arXiv subjects

Peter J. Green

Publications and source records attributed to Peter J. Green.

15 recordsLinked to original sources

arXiv2021

Inference about complex relationships using peak height data from DNA mixtures

In both criminal cases and civil cases there is an increasing demand for the analysis of DNA mixtures involving relationships. The goal might be, for example, to identify the contributors to a DNA mixture where the donors may be related, or to infer the relationship between individuals based on a mixture. This paper introduces an approach to modelling and computation for DNA mixtures involving contributors with arbitrarily complex relationships. It builds on an extension of Jacquard's condensed coefficients of identity, to specify and compute with joint relationships, not only pairwise ones, including the possibility of inbreeding. The methodology developed is applied to two casework examples involving a missing person, and simulation studies of performance, in which the ability of the methodology to recover complex relationship information from synthetic data with known `true' family structure is examined. The methods used to analyse the examples are implemented in the new KinMix R package, that extends the DNAmixtures package to allow for modelling DNA mixtures with related contributors. KinMix inherits from DNAmixtures the capacity to deal with mixtures with many contributors, in a time- and space-efficient way.

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arXiv2021

Casework applications of probabilistic genotyping methods for DNA mixtures that allow relationships between contributors

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arXiv2020

COVID-19 and the difficulty of inferring epidemiological parameters from clinical data

Knowing the infection fatality ratio (IFR) is of crucial importance for evidence-based epidemic management: for immediate planning; for balancing the life years saved against the life years lost due to the consequences of management; and for evaluating the ethical issues associated with the tacit willingness to pay substantially more for life years lost to the epidemic, than for those to other diseases. Against this background Verity et al. (2020, Lancet Infections Diseases) have rapidly assembled case data and used statistical modelling to infer the IFR for COVID-19. We have attempted an in-depth statistical review of their approach, to identify to what extent the data are sufficiently informative about the IFR to play a greater role than the modelling assumptions, and have tried to identify those assumptions that appear to play a key role. Given the difficulties with other data sources, we provide a crude alternative analysis based on the Diamond Princess Cruise ship data and case data from China, and argue that, given the data problems, modelling of clinical data to obtain the IFR can only be a stop-gap measure. What is needed is near direct measurement of epidemic size by PCR and/or antibody testing of random samples of the at risk population.

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arXiv2018

Introduction to finite mixtures

Mixture models have been around for over 150 years, as an intuitively simple and practical tool for enriching the collection of probability distributions available for modelling data. In this chapter we describe the basic ideas of the subject, present several alternative representations and perspectives on these models, and discuss some of the elements of inference about the unknowns in the models. Our focus is on the simplest set-up, of finite mixture models, but we discuss also how various simplifying assumptions can be relaxed to generate the rich landscape of modelling and inference ideas traversed in the rest of this book.

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arXiv2018

Julian Ernst Besag, 26 March 1945 -- 6 August 2010, a biographical memoir

Julian Besag was an outstanding statistical scientist, distinguished for his pioneering work on the statistical theory and analysis of spatial processes, especially conditional lattice systems. His work has been seminal in statistical developments over the last several decades ranging from image analysis to Markov chain Monte Carlo methods. He clarified the role of auto-logistic and auto-normal models as instances of Markov random fields and paved the way for their use in diverse applications. Later work included investigations into the efficacy of nearest neighbour models to accommodate spatial dependence in the analysis of data from agricultural field trials, image restoration from noisy data, and texture generation using lattice models.

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arXiv2017

Conjugate generalized linear mixed models for clustered data

This article concerns a class of generalized linear mixed models for clustered data, where the random effects are mapped uniquely onto the grouping structure and are independent between groups. We derive necessary and sufficient conditions that enable the marginal likelihood of such class of models to be expressed in closed-form. Illustrations are provided using the Gaussian, Poisson, binomial and gamma distributions. These models are unified under a single umbrella of conjugate generalized linear mixed models, where "conjugate" refers to the fact that the marginal likelihood can be expressed in closed-form, rather than implying inference via the Bayesian paradigm. Having an explicit marginal likelihood means that these models are more computationally convenient, which can be important in big data contexts. Except for the binomial distribution, these models are able to achieve simultaneous conjugacy, and thus able to accommodate both unit and group level covariates.

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arXiv2017

On the "Poisson Trick" and its Extensions for Fitting Multinomial Regression Models

This article is concerned with the fitting of multinomial regression models using the so-called "Poisson Trick". The work is motivated by Chen & Kuo (2001) and Malchow-Møller & Svarer (2003) which have been criticized for being computationally inefficient and sometimes producing nonsense results. We first discuss the case of independent data and offer a parsimonious fitting strategy when all covariates are categorical. We then propose a new approach for modelling correlated responses based on an extension of the Gamma-Poisson model, where the likelihood can be expressed in closed-form. The parameters are estimated via an Expectation/Conditional Maximization (ECM) algorithm, which can be implemented using functions for fitting generalized linear models readily available in standard statistical software packages. Compared to existing methods, our approach avoids the need to approximate the intractable integrals and thus the inference is exact with respect to the approximating Gamma-Poisson model. The proposed method is illustrated via a reanalysis of the yogurt data discussed by Chen & Kuo (2001).

