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Professor Peter Green

Biography

I am a statistical scientist, principally interested in Bayesian inference in complex stochastic systems, Markov chain Monte Carlo methodology, forensic genetics, Bayesian nonparametrics, Bayesian inverse problem and graphical models.

Before moving to Bristol to a Chair of Statistics in 1989, I had been lecturing at the Universities of Bath (1974-1978) and Durham (1978-1989). I have now retired from my full-time role at Bristol, but hold an Emeritus Professorship and Professorial Research Fellowship there, as well as a Distinguished Professorship at UTS.

I have been awarded a Royal Society Wolfson Research Merit Award (2006-11), Fellowship of the Royal Society (2003), Chartered Statistician (2001), Guy Medal in Silver, Royal Statistical Society (1999), Fellowship of the Institute of Mathematical Statistics (1991) and Guy Medal in Bronze, Royal Statistical Society (1987).

I have collaborations with researchers in the USA, in Italy and in Australia, and, formerly, in Norway, Canada and Singapore.

My main web page is at http://www.stats.bris.ac.uk/~peter/ (ths opens an external site).

Professional

Professional associations:
I am a Fellow of the Royal Society, of the Institute of Mathematical Statistics and of the Royal Statistical Society, and a Chartered Statistician.

I was President of the Royal Statistical Society (2001-2003) and of the International Society for Bayesian Analysis (2007).
 

From 2014 to 2016 I was Editor of the journal Statistical Science.

Image of Peter Green
Distinguished Professor, School of Mathematical and Physical Sciences
BA (Hons), M.Sc, PhD
 
Phone
+61 2 9514 1742

Research Interests

I work principally in the area of complex systems, where my aim has been to investigate full Bayesian inference in much more complex models than was possible a few years ago. I have been actively exploiting the increased computing power now available, together with advances in the discipline of graphical modelling, and the use of methodologies such as EM and Markov chain Monte Carlo.

While I have developed some highly non-trivial implementations, the real objective is not computational. Rather, it is to investigate issues raised by Bayesian inference in complex models, going beyond point estimates, presenting richer aspects of complex posterior distributions, and studying issues of prior sensitivity, simultaneous inference, model uncertainty, model criticism, etc. Within this general framework, I have made contributions both to generic methodological issues such as mixture modelling, Markov random fields, and graphical models, and also to specific applications, especially in complex biomedical systems.

Earlier in my career my focus was on Re-weighted least squares, smoothing and penalized likelihood, Computational geometry and applications, and Branching processes and applications.

For further details, see http://www.stats.bris.ac.uk/~peter/Research.html (opens an external site).

In the past, I have taught courses across most of statistics and probability, and undergraduate and postgraduate level.

Chapters

Green, P.J. 2015, 'MAD-Bayes matching and alignment for labelled and unlabelled configurations' in Dryden, I.L. & Kent, J.T. (eds), Geometry Driven Statistics, John Wiley & Sons, Chichester, pp. 365-375.
Editors Dryden Kent Geometry Driven Statistics Geometry Driven Statistics Edited by Ian L. Geometry Driven Statistics Editors Ian L. Dryden University of Nottingham, UK John T. Kent University of Leeds, UK A timely collection of advanced, ...
Green, P.J. 2015, 'MAD-Bayes matching and alignment for labelled and unlabelled configurations' in Dryden, I.L. & Kent, J.T. (eds), Geometry Driven Statistics, John Wiley & Sons, Chichester, pp. 377-390.
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Professor Kanti Mardia has made numerous original contributions to an area of unlabelled shape analysis inspired by matching and alignment problems arising in protein bioinformatics. In this chapter, a Bayesian model proposed by Mardia and Green in 2006, and others related to it, are revisited to investigate the potential for using modern optimisation algorithms to expedite calculations of properties of the posterior distribution, in place of the Monte Carlo computational methods originally proposed.
Chopin, N., Jacob, P. & Green, P.J. 2012, 'Free Energy Sequential Monte Carlo, Application to Mixture Modelling' in Bayesian Statistics 9.
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© Oxford University Press 2011. All rights reserved. We introduce a new class of Sequential Monte Carlo (SMC) methods, which we call free energy SMC. This class is inspired by free energy methods, which originate from physics, and where one samples from a biased distribution such that a given function () of the state is forced to be uniformly distributed over a given interval. From an initial sequence of distributions ( t) of interest, and a particular choice of (), a free energy SMC sampler computes sequentially a sequence of biased distributions (t) with the following properties: (a) the marginal distribution of () with respect to t is approximatively uniform over a specified interval, and (b) t and t have the same conditional distribution with respect to . We apply our methodology to mixture posterior distributions, which are highly multimodal. In the mixture context, forcing certain hyper-parameters to higher values greatly facilitates mode swapping, and makes it possible to recover a symmetric output. We illustrate our approach with univariate and bivariate Gaussian mixtures and two real-world datasets.
Green, P.J., Mardia, K.V., Nyirongo, V.B. & Ruffieux, Y. 2010, 'Bayesian modelling for matching and alignment of biomolecules' in O'Hagan, A. & West, M. (eds), The Oxford Handbook of Applied Bayesian Analysis, Oxford University Press (OUP), United Kingdom, pp. 27-50.
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Green, P.J. 2010, 'Colouring and breaking sticks: random distributions and heterogeneous clustering' in Bingham, N.H. & Goldie, C.M. (eds), Probability and Mathematical Genetics: Papers in Honour of Sir John Kingman, Cambridge University Press, United Kingdom, pp. 319-344.
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Conferences

