CRiSM seminars 2007 Feb-June
February 1st
Ian White (MRC, Cambridge)
Causal inference from randomised clinical trials: when is it needed and why isn’t it done?
February 15th
Raazesh Sainudin (University of Oxford)
Title: An Auto-validating Rejection Sampler
Abstract: In Bayesian statistical inference and computationally intensive frequentist inference, one is interested in obtaining samples from densities which may be many dimensional, multi-modal, scale-sensitive, and unnormalized. One of the simplest Monte Carlo methods is the rejection sampler due to von Neumann. Here we introduce an auto-validating version of the rejection sampler via interval analysis. In particular, we adaptively partition the domain into boxes, use interval analysis to obtain an upper bound on the density's shape in each box, and thus construct a rigorous envelope function and a corresponding density which is easily sampled from.
We show that such a sampler does provide us with independent samples from a large class of target densities (Elementary Lipschitz class, ie densities that can be represented as a finite composition of standard arithmetic operations, standard functions and simple functions with known singularities) in a guaranteed manner. The guarantees here are synonyms for computer-assisted proofs.
We illustrate the efficiency of the sampler by theory and by examples in up to 10 dimensions. Our sampler is immune to the `pathologies' of some infamous densities, including the multi-variate versions of highly spiky witch's hat and multi-modal Rosenbrock, that primarily arise from the assumption that real numbers are representable and manipulable in computers with finite memory. We also draw provably i.i.d. samples from the posterior distribution over piece-wise Euclidean spaces of small primate phylogenies and thereby solve an open problem in statistical molecular evolution.
February 22nd
Helene Massam (York University, Toronto)
Title: Wishart distributions for decomposable graphs
Abstract: When considering a graphical Gaussian model NG Markov with respect to a decomposable graph G, the parameter space of interest for the precision parameter is the cone PG of positive definite matrices with fixed zeros corresponding to the missing edges of G. The parameter space for the scale parameter of NG is the cone QG, dual to PG, of incomplete matrices with submatrices corresponding to the cliques of G being positive definite. We construct on the cones QG and PG two families of Wishart distributions, namely the type I and type II Wisharts. They can be viewed as a generalization of the hyper Wishart and the inverse of the hyper inverse Wishart as defined by Dawid and Lauritzen (1993). The type I and II Wisharts have properties similar to those of the hyper and hyper inverse Wishart. Indeed, the inverse of the type II Wishart forms a conjugate family of priors for the covariance parameter of the graphical Gaussian model and is strong directed hyper Markov for every direction given to the graph by a perfect order of its cliques, while the type I Wishart is weak hyper Markov. Moreover, the inverse type II Wishart as a conjugate family presents the advantage of having a multi-dimensional shape parameter, thus offering flexibility for the choice of a prior. The type II Wishart can be viewed as the analog for graphical Gaussian model of the enriched conjugate prior defined by Consonni and Veronese (2003). This is joint work with G. Letac.
February 26th - Joint Stats/Econometrics Seminar, venue: S0.20 (social studies ground floor), time: 17.00-18.30
Jean-Pierre Florens (University of Toulouse and IUF)
Title: two related papes will be presented, Non Parametric Instrumental Regression (with S. Darolles and E. Renault) and The Practice of Non Parametric Instrumental Variables Estimation ( with R. Lestringant)
Abstract: The objective of the paper is to show that non parametric instrumental variables estimation is easy to implement and is a very powerful tool for the estimation of structural econometric models. A particular attention will be devoted to the choice of the regularization parameters and we propose some data driven methods. The basic model is extended to models containing both endogenous and exogenous variables and to semi parametric models. We conclude the paper by considering two non linear problems: the case of non separable models (with application to endogeneity in duration models) and the game theoretic models. The paper presents some applications of the theory of ill posed inverse problems to econometrics and is based on simulations. Some theoretical results (Bayesian or frequentist) will complete this presentation.
Dan Gamerman
Title: Bayesian Analysis of the Item Response Theory: A Generalized Approach
Abstract: The Item Response Theory (IRT) is a psychometric theory world wide used in education evaluation and cognitive psychology. Along with the increasing of this theory's application, new issues with great practical importance arise, like considering the Differential Item Functioning (DIF). Because of the need of IRT generalization by models that consider these important issues, two IRT models that consider DIF existence are proposed. They are both generalizations of the three parameters logistic model and differ from each other on the hypothesis of knowing which items have DIF, that is: one of them incorporate DIF detection. It is presented a Bayesian approach based on MCMC methods for the inference procedure. Later, simulated studies with both models are presented in order to study their characteristics and the efficiency of the Bayesian methods on the parameters estimation. Finally, it is also presented an example with real data concerning an educational program in the state of Rio de Janeiro, Brazil.
