Abstract

Results on the behaviour of the rightmost particle in the nth generation in the branching random walk are reviewed and the phenomenon of anomalous spreading speeds, noticed recently in related deterministic models, is considered. The relationship between such results and certain coupled reaction-diffusion equations is indicated.

AMS subject classification (MSC2010) 60J80

Introduction

I arrived at the University of Oxford in the autumn of 1973 for postgraduate study. My intention at that point was to work in Statistics. The first year of study was a mixture of taught courses and designated reading on three areas (Statistics, Probability, and Functional Analysis, in my case) in the ratio 2:1:1 and a dissertation on the main area. As part of the Probability component, I attended a graduate course that was an exposition, by its author, of the material in Hammersley (1974), which had grown out of his contribution to the discussion of John's invited paper on subadditive ergodic theory (Kingman, 1973). A key point of Hammersley's contribution was that the postulates used did not cover the time to the first birth in the nth generation in a Bellman–Harris process. Hammersley (1974) showed, among other things, that these quantities did indeed exhibit the anticipated limit behaviour in probability. I decided not to be examined on this course, which was I believe a wise decisin but I was intrigued by the material. That interest turned out to be critical a few months later.

Abstract

We consider perfect simulation algorithms for locally stable point processes based on dominated coupling from the past, and apply these methods in two different contexts. A new version of the algorithm is developed which is feasible for processes which are neither purely attractive nor purely repulsive. Such processes include multiscale area-interaction processes, which are capable of modelling point patterns whose clustering structure varies across scales. The other topic considered is nonparametric regression using wavelets, where we use a suitable area-interaction process on the discrete space of indices of wavelet coefficients to model the notion that if one wavelet coefficient is non-zero then it is more likely that neighbouring coefficients will be also. A method based on perfect simulation within this model shows promising results compared to the standard methods which threshold coefficients independently.

Keywords coupling from the past (CFTP), dominated CFTP, exact simulation, local stability, Markov chain Monte Carlo, perfect simulation, Papangelou conditional intensity, spatial birth-and-death process

AMS subject classification (MSC2010) 62M30, 60G55, 60K35

Introduction

Markov chain Monte Carlo (MCMC) is now one of the standard approaches of computational Bayesian inference. A standard issue when using MCMC is the need to ensure that the Markov chain we are using for simulation has reached equilibrium. For certain classes of problem, this problem was solved by the introduction of coupling from the past (CFTP) (Propp and Wilson, 1996, 1998).

Abstract

The paper considers the situation of transmission over a memoryless noisy channel with feedback, which can be given a number of interpretations. The criteria for achieving the maximal rate of information transfer are well known, but examples of a simple and meaningful coding meeting these are few. Such a one is found for the Gaussian channel.

Keywords feedback channel, Gaussian channel

AMS subject classification (MSC2010) 94A24

Interrogation, transmission and coding

In this section we set out material which is classic in high degree, harking back to Shannon's seminal paper (1948), and presented in some form in texts such as those of Blahut (1987), Cover and Thomas (1991) and MacKay (2003). However, some exposition is necessary if we are to distinguish clearly between three versions of the model.

Suppose that an experimenter wishes to determine the value of a random variable U of which he knows only the probability distribution P(U). The formal argument is conveyed well enough for the moment if we suppose all distributions discrete and use the notation P(·) generically for such distributions. We may also abuse this convention by occasionally using P(U) to denote the function of U defined by the distribution.

The outcome of the experiment, if errorless, might be written x(U), where the form of the function x(U) reflects the design of the experiment. The experimenter will choose this, subject to practical constraints, so as to make the experiment as informative as possible.

Abstract

Computer-based tests with randomly generated questions allow a large number of different tests to be generated. Given a fixed number of alternatives for each question, the number of tests that need to be generated before all possible questions have appeared is surprisingly low.

AMS subject classification (MSC2010) 60G70, 60K99

Introduction

The use of computer-based tests in which questions are randomly generated in some way provides a means whereby a large number of different tests can be generated; many universities currently use such tests as part of the student assessment process. In this paper we present findings that illustrate that, although the number of different possible tests is high and grows very rapidly as the number of alternatives for each question increases, the average number of tests that need to be generated before all possible questions have appeared at least once is surprisingly low. We presented preliminary findings along these lines in Cornish et al. (2006).

A computer-based test consists of q questions, each (independently) selected at random from a separate bank of a alternatives. Let Nq be the number of tests one needs to generate in order to see all the aq questions in the q question banks at least once. We are interested in how, for fixed a, the random variable Nq grows with the number of questions q in the test.

Abstract

We prove a long-standing conjecture which characterizes the Ewens—Pitman two-parameter family of exchangeable random partitions, plus a short list of limit and exceptional cases, by the following property: for each n = 2, 3, …, if one of n individuals is chosen uniformly at random, independently of the random partition πn of these individuals into various types, and all individuals of the same type as the chosen individual are deleted, then for each r > 0, given that r individuals remain, these individuals are partitioned according to for some sequence of random partitions which does not depend on n. An analogous result characterizes the associated Poisson—Dirichlet family of random discrete distributions by an independence property related to random deletion of a frequency chosen by a size-biased pick. We also survey the regenerative properties of members of the two-parameter family, and settle a question regarding the explicit arrangement of intervals with lengths given by the terms of the Poisson–Dirichlet random sequence into the interval partition induced by the range of a homogeneous neutral-to-the right process.

