is performed using the DIC and it is demonstrated that the 9 transition probabilities can be modeled in terms of 6 rates, and that a single parameter, acting on all backward transitions, is a sufficient representation of the difference between the effects of the treatments in each trial.

Vilda Purutçuoglu* and Ernst Wit: Bayesian inference for the MAPK/ERK pathway by considering the dependency of the kinetic parameters
The MAPK (mitogen-activated protein kinase) or its synonymous ERK (extracellular signal regulated kinase) pathway is one of the major signal transduction systems which regulates the cellular growth control of all eukaryotes like cell proliferation and apoptosis. Because of its importance in cellular lifecycle, it has been studied intensively, resulting in a number of qualitative descriptions of this regulatory mechanism. In this study we describe the MAPK/ERK pathway as an explicit set of (quasi) reactions by combining these studies. Then we estimate in a Bayesian setting the model parameters of the network, i.e. the stochastic rate constants, via MCMC and data augmentation. In the estimation we apply the Euler approximation, which is the discretized version of diffusion approximation. Additionally in our inference algorithm we consider all possible kinds of dependency coming from distinct stages of updates of time states and parameters for such a realistic and complex system. We use the simulated data from the diffusion approximation according to our (quasi) biochemical reactions to test the inference method. Our reaction set takes into account the localization and different binding sites of the molecules in the cell by implementing the multiple parametrizations. From the simulations it is clear that the sampler converges well and is able to identify the dynamics of the MAPK/ERK pathway.

Matteo Ruggiero* and Stephen G. Walker : Urn-based Particle Processes for Fleming-Viot Models in Bayesian Nonparametrics
The Fleming-Viot process is a probability-measure-valued diffusion which arises as the large population limit of a wide class of population genetics models. In the selectively neutral case its stationary distribution is known to be the Dirichlet process, but its connections with Bayesian nonparametrics in more complex cases are still to be explored. In this work, by means of known and newly defined generalised P`olya-urn schemes, several types of pure jump particle processes are introduced, describing the evolution in time of an exchangeable population. By considering the empirical measure of the individuals at each time point, the jump processes provide an explicit construction of four different formulations of the Fleming-Viot process, corresponding to the cases of neutrality, viability selection, haploid and diploid fertility selection. Weak convergence of the constructed measure-valued processes to the respective Fleming-Viot diffusion holds. The stationary distribution of each case is derived and shown to be the de Finetti measure of the infinite sequence of individuals. In presence of viability selection the stationary distribution turns out to be the two-parameter Poisson-Dirichlet process.

Refik Soyer* and M. Murat Tarimcilar: Modeling and Analysis of Call Center Arrival Data: A Bayesian Approach
In this paper we present a modulated nonhomogeneous Poisson process model to describe and analyze arrival data to a call center. The attractive feature of this model is that it takes into account both covariate and time effects on the call volume intensity and in so doing enables us to assess the effectiveness of different advertising strategies along with predicting the arrival patterns. A Bayesian analysis of the model is developed and an extensions of the model are considered to describe potential heterogeneity in arrival patterns and to predict aggregated arrivals based on all advertisements. The latter yields superposition of modulated nonhomogeneous Poisson processes. The proposed models and the methodology are implemented using real call center arrival data.

Matt Taddy* and Athanasios Kottas: Bayesian Nonparametric Markov Switching Regression Models
Markov switching regression models can be used to study heterogeneous populations that depend on covariates and are observed over time. The model formulation involves a mixture of regressions with a Markov chain as the model for the mixing distribution. Applications are found in several fields including, for instance, ecology and econometrics. We focus on applications where the number of homogeneous subpopulations (groups) is small and can be assumed to be known, and our interest lies on modeling for the individual group regression functions and/or response distributions. We propose flexible prior probability models, based on Dirichlet process mixtures, for the joint distribution of individual group responses and covariates. The implied conditional distribution of the response given the covariates is then used for inference. This model allows both non-linearities in the resulting regression functions and non-normalities in the response distributions. Full posterior inference is implemented using posterior simulation techniques for Dirichlet process mixtures. The modeling approach is illustrated with simulated data and also applied to fisheries data to obtain inference for stock-recruitment relationships subject to environmental regime shifts.

George W. Woodworth and Jay Kadane*: Age and Time-Varying Proportional Hazards Model for Employment Discrimination
We use a discrete-time proportional hazards model of time to involuntary employment termination. This model enables us to examine both the continuous effect of age of an employee and whether that effect has varied over time, generalizing earlier work (Kadane and Woodworth, 2004). We model the log hazard surface (over age and time) as a thin-plate spline, a Bayesian smoothness-prior implementation of penalized likelihood methods of surface-fitting (Wahba, 1990). The nonlinear component of the surface has only two parameters, smoothness and anisotropy. The first, a scale parameter, governs the overall smoothness of the surface, and the second, anisotropy controls the relative smoothness over time and over age. For any fixed value of the anisotropy parameter, the prior is equivalent to a Gaussian process with linear drift over the time-space plane with easily computed eigenvectors and eigenvalues that depend only on the configuration of data in the time-age plane and the anisotropy parameter. This model has application to legal cases in which a company is charged with disproportionately disadvantaging older workers is deciding whom to terminate. A numerical example is computed.

* denotes the speaker