Green, P.J., Noad, R. & Smart, N. 2005, 'Further hidden Markov model cryptanalysis', Cryptographic Hardware And Embedded Systems - CHES 2005, Proceedings, Workshop on Cryptographic Hardware and Embedded Systems, Springer-Verlag Berlin, Edinburgh, UK, pp. 61-74.
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We extend the model of Karlof and Wagner for modelling side channel attacks via Input Driven Hidden Markov Models (IDHMM) to the case where not every state corresponds to a single observable symbol. This allows us to examine algorithms where errors in me

Journal articles

Green, P.J. & Mortera, J. 2017, 'Paternity testing and other inference about relationships from DNA mixtures', Forensic Science International: Genetics.
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Green, P.J. 2015, 'Discussion of "Analysis of forensic DNA mixtures with artefacts"', Journal of the Royal Statistical Society: Series C (Applied Statistics), vol. 64, pp. 41-41.
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DNA is now routinely used in criminal investigations and court cases, although DNA samples taken at crime scenes are of varying quality and therefore present challenging problems for their interpretation. We present a statistical model for the quantitative peak information obtained from an electropherogram of a forensic DNA sample and illustrate its potential use for the analysis of criminal cases. In contrast with most previously used methods, we directly model the peak height information and incorporate important artefacts that are associated with the production of the electropherogram. Our model has a number of unknown parameters, and we show that these can be estimated by the method of maximum likelihood in the presence of multiple unknown individuals contributing to the sample, and their approximate standard errors calculated; the computations exploit a Bayesian network representation of the model. A case example from a UK trial, as reported in the literature, is used to illustrate the efficacy and use of the model, both in finding likelihood ratios to quantify the strength of evidence, and in the deconvolution of mixtures for finding likely profiles of the individuals contributing to the sample. Our model is readily extended to simultaneous analysis of more than one mixture as illustrated in a case example. We show that the combination of evidence from several samples may give an evidential strength which is close to that of a single-source trace and thus modelling of peak height information provides a potentially very efficient mixture analysis.
Green, P.J., Łatuszyński, K., Pereyra, M. & Robert, C.P. 2015, 'Bayesian computation: a summary of the current state, and samples backwards and forwards', Statistics and Computing, vol. 25, no. 4, pp. 835-862.
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© 2015, The Author(s). Recent decades have seen enormous improvements in computational inference for statistical models; there have been competitive continual enhancements in a wide range of computational tools. In Bayesian inference, first and foremost, MCMC techniques have continued to evolve, moving from random walk proposals to Langevin drift, to Hamiltonian Monte Carlo, and so on, with both theoretical and algorithmic innovations opening new opportunities to practitioners. However, this impressive evolution in capacity is confronted by an even steeper increase in the complexity of the datasets to be addressed. The difficulties of modelling and then handling ever more complex datasets most likely call for a new type of tool for computational inference that dramatically reduces the dimension and size of the raw data while capturing its essential aspects. Approximate models and algorithms may thus be at the core of the next computational revolution.
Green, P.J. & Bochkina, N.A. 2014, 'The Bernstein-von Mises theorem and non-regular models', Annals of Statistics, vol. 42, no. 5, pp. 1850-1878.
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We study the asymptotic behaviour of the posterior distribution in a broad class of statistical models where the "true" solution occurs on the boundary of the parameter space. We show that in this case Bayesian inference is consistent, and that the posterior distribution has not only Gaussian components as in the case of regular models (the Bernstein-von Mises theorem) but also has Gamma distribution components whose form depends on the behaviour of the prior distribution near the boundary and have a faster rate of convergence. We also demonstrate a remarkable property of Bayesian inference, that for some models, there appears to be no bound on efficiency of estimating the unknown parameter if it is on the boundary of the parameter space. We illustrate the results on a problem from emission tomography
Green, P.J. 2013, 'Discussion of `Beyond mean regression'', Statistical Modelling, vol. 13, no. 4, pp. 305-315.
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Methodology for regression beyond the mean has been a goal of researchers for many years. This discussion provides some additional context for the important ideas in the present paper, by recounting some of the historical background to the GAMLSS approach and pointing to the power and appeal of fully probabilistic regression analysis in the setting of Bayesian nonparametrics
Vincent, T.L., Green, P.J. & Woolfson, D.N. 2013, 'LOGICOIL - multi-state prediction of coiled-coil oligomeric state', Bioinformatics, vol. 29, no. 1, pp. 69-76.