March 15th
This talk focuses on multiagent Bayesian reasoning through graphical models. The computational challenge of probabilistic reasoning and how Bayesian networks meet the challenge will be reviewed. After motivating multiagent systems, multiply sectioned Bayesian networks (MSBNs) will be introduced as a class of graphical models for multiagent uncertain knowledge representation. The fundamental assumptions that logically lead to MSBNs will be discussed. How multiple agents reason probabilistically using MSBNs will be presented algorithmically, and key computational properties of the framework will be discussed. The computation process will be illustrated through equipment monitoring and fault isolation.
Title: Bayesian nonparametric methods for prediction in EST analysis
Abstract: Expressed sequence tags (ESTs) analyses are an important tool for gene identification in organisms. Given a preliminary EST survey from a certain cDNA library, various features of a possible additional sample have to be predicted. For instance, interest may rely on estimating the number of new genes to be detected, the gene discovery rate at each additional read and the probability of not re-observing certain specific genes present in the initial sample. We propose a Bayesian nonparametric approach for prediction in EST analysis based on nonparametric priors inducing Gibbs-type exchangeable random partitions and derive estimators for the relevant quantities. Several EST datasets are analysed by resorting to the two parameter Poisson-Dirichlet process, which represents the most remarkable Gibbs-type prior. Our proposal has appealing properties over frequentist nonparametric methods, which become unstable when prediction is required for large future samples.
Title: Dynamic Covariates and Frailty for Recurrent Event Data
Abstract: We examine methods for modelling of longitudinal binary data subject to both intermittent missingness and dropout, based around the analysis of data from a study into the health impact of a sanitation programme carried out in Salvador, Brazil. In total 926 children were followed up at home twice a week from October 2000 to January 2002, from which daily occurrence of diarrhoea was recorded for each child being followed up. A challenging factor in analysing these data is the presence of between subject heterogeneity not explained by known risk factors, combined with significant loss of observed data through either intermittent missingness (average of 78 days per child) or dropout (21% of children). We compare dynamic modelling and frailty strategies and show the advantages of taking an event history approach with an additive discrete time regression model.
Title: MCMC algorithms for distributions with intractable normalizing constants, with a view to perfect simulation and non-parametric Bayesian inference for inhomogeneous Markov point processes
Abstract: Maximum likelihood parameter estimation and sampling from Bayesian posterior distributions are problematic when the probability distribution for the parameter of interest involves an intractable normalizing constant which is also a function of that parameter. Most methods to date have used various approximations to estimate or eliminate such normalizing constants. In [4] we present new methodology for drawing samples from such a distribution without approximation. The novelty lies in the introduction of an auxiliary variable in a Metropolis-Hastings algorithm and the choice of proposal distribution so that the algorithm does not depend upon the unknown normalizing constant. The method requires the auxiliary variable to be simulated from the distribution which defines the normalizing constant, for which perfect (or exact) simulation as exemplified by the Propp-Wilson algorithm [7] and the dominated coupling from the past (dominating CFTP) algorithm [2, 3] becomes useful.
In [1] we illustrate the method by the following application example. An inhomogeneous point pattern showing the location of cells in a section of the mocous membrane of the stomach of a healthy rat is modelled by a Markov point process, with a location dependent first order term and pairwise interaction only. We consider a flexible non-parametric Bayesian setting, assuming a priori that the first order term is a shot noise process, the interaction function for each pair of points depends only on the distance between the two points, and the interaction function is a piecewise linear function modelled by a marked Poisson process. Since the Markov point process density involves a normalizing constant where no closed form expression is known, we apply the auxiliary variable technique from [4] for posterior simulation. The auxiliary variable in [1] is specified by a partially ordered Markov point process model.
Next, we briefly discuss the dominating CFTP algorithm due to Kendall [2]. In [3] we give a general formulation of dominated CFTP, which applies for certain stochastic models on ordered spaces, and discuss in particular how to make perfect simulation of general locally stable point processes. In the present talk we consider only the case of perfect simulation of pairwise interaction point processes as equilibrium distributions of spatial birth-and-death processes.
For those interested in further details of spatial point process models, simulation and inference, see [5] and [6].
Finally, if time allows, current research on extensions of the auxiliary variable technique will be discussed.
References:
[1] K.K. Berthelsen and J. Møller (2007). Non-parametric Bayesian inference for inhomogeneous Markov point processes. Research Report R-2007-9, Department of Mathematical Sciences, Aalborg University.
[2] W.S. Kendall (1998). Perfect simulation for the area-interaction point process. In L. Accardi and C.C. Heyde, Probability Towards 2000, Springer Lecture Notes in Statistics 128, Springer Verlag, New York.
[3] W.S. Kendall and J. Møller (2000). Perfect simulation using dominating processes on ordered spaces, with application to locally stable point processes. Advances in Applied Probability, 32, 844-865.
[4] J. Møller, A.N. Pettitt, K.K. Berthelsen and R.W. Reeves (2006). An efficient Markov chain Monte Carlo method for distributions with intractable normalising constants. Biometrika, 93, 451-458.