AMS subject classification (MSC2010) 60C05, 60G09, 05A18

Introduction

Kingman introduced the concept of a partition structure, that is a family of probability distributions for random partitions πn of a positive integer n, with a sampling consistency property as n varies.

Abstract

The volume of a Wiener sausage constructed from a diffusion process with periodic, mean-zero, divergence-free velocity field, in dimension 3 or more, is shown to have a non-random and positive asymptotic rate of growth. This is used to establish the existence of a homogenized limit for such a diffusion when subject to Dirichlet conditions on the boundaries of a sparse and independent array of obstacles. There is a constant effective long-time loss rate at the obstacles. The dependence of this rate on the form and intensity of the obstacles and on the velocity field is investigated. A Monte Carlo algorithm for the computation of the volume growth rate of the sausage is introduced and some numerical results are presented for the Taylor–Green velocity field.

AMS subject classification (MSC2010) 60G60, 60G65, 35B27, 65C05

Introduction

We consider the problem of the existence and characterization of a homogenized limit for advection-diffusion in a perforated domain. This problem was initially motivated for us as a model for the transport of water vapour in the atmosphere, subject to molecular diffusion and turbulent advection, where the vapour is also lost by condensation on suspended ice crystals. It is of interest to determine the long-time rate of loss and in particular whether this is strongly affected by the advection. In this article we address a simple version of this set-up, where the advection is periodic in space and constant in time and where the ice crystals remain fixed in space.

Abstract

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.

Keywords Bayesian nonparametrics, gene expression profiles, hierarchical models, loss functions, MCMC samplers, optimal clustering, partition models, Pólya urn, stick breaking

AMS subject classification (MSC2010) 60G09, 62F15, 62G99, 62H30, 62M99

Introduction

The purpose of this note is four-fold: to remind some Bayesian nonparametricians gently that closer study of some probabilistic literature might be rewarded, to encourage probabilists to think that there are statistical modelling problems worth of their attention, to point out to all another important connection between the work of John Kingman and modern statistical methodology (the role of the coalescent in population genetics approaches to statistical genomics being the most important example; see papers by Donnelly, Ewens and Griffiths in this volume), and finally to introduce a modest generalisation of the Dirichlet process.

John Frank Charles Kingman was born on 28th August 1939, a few days before the outbreak of World War II. This Festschrift is in honour of his seventieth birthday.

John Kingman was born in Beckenham, Kent, the son of the scientist Dr F. E. T. Kingman FRSC and the grandson of a coalminer. He was brought up in north London, where he attended Christ's College, Finchley. He was an undergraduate at Cambridge, where at age 19 at the end of his second year he took a First in Part II of the Mathematical Tripos, following it with a Distinction in the graduate-level Part III a year later, for his degree. He began postgraduate work as a research student under Peter Whittle, but transferred to David Kendall in Oxford when Peter left for Manchester in 1961, returning to Cambridge when Kendall became the first Professor of Mathematical Statistics there in 1962.

John's early work was on queueing theory, a subject he had worked on with Whittle, but was also an interest of Kendall's. His lifelong interest in mathematical genetics also dates back to this time (1961). His next major interest was in Markov chains, and in a related matter—what happens to Feller's theory of recurrent events in continuous time. His first work here dates from 1962, and led to his landmark 1964 paper on regenerative phenomena, where we meet (Kingman) p-functions.

Abstract

The dynamics of tumour evolution are not well understood. In this paper we provide a statistical framework for evaluating the molecular variation observed in different parts of a colorectal tumour. A multi-sample version of the Ewens Sampling Formula forms the basis for our modelling of the data, and we provide a simulation procedure for use in obtaining reference distributions for the statistics of interest. We also describe the large-sample asymptotics of the joint distributions of the variation observed in different parts of the tumour. While actual data should be evaluated with reference to the simulation procedure, the asymptotics serve to provide theoretical guidelines, for instance with reference to the choice of possible statistics.

AMS subject classification (MSC2010) 92D20; 92D15, 92C50, 60C05, 62E17

Introduction

Cancers are thought to develop as clonal expansions from a single transformed, ancestral cell. Large-scale sequencing studies have shown that cancer genomes contain somatic mutations occurring in many genes; cf. Greenman et al., Sjöblom et al., Shah et al. Many of these mutations are thought to be passenger mutations (those that are not driving the behaviour of the tumour), and some are pathogenic driver mutations that influence the growth of the tumour. The dynamics of tumour evolution are not well understood, in part because serial observation of tumour growth in humans is not possible.

In an attempt to better understand tumour growth and structure, a number of evolutionary approaches have been described.