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Motivation: The coiled coil is a ubiquitous -helical protein-structure domain that directs and facilitates proteinprotein interactions in a wide variety of biological processes. At the protein-sequence level, the coiled coil is readily recognized via a conspicuous heptad repeat of hydrophobic and polar residues. However, structurally coiled coils are more complicated, existing in a wide range of oligomer states and topologies. As a consequence, predicting these various states from sequence remains an unmet challenge. Results: This work introduces LOGICOIL, the first algorithm to address the problem of predicting multiple coiled-coil oligomeric states from protein-sequence information alone. By covering 490% of the known coiled-coil structures, LOGICOIL is a net improvement compared with other existing methods, which achieve a predictive coverage of 31% of this population. This leap in predictive power offers better opportunities for genome-scale analysis, and analyses of coiled-coil containing protein assemblies.
Green, P.J. & Thomas, A. 2013, 'Sampling decomposable graphs using a Markov chain on junction trees', Biometrika, vol. 100, no. 1, pp. 91-110.
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Full Bayesian computational inference for model determination in undirected graphical models is currently restricted to decomposable graphs or other special cases, except for small-scale problems, say up to 15 variables. In this paper we develop new, more efficient methodology for such inference, by making two contributions to the computational geometry of decomposable graphs. The first of these provides sufficient conditions under which it is possible to completely connect two disconnected complete subsets of vertices, or perform the reverse procedure, yet maintain decomposability of the graph. The second is a new Markov chainMonte Carlo sampler for arbitrary positive distributions on decomposable graphs, taking a junction tree representing the graph as its state variable. The resulting methodology is illustrated with numerical experiments on three models
Hastie, D.I. & Green, P.J. 2012, 'Model choice using reversible jump Markov chain Monte Carlo', Statistica Neerlandica, vol. 66, no. 3, pp. 309-338.
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We review the across-model simulation approach to computation for Bayesian model determination, based on the reversible jump Markov chain Monte Carlo method. Advantages, difficulties and variations of the methods are discussed. We also discuss some limitations of the ideal Bayesian view of the model determination problem, for which no computational methods can provide a cure.
Scheel, I., Green, P.J. & Rougier, J.C. 2011, 'A graphical diagnostic for identifying influential model choices in bayesian hierarchical models', Scandinavian Journal of Statistics, vol. 38, no. 3, pp. 1-22.
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Real-world phenomena are frequently modelled by Bayesian hierarchical models. The building-blocks in such models are the distribution of each variable conditional on parent and/or neighbour variables in the graph. The specifications of centre and spread of these conditional distributions may be well motivated, whereas the tail specifications are often left to convenience. However, the posterior distribution of a parameter may depend strongly on such arbitrary tail specifications. This is not easily detected in complex models. In this article, we propose a graphical diagnostic, the Local critique plot, which detects such influential statistical modelling choices at the node level. It identifies the properties of the information coming from the parents and neighbours (the local prior) and from the children and co-parents (the lifted likelihood) that are influential on the posterior distribution, and examines local conflict between these distinct information sources. The Local critique plot can be derived for all parameters in a chain graph model.
Armstrong, C.T., Vincent, T.L., Green, P.J. & Woolfson, D.N. 2011, 'SCORER 2.0: an algorithm for distinguishing parallel dimeric and trimeric coiled-coil sequences', Bioinformatics, vol. 27, no. 14, pp. 1908-1914.
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Motivation: The coiled coil is a ubiquitous a-helical protein structure domain that directs and facilitates proteinprotein interactions in a wide variety of biological processes. At the protein-sequence level, coiled coils are quite straightforward and readily recognized via the conspicuous heptad repeats of hydrophobic and polar residues. However, structurally they are more complicated, existing in a range of oligomer states and topologies. Here, we address the issue of predicting coiled-coil oligomeric state from protein sequence. Results: The predominant coiled-coil oligomer states in Nature are parallel dimers and trimers. Here, we improve and retrain the first-published algorithm, SCORER, that distinguishes these states, and test it against the current standard, MultiCoil. The SCORER algorithm has been revised in two key respects: first, the statistical basis for SCORER is improved markedly. Second, the training set for SCORER has been expanded and updated to include only structurally validated coiled coils. The result is a much-improved oligomer state predictor that outperforms MultiCoil, particularly in assigning oligomer state to short coiled coils, and those that are diverse from the training set.
Thomas, A. & Green, P.J. 2009, 'Enumerating The Junction Trees Of A Decomposable Graph', Journal of Computational and Graphical Statistics, vol. 18, no. 4, pp. 930-940.
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We derive methods for enumerating the distinct junction tree representations for any given decomposable graph. We discuss the relevance of the method to estimating conditional independence graphs of graphical models and give an algorithm that, given a ju
Green, P.J. & Mortera, J. 2009, 'Sensitivity of inferences in forensic genetics to assumptions about founding genes', Annals of Applied Statistics, vol. 3, no. 2, pp. 731-763.