[5] J. Møller and R.P. Waagepetersen (2003). Statistical Inference and Simulation for Spatial Point Processes. Chapman and Hall/CRC, Boca Raton.
[6] J. Møller and R.P. Waagepetersen (2007). Modern statistics for spatial point processes (with discussion). Scandinavian Journal of Statistics (to appear).
[7] J.G. Propp and D.B. Wilson (1996). Exact sampling with coupled Markov chains and applications to statistical mechanics. Random Structures and Algorithms, 9, 223-252.
Title: Hyperanalytic Denoising
Abstract: Image estimation requires a compromise between reconstruction flexibility and computational tractability. A commonly adopted approach to the estimation problem is to decompose the observed image in a suitable basis, where the compression of the decomposition simplifies subsequent analysis. Much effort has been expanded in designing appropriate 2-D decompositions, but of great importance is also the statistical estimation procedure chosen, and this talk will focus on the estimation of any set of decomposition coefficients. A new estimation method that can be combined with a local decomposition method is introduced. Under the assumption that structured features correspond to highly concentrated and connected regions of the spatial and spatial frequency space additional image replicates with the same local structure as the observed image are constructed from the observed image. The decomposition of the image is estimated not only using the observed image decomposition coefficients, but also using a set of local coefficients constructed from the replicate images. Given the tractable form of the first and second order structure of the full set of decomposition coefficients of both the image and replicate images at any given scale and spatial position, the full procedure can be specified analytically, and its risk calculated explicitly. The proposed method is implemented on several examples, and theoretical risk calculations substantiated, as well as visually appealing reconstructions presented.
Title: A multivariate semiparametric Bayesian spatial modeling framework for hurricane surface wind fields
Abstract: Storm surge, the onshore rush of sea water caused by the high
winds and low pressure associated with a hurricane, can compound the effects of inland flooding caused by rainfall, leading to loss of property and loss of life for residents of coastal areas. Numerical ocean models are essential for creating storm surge forecasts for coastal areas. These models are driven primarily by the surface wind forcings. Currently, the gridded wind fields used by ocean models are specified by deterministic formulas that are based on the central pressure and location of the storm center. While these equations incorporate important physical knowledge about the structure of hurricane surface wind fields they cannot always capture the asymmetric and dynamic nature of a hurricane. A new Bayesian multivariate spatial statistical modeling framework is introduced combining data with physical knowledge about the wind fields to improve the estimation of the wind vectors. Many spatial models assume the data follow a Gaussian distribution. However, this may be overly-restrictive for wind fields data which often display erratic behavior, such as sudden changes in time or space. In this paper, we develop a semiparametric multivariate spatial model for these data. Our model builds on the stick-breaking prior, which is frequently used in Bayesian modelling to capture uncertainty in the parametric form of an outcome. The stick-breaking prior is extended to the spatial setting by assigning each location a different, unknown distribution, and smoothing the distributions in space with a series of kernel functions. This semiparametric spatial model is shown to improve prediction compared to usual Bayesian Kriging methods for the wind field of Hurricane Ivan.
Title: Limit theorems for max-compound Cox processes and their application to the estimation of risks of catastrophic events in non-homogeneous flows of extremal events
Abstract: Max-compound Cox processes are introduced. Limit theorems for their distributions is proved. These results are applied to the estimation of risks of catastrophic events in non-homogeneous flows of extremal events. As an example, the problem of estimation of the risk of the impact of the Earth and a large asteroid is considered.
Title: STEIN ESTIMATION AND PREDICTION: A SYNTHESIS
Abstract: Stein (1956, Proc. 3rd Berkeley Symposium), in his seminal paper, came with the surprising discovery that the sample mean is an inadmissible estimator of the population mean in three or higher dimensions under squared error loss. The past five decades have witnessed multiple extensions and variations of Stein's results. Extension of Stein's results to prediction problems is of more recent origin, beginning with Komaki (2001, Biometrika), George, Liang and Yu (2006, Annals of Statistics) and Ghosh, Mergel and Datta (2006). The present article shows how both the estimation and prediction problems go hand in hand under certain \intrinsic losses", which includes both the Kullback-Leibler and Bhattacharyya-Hellinger divergence losses. The estimators dominating the sample mean under such losses are motivated both from the Bayesian and empirical Bayes point of view.
Title: On distortions induced by truncation in linear regression systems
Abstract: We consider the effects induced by truncation in multivariate Gaussian models, with special attention to the modification produced on the concentration matrix and, therefore, on the linear regression coefficients. Truncation mechanisms on some of the variables, which may or may not be observed, have an important role in construction of selection type models. The derivations allow (a) to recover exhisting results in a unified framework and (b) to provide some insights on the distortion induced in linear regression coefficients by the selection mechanism. In some cases, this bias can also be consistently estimated from the observed data. Connections to the Extended Skew Normal distributions are also provided and used to construct models for binary response models with selection.