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arXiv2017

Paternity testing and other inference about relationships from DNA mixtures

We present methods for inference about relationships between contributors to a DNA mixture and other individuals of known genotype: a basic example would be testing whether a contributor to a mixture is the father of a child of known genotype. The evidence for such a relationship is evaluated as the likelihood ratio for the specified relationship versus the alternative that there is no such relationship. We analyse real casework examples from a criminal case and a disputed paternity case; in both examples part of the evidence was from a DNA mixture. DNA samples are of varying quality and therefore present challenging problems in interpretation. Our methods are based on a recent statistical model for DNA mixtures, in which a Bayesian network (BN) is used as a computational device; the present work builds on that approach, but makes more explicit use of the BN in the modelling. The R code for the analyses presented is freely available as supplementary material. We show how additional information of specific genotypes relevant to the relationship under analysis greatly strengthens the resulting inference. We find that taking full account of the uncertainty inherent in a DNA mixture can yield likelihood ratios very close to what one would obtain if we had a single source DNA profile. Furthermore, the methods can be readily extended to analyse different scenarios as our methods are not limited to the particular genotyping kits used in the examples, to the allele frequency databases used, to the numbers of contributors assumed, to the number of traces analysed simultaneously, nor to the specific hypotheses tested.

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arXiv2015

Bayesian computation: a perspective on the current state, and sampling backwards and forwards

The past decades have seen enormous improvements in computational inference based on statistical models, with continual enhancement in a wide range of computational tools, in competition. In Bayesian inference, first and foremost, MCMC techniques continue to evolve, moving from random walk proposals to Langevin drift, to Hamiltonian Monte Carlo, and so on, with both theoretical and algorithmic inputs opening wider access to practitioners. However, this impressive evolution in capacity is confronted by an even steeper increase in the complexity of the models and 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 reduce 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.

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arXiv2014

The Bernstein-von Mises theorem and nonregular models

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.

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arXiv2012

Consistency and efficiency of Bayesian estimators in generalised linear inverse problems

Formulating a statistical inverse problem as one of inference in a Bayesian model has great appeal, notably for what this brings in terms of coherence, the interpretability of regularisation penalties, the integration of all uncertainties, and the principled way in which the set-up can be elaborated to encompass broader features of the context, such as measurement error, indirect observation, etc. The Bayesian formulation comes close to the way that most scientists intuitively regard the inferential task, and in principle allows the free use of subject knowledge in probabilistic model building. However, in some problems where the solution is not unique, for example in ill-posed inverse problems, it is important to understand the relationship between the chosen Bayesian model and the resulting solution. Taking emission tomography as a canonical example for study, we present results about consistency of the posterior distribution of the reconstruction, and a general method to study convergence of posterior distributions. To study efficiency of Bayesian inference for ill-posed linear inverse problems with constraint, we prove a version of the Bernstein-von Mises theorem for nonregular Bayesian models.

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arXiv2012

Sampling decomposable graphs using a Markov chain on junction trees

Full Bayesian computational inference for model determination in undirected graphical models is currently restricted to decomposable graphs, except for problems of very small scale. 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 chain Monte 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 specific models.

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arXiv2010

Colouring and breaking sticks: random distributions and heterogeneous clustering

We begin by reviewing some probabilistic results about the Dirichlet Process and its close relatives, focussing on their implications for statistical modelling and analysis. We then introduce a class of simple mixture models in which clusters are of different `colours', with statistical characteristics that are constant within colours, but different between colours. Thus cluster identities are exchangeable only within colours. The basic form of our model is a variant on the familiar Dirichlet process, and we find that much of the standard modelling and computational machinery associated with the Dirichlet process may be readily adapted to our generalisation. The methodology is illustrated with an application to the partially-parametric clustering of gene expression profiles.

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arXiv2009

Sensitivity of inferences in forensic genetics to assumptions about founding genes

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, when standard assumptions are violated--for example, when allele frequencies are unknown, there is identity by descent or the population is heterogeneous--dependence is generated among founding genes, that makes exact calculation of conditional probabilities by propagation methods less straightforward. Here we illustrate different methodologies for assessing sensitivity to assumptions about founders in forensic genetics problems. These include constrained steepest descent, linear fractional programming and representing dependence by structure. We illustrate these methods on several forensic genetics examples involving criminal identification, simple and complex disputed paternity and DNA mixtures.

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arXiv2005

Bayesian alignment using hierarchical models, with applications in protein bioinformatics

An important problem in shape analysis is to match configurations of points in space filtering out some geometrical transformation. In this paper we introduce hierarchical models for such tasks, in which the points in the configurations are either unlabelled, or have at most a partial labelling constraining the matching, and in which some points may only appear in one of the configurations. We derive procedures for simultaneous inference about the matching and the transformation, using a Bayesian approach. Our model is based on a Poisson process for hidden true point locations; this leads to considerable mathematical simplification and efficiency of implementation. We find a novel use for classic distributions from directional statistics in a conditionally conjugate specification for the case where the geometrical transformation includes an unknown rotation. Throughout, we focus on the case of affine or rigid motion transformations. Under a broad parametric family of loss functions, an optimal Bayesian point estimate of the matching matrix can be constructed, that depends only on a single parameter of the family. Our methods are illustrated by two applications from bioinformatics. The first problem is of matching protein gels in 2 dimensions, and the second consists of aligning active sites of proteins in 3 dimensions. In the latter case, we also use information related to the grouping of the amino acids. We discuss some open problems and suggest directions for future work.

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