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Many forensic genetics problems can be handled using structured systems of discrete variables, for which Bayesian networks offer an appealing practical modeling framework, and allow inferences to be computed by probability propagation methods. However, w
Ruffieux, Y. & Green, P.J. 2009, 'Alignment of multiple configurations using hierarchical models', Journal Of Computational And Graphical Statistics, vol. 18, no. 3, pp. 756-773.
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We describe a method for aligning multiple unlabeled configurations simultaneously, Specifically. we extend the two-con figuration matching approach of Green and Mardia (2006) to the multiple configuration setting. Our approach is based on the introducti
Thomas, A. & Green, P.J. 2009, 'Enumerating the decomposable neighbors of a decomposable graph under a simple perturbation scheme', Computational Statistics & Data Analysis, vol. 53, no. 4, pp. 1232-1238.
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Given a decomposable graph, we characterize and enumerate the set of pairs of vertices whose connection or disconnection results in a new graph that is also decomposable. We discuss the relevance of these results to Markov chain Monte Carlo methods that
Hosking, F.J., Sterne, J.A., Smith, G.D. & Green, P.J. 2008, 'Inference from genome-wide association studies using a novel Markov model', Genetic Epidemiology, vol. 32, no. 6, pp. 497-504.
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In this paper we propose a Bayesian modeling 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)
Hurn, M., Green, P.J. & Al-awadhi, F. 2008, 'A Bayesian hierarchical model for photometric red shifts', Journal of the Royal Statistical Society Series C: Applied Statistics, vol. 57, no. 4, pp. 487-504.
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The Sloan digital sky survey is an extremely large astronomical survey that is conducted with the intention of mapping more than a quarter of the sky. Among the data that it is generating are spectroscopic and photometric measurements, both containing in
Mardia, K.V., Nyirongo, V.B., Green, P.J., Gold, N.D. & Westhead, D.R. 2007, 'Bayesian refinement of protein functional site matching', BMC Bioinformatics, vol. 8, no. 1, pp. 1-18.
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Background: Matching functional sites is a key problem for the understanding of protein function and evolution. The commonly used graph theoretic approach, and other related approaches, require adjustment of a matching distance threshold a priori accordi
Lau, J. & Green, P.J. 2006, 'Bayesian model-based clustering procedures', Journal of Computational and Graphical Statistics, vol. 16, no. 3, pp. 526-558.
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This paper establishes a general framework for Bayesian model-based clustering, in which subset labels are exchangeable, and items are also exchangeable, possibly up to covariate e®ects. It is rich enough to encompass a variety of existing procedures, including some recently discussed methodologies involving stochastic search or hierarchical clustering, but more importantly allows the formulation of clustering procedures that are optimal with respect to a specied loss function. Our focus is on loss functions based on pairwise coincidences, that is, whether pairs of items are clustered into the same subset or not.
Green, P.J. & Mardia, K.V. 2006, 'Bayesian alignment using Hierarchical models, with applications in protein bioinformatics', Biometrika, vol. 93, no. 2, pp. 235-254.
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An important problem in shape analysis is to match configurations of points in space after filtering out some geometrical transformation. In this paper we introduce hierarchical models for such tasks, in which the points in the configurations are either
Hein, A.K., Richardson, S., Causton, H.C., Ambler, G.K. & Green, P.J. 2005, 'BGX: A fully Bayesian integrated approach to the analysis of affymetrix genechip data', Biostatistics, vol. 6, no. 3, pp. 349-373.
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We present Bayesian hierarchical models for the analysis of Affymetrix GeneChip data. The approach we take differs from other available approaches in two fundamental aspects. Firstly, we aim to integrate all processing steps of the raw data in a common s
Best, N. & Green, P. 2005, 'Structure and uncertainty: Graphical models for understanding complex data', Significance, vol. 2, no. 4, pp. 177-181.
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© 2005 The Royal Statistical Society. Statistics is fundamental to making sense of the complexity of modern science. From the micro-level of the human genome to the macro-level of the universe, scientists need statistical models to help them extract meaning from empirical observations. Graphical models have been used across a wide variety of disciplines for building multivariate probabilistic models to represent, and draw inference about, complex phenomena. Nicky Best and Peter Green explain the ideas behind graphical models and show how they can be used to help tackle the challenges of complex statistical problems.
Nott, D. & Green, P.J. 2004, 'Bayesian variable selection and the Swendsen-Wang slgorithm', Journal Of Computational And Graphical Statistics, vol. 13, no. 1, pp. 141-157.
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The need to explore model uncertainty in linear regression models with many predictors has motivated improvements in Markov chain Monte Carlo sampling algorithms for Bayesian variable selection. Currently used sampling algorithms for Bayesian variable se
Green, P.J. 2003, 'Diversities of gifts, but the same spirit', Journal Of The Royal Statistical Society Series D-The Statistician, vol. 52, pp. 423-435.
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This address reviews the great diversity of the discipline of statistics, seeking an essential unity among its various aspects. The role of statistical modelling in underpinning the subject is stressed. To safeguard the discipline in the future, it is se
Scaccia, L. & Green, P.J. 2003, 'Bayesian growth curves using normal mixtures with nonparametric weights', Journal Of Computational And Graphical Statistics, vol. 12, no. 2, pp. 308-331.
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Reference growth curves estimate the distribution of a measurement as it changes according to some covariate, often age. We present a new methodology to estimate growth curves based on mixture models and splines. We model the distribution of the measurem
Robert, C., Meng, X., Moller, J., Rosenthal, J., Jennison, C., Hurn, M., Al-awadhi, F., Mccullagh, P., Andrieu, C., Doucet, A., Dellaportas, P., Papageorgiou, I., Ehlers, R., Erosheva, E., Fienberg, S., Forster, J., Gill, R., Friel, N., Green, P.J. & Hastie, D. 2003, 'Efficient Construction Of Reversible Jump Markov Chain Monte Carlo Proposal Distributions - Discussion On The Paper By Brooks, Giudici And Roberts', Journal Of The Royal Statistical Society Series B-Statistical Methodology, vol. 65, no. 1, pp. 3-55.
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Robert, C.P., Meng, X.L., Møller, J., Rosenthal, J.S., Jennison, C., Hurn, M.A., Al-Awadhi, F., McCullagh, P., Andrieu, C., Doucet, A., Dellaportas, P., Papageorgiou, I., Ehlers, R.S., Erosheva, E.A., Fienberg, S.E., Forster, J.J., Gill, R.C., Friel, N., Green, P., Hastie, D., King, R., Künsch, H.R., Lazar, N.A. & Osinski, C. 2003, 'Discussion on the paper by Brooks, Giudici and Roberts', Journal of the Royal Statistical Society. Series B: Statistical Methodology, vol. 65, no. 1, pp. 39-55.
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Green, P.J. & Richardson, S. 2002, 'Hidden Markov models and disease mapping', Journal Of The American Statistical Association, vol. 97, no. 460, pp. 1055-1070.
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We present new methodology to extend hidden Markov models to the spatial domain, and use this class of models to analyze spatial heterogeneity of count data on a rare phenomenon. This situation occurs commonly in many domains of application, particularly
Fernandez, C. & Green, P.J. 2002, 'Modelling spatially correlated data via mixtures: A Bayesian approach', Journal Of The Royal Statistical Society Series B-Statistical Methodology, vol. 64, no. 1, pp. 805-826.
The paper develops mixture models for spatially indexed data. We confine attention to the case of finite, typically irregular, patterns of points or regions with prescribed spatial relationships, and to problems where it is only the weights in the mixtur
Richardson, S., Leblond, L., Jaussent, I. & Green, P.J. 2002, 'Mixture models in measurement error problems, with reference to Epidemiological studies', Journal Of The Royal Statistical Society Series A-Statistics In Society, vol. 165, no. 1, pp. 549-566.
The paper focuses on a Bayesian treatment of measurement error problems and on the question of the specification of the prior distribution of the unknown covariates. It presents a flexible semiparametric model for this distribution based on a mixture of
Dawid, A., Cox, D., Kreiner, S., Green, P.J., Shipley, B., Kent, J., Smith, J., Koster, J., Madigan, D., Andersson, S., Perlman, M., Robert, C., Marin, J., Rosenbaum, P., Roverato, A., Consonni, G. & Studeny, M. 2002, 'Chain Graph Models And Their Causal Interpretations - Discussion On The Paper By Lauritzen And Richardson', Journal Of The Royal Statistical Society Series B-Statistical Methodology, vol. 64, no. 1, pp. 348-361.
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Viallefont, V., Richardson, S. & Green, P.J. 2002, 'Bayesian analysis Of Poisson mixtures', Journal Of Nonparametric Statistics, vol. 14, no. 1-Feb, pp. 181-202.
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The modelling of rare events via a Poisson distribution sometimes reveals substantial over-dispersion, indicating that some unexplained discontinuity arises in the data. We suggest modelling this over-dispersion by a Poisson mixture. In a hierarchical Ba
Brooks, S.P., Smith, J., Vehtari, A., Plummer, M., Stone, M., Robert, C.P., Titterington, D.M., Nelder, J.A., Atkinson, A., Dawid, A.P., Lawson, A., Clark, A., Bernardo, J.M., Sahu, S.K., Richardson, S., Green, P., Burnham, K.P., DeIorio, M., Robert, C.P., Draper, D., Gelfand, A.E., Trevisani, M., Hodges, J., Lee, Y., De Luna, X. & Meng, X.L. 2002, 'Discussion on the paper by Spiegelhalter, Best, Carlin and van der Linde', Journal of the Royal Statistical Society. Series B: Statistical Methodology, vol. 64, no. 4, pp. 616-639.
Green, P.J. & Mira, A. 2001, 'Delayed Rejection In Reversible Jump Metropolis-Hastings', Biometrika, vol. 88, no. 4, pp. 1035-1053.
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In a Metropolis-Hastings algorithm, rejection of proposed moves is an intrinsic part of ensuring that the chain converges to the intended target distribution. However, persistent rejection, perhaps in particular parts of the state space, may indicate tha
Green, P.J. & Richardson, S. 2001, 'Modelling heterogeneity with and without the dirichlet process', Scandinavian Journal Of Statistics, vol. 28, no. 2, pp. 355-375.
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We investigate the relationships between Dirichlet process (DP) based models and allocation models for a variable number of components, based on exchangeable distributions. It is shown that the DP partition distribution is a Limiting case of a Dirichlet-
Mira, A. & Green, P.J. 2001, 'Discussion', Journal of Computational and Graphical Statistics, vol. 10, no. 1, pp. 94-97.
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Cook, R., Pardoe, L., Gelfand, A., Green, P.J., Hastie, T. & Tibshirani, R. 2000, 'Bayesian backfitting - Comments and rejoinder', Statistical Science, vol. 15, no. 3, pp. 213-223.
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Davino, F., Frigessi, A. & Green, P.J. 2000, 'Penalized pseudolikelihood inference in spatial interaction models with covariates', Scandinavian Journal Of Statistics, vol. 27, no. 3, pp. 445-458.
Given spatially located observed random variables ((x) under bar, (z) under bar) = {(x(i), z(i))}(i), we propose a new method for non-parametric estimation of the potential functions of a Markov random field p((x) under bar\(z) under bar), based on a rou
Nobile, A. & Green, P.J. 2000, 'Bayesian analysis of factorial experiments by mixture modelling', Biometrika, vol. 87, no. 1, pp. 15-35.
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A Bayesian analysis for factorial experiments is presented, using finite mixture distributions to model the main effects and interactions. This allows both estimation and an analogue of hypothesis testing in a posterior analysis using a single prior spec
Giudici, P. & Green, P.J. 1999, 'Decomposable graphical Gaussian model determination', Biometrika, vol. 86, no. 4, pp. 785-801.
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We propose a methodology for Bayesian model determination in decomposable graphical Gaussian models. To achieve this aim we consider a hyper inverse Wishart prior distribution on the concentration matrix for each given graph. To ensure compatibility acro
Glasbey, C., Preece, D., Diggle, P., Pearce, S., Kunsch, H., Gilmour, S., Fernandez, C., Green, P.J., Butler, N., Bailey, R., Smith, R., Silverman, B., Jennison, C., Barnard, G., Bartlett, M., Best, N., Ickstadt, K., Wolpert, R., Byers, S. & Davison, A. 1999, 'Bayesian Analysis Of Agricultural Field Experiments - Discussion', Journal Of The Royal Statistical Society Series B-Statistical Methodology, vol. 61, no. 1, pp. 717-746.
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Hodgson, M. & Green, P.J. 1999, 'Bayesian Choice Among Markov Models Of Ion Channels Using Markov Chain Monte Carlo', Proceedings Of The Royal Society Of London Series A-Mathematical Physical And Engineering Sciences, vol. 455, no. 1989, pp. 3425-3448.
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Hodgson formulates a Bayesian method of ion-channel analysis, using an alternating renewal model of channel kinetics and an autoregressive process to model patch-clamp noise. The resulting posterior is computed by a reversible jump Markov chain Monte Car
Kempton, R., Mead, R., Engel, B., Ter Braak, C., Nelder, J., Morton, R., Green, P.J., Molenberghs, G., Basford, K., Longford, N., Gilmour, S., Butler, N., Eilers, P. & Pettitt, T. 1999, 'The Analysis Of Designed Experiments And Longitudinal Data By Using Smoothing Splines - Discussion', Journal Of The Royal Statistical Society Series C-Applied Statistics, vol. 48, no. 1, pp. 300-311.
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Murdoch, D. & Green, P.J. 1998, 'Exact sampling from a continuous state space', Scandinavian Journal Of Statistics, vol. 25, no. 3, pp. 483-502.
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Propp & Wilson (1996) described a protocol, called coupling from the past, for exact sampling from a target distribution using a coupled Markov chain Monte Carlo algorithm. In this paper,ve extend coupling from the past to various MCMC samplers on a cont
Ganesh, A., Green, P.J., Oconnell, N. & Pitts, S. 1998, 'Bayesian network management', Queueing Systems, vol. 28, no. 1-Mar, pp. 267-282.
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We formulate some general network (and risk) management problems in a Bayesian context, and point out some of the essential features. We argue and demonstrate that, when one is interested in rare events, the Bayesian and frequentist approaches can lead t
Pievatolo, A. & Green, P.J. 1998, 'Boundary detection through dynamic polygons', Journal Of The Royal Statistical Society Series B-Statistical Methodology, vol. 60, no. 1, pp. 609-626.
A method for the Bayesian restoration of noisy binary images portraying an object with constant grey level on a background is presented. The restoration, performed by fitting a polygon with any number of sides to the objects outline, is driven by a new p
Pievatolo, A. & Green, P.J. 1998, 'Boundary detection through dynamic polygons', Journal of the Royal Statistical Society. Series B: Statistical Methodology, vol. 60, no. 3, pp. 609-626.
A method for the Bayesian restoration of noisy binary images portraying an object with constant grey level on a background is presented. The restoration, performed by fitting a polygon with any number of sides to the object's outline, is driven by a new probabilistic model for the generation of polygons in a compact subset of R 2 , which is used as a prior distribution for the polygon. Some measurability issues raised by the correct specification of the model are addressed. The simulation from the prior and the calculation of the a posteriori mean of grey levels are carried out through reversible jump Markov chain Monte Carlo computation, whose implementation and convergence properties are also discussed. One example of restoration of a synthetic image is presented and compared with existing pixel-based methods. © 1998 Royal Statistical Society.
Ganesh, A., Green, P., O'Connell, N. & Pitts, S. 1998, 'Bayesian network management', Queueing Systems, vol. 28, no. 1-3, pp. 267-282.
We formulate some general network (and risk) management problems in a Bayesian context, and point out some of the essential features. We argue and demonstrate that, when one is interested in rare events, the Bayesian and frequentist approaches can lead to very different strategies: the former typically leads to strategies which are more conservative. We also present an asymptotic formula for the predictive probability of ruin (for a random walk with positive drift) for large initial capital and large number of past observations. This is a preliminary investigation which raises many interesting questions for future research.
Rubin, D., Titterington, D., Gilks, W., Diebolt, J., Aitkin, M., Smith, C., Hinde, J., Kent, J., Tyler, D., Damien, P., Walker, S., Chauveau, D., Draper, D., Dupuis, J., Fessler, J., Gelman, A., Green, P.J., Hero, A., Lavielle, M., Liu, C., Liu, J. & Roberts, G. 1997, 'The Em Algorithm - An Old Folk-Song Sung To A Fast New Tune - Discussion', Journal Of The Royal Statistical Society Series B-Statistical Methodology, vol. 59, no. 3, pp. 541-567.
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Richardson, S. & Green, P.J. 1997, 'On bayesian analysis of mixtures with an unknown number of components', Journal Of The Royal Statistical Society Series B-Methodological, vol. 59, no. 4, pp. 731-758.
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New methodology for fully Bayesian mixture analysis is developed, making use of reversible jump Markov chain Monte Carlo methods that are capable of jumping between the parameter subspaces corresponding to different numbers of components in the mixture.
Green, P.J. 1995, 'Reversible jump Markov chain Monte Carlo computation and Bayesian model determination', Biometrika, vol. 82, no. 4, pp. 711-732.
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Markov chain Monte Carlo methods for Bayesian computation have until recently been restricted to problems where the joint distribution of all variables has a density with respect to some fixed standard underlying measure. They have therefore not been ava
Besag, J., Green, P.J., Higdon, D. & Mengersen, K. 1995, 'Bayesian computation and stochastic-systems', Statistical Science, vol. 10, no. 1, pp. 3-41.
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Markov chain Monte Carlo (MCMC) methods have been used extensively in statistical physics over the last 40 years, in spatial statistics for the past 20 and in Bayesian image analysis over the last decade. In the last five years, MCMC has been introduced
Besag, J., Green, P., Higdon, D. & Mengersen, K. 1995, 'Rejoinder', Statistical Science, vol. 10, no. 1, pp. 58-67.
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Green, P. 1994, 'Editorial', Statistical Methods in Medical Research, vol. 3, no. 1, pp. 1-3.
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Green, P.J. & Weir, L.S. 1994, 'Modelling data from single-photon emission computerized tomography', Journal of Applied Statistics, vol. 21, no. 1-2, pp. 313-337.
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The completely random character of radioactive disintegration provides the basis of a strong justification for a Poisson linear model for single-photon emission computed tomography data, which can be used to produce reconstructions of isotope densities, whether by maximum likelihood or Bayesian methods. However, such a model requires the construction of a matrix of weights, which represent the mean rates of arrival at each detector of photons originating from each point within the body space. Two methods of constructing these weights are discussed, and reconstructions resulting from phantom and real data are presented. © 1994, Taylor & Francis Group, LLC. All rights reserved.
Green, P. 1994, 'Statistical aspects of imaging.', Stat Methods Med Res, vol. 3, no. 1, pp. 1-3.
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Besag, J. & Green, P.J. 1993, 'Spatial statistics and bayesian computation', Journal Of The Royal Statistical Society Series B-Methodological, vol. 55, no. 1, pp. 25-37.
Markov chain Monte Carlo (MCMC) algorithms, such as the Gibbs sampler, have provided a Bayesian inference machine in image analysis and in other areas of spatial statistics for several years, founded on the pioneering ideas of Ulf Grenander. More recentl
Clifford, P., Jennison, C., Wakefield, J., Phillips, D., Frigessi, A., Gray, A., Lawson, A., Forster, J., Ramgopal, P., Arslan, O., Constable, P., Kent, J., Wolff, R., Harding, E., Middleton, R., Diggle, P., Aykroyd, R., Berzuini, C., Brewer, M. & Aitken, C. 1993, 'Discussion On The Meeting On The Gibbs Sampler And Other Markov Chain-Monte Carlo Methods', Journal Of The Royal Statistical Society Series B-Methodological, vol. 55, no. 1, pp. 53-102.
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Cole, T. & Green, P.J. 1992, 'Smoothing reference centile curves - The LMS method and penalized likelihood', Statistics In Medicine, vol. 11, no. 10, pp. 1305-1319.
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Reference centile curves show the distribution of a measurement as it changes according to some covariate, often age. The LMS method summarizes the changing distribution by three curves representing the median, coefficient of variation and skewness, the
Aykroyd, R. & Green, P.J. 1991, 'Global and local priors, and the location of lesions using gamma-camera imagery', Philosophical Transactions Of The Royal Society Of London Series A-Mathematical Physical And Engineering Sciences, vol. 337, no. 1647, pp. 323-342.
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After a brief review of the paradigm of bayesian image restoration, we pose the question: If high-level prior information is available and usable, what is lost by modelling at the pixel level instead? Our discussion is based on a real application where t
Green, P.J. 1991, 'Bayesian image-restoration, with 2 applications in spatial statistics - Discussion', Annals Of The Institute Of Statistical Mathematics, vol. 43, no. 1, pp. 22-24.
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Green, P.J. 1990, 'On use of the EM algorithm for penalized likelihood estimation', Journal of The Royal Statistical Society Series B-methodological, vol. 52, no. 3, pp. 443-452.
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The EM algorithmis a popular approach to maximuml ikelihoode stimationb ut has not been muchu sed forp enalizedl ikelihoodo r maximuma posteriori estimation.T his paper discussesp ropertieos f theE M algorithmin suchc ontextsc, oncentratinogn rateso f convergence, and presentsa n alternativet hati s usuallym ore practicala nd convergesa t least as quickly
Green, P.J. 1990, 'Bayesian reconstructions from emission tomography data using a modified EM algorithm', IEEE Transactions On Medical Imaging, vol. 9, no. 1, pp. 84-93.
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Green, P.J. 1988, 'Regression, curvature and weighted least-squares', Mathematical Programming, vol. 42, no. 1, pp. 41-51.
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Green, P.J. 1987, 'Penalized likelihood for general semiparametric regression-models', International Statistical Review, vol. 55, no. 3, pp. 245-259.
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Lenth, R.V. & Green, P.J. 1987, 'Consistency of deviance-based M estimators', Journal of The Royal Statistical Society Series B-methodological, vol. 49, no. 3, pp. 326-330.
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In a general estimation problem, the deviance function generates statistics that are similar to squared standardized residuals. A deviance-based M estimator (DBME) is defined as an adaptively weighted maximum-likelihood estimator, where the weights depend upon the deviances. In both a single-parameter and a regression setting, we give some general conditions under which a DMBE is consistent. For a suitable weighting scheme, these conditions are satisfied in many continuous Cramer-Rao-regular families and in related linear or nonlinear regression cases. The conditions fail (and the estimator is inconsistent) in most discrete families.
Green, P.J. 1985, 'Linear-models for field trials, smoothing and cross-validation', Biometrika, vol. 72, no. 3, pp. 527-537.
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Green, P.J., Jennison, C. & Seheult, A. 1985, 'Analysis of field experiments by least-squares smoothing', Journal of The Royal Statistical Society Series B-methodological, vol. 47, no. 2, pp. 299-315.
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Assuminga smooth trendp lus independente rrorm odel fort he environmentael ffects in the yields of a fieldp lot experiment,le ast squares smoothingm ethodsa re developed to estimate both the treatmente ffectsa nd the unknown trend. Treatmente stimates are closely related to those resultingf roma generalizedl east squares analysisi n which the covariance structuref or the environmentael ffectsh as a particularf orm.H owever, the main emphases are on the accuracy of treatmente stimatesu nder a fixed smooth trend plus error model and the exploratory power of the basic method to isolate trende ffectso f unknownf orm. Althought he detailed developmenti s for the one-dimensionalc ase, generalizations of the smoothness concept and extensions to two dimensions are also discussed. Application of the basic method is illustrated on three data sets and the results compared with other analyses.
Green, P.J. 1984, 'Iteratively reweighted least-squares for maximum-likelihood estimation, and some robust and resistant alternatives', Journal Of The Royal Statistical Society Series B-Methodological, vol. 46, no. 2, pp. 149-192.
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Wright, B.D., Green, P.J. & Braiden, P. 1982, 'Quantitative-analysis of delayed fracture observed in stress rate tests on brittle materials', Journal Of Materials Science, vol. 17, no. 11, pp. 3227-3234.
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Green, P.J. 1981, 'Modeling yeast-cell growth using stochastic branching-processes', Journal Of Applied Probability, vol. 18, no. 4, pp. 799-808.
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Nelson, S. & Green, P.J. 1981, 'The random transition model of the cell-cycle - A critical-review', Cancer Chemotherapy And Pharmacology, vol. 6, no. 1, pp. 11-18.
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Green, P.J. 1980, 'A 'random transition' in the cell cycle?', Nature, vol. 285, no. 5760, p. 116.
Green, P.J. & Silverman, B. 1979, 'Constructing the convex hull of a set of points in the plane', Computer Journal, vol. 22, no. 3, pp. 262-266.
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Green, P.J. & Sibson, R. 1978, 'Computing dirichlet tessellations in plane', Computer Journal, vol. 21, no. 2, pp. 168-173.
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Green, P.J. 1977, 'Conditional limit-theorems for general branching-processes', Journal Of Applied Probability, vol. 14, no. 3, pp. 451-463.
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Green, P.J. 1977, 'Conditioning a branching-process on non-extinction', Mathematical Biosciences, vol. 35, no. 3-4, pp. 261-265.
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Green, P.J. 1976, 'The maximum and time to absorption of a left-continuous random walk', Journal Of Applied Probability, vol. 13, no. 3, pp. 444-454.
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For a left-continuous random walk, absorbing at 0, the joint distribution of the maximum and time to absorption is derived. A description of the tails of the distributions and a conditional limit theorem are obtained for the cases where absorption is certain.
Barnett, V., Green, P.J. & Robinson, A. 1976, 'Concomitants and correlation estimates', Biometrika, vol. 63, no. 2, pp. 323-328.
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Professor Louise Ryan and Jarod Lee, UTS.

Professor AlunThomas, University of Utah.

Professr Juli Mortera, Universita Roma Tre

Dr Natalia Bochkina, University of Edinburgh