Walter R. Gilks

Adaptive Rejection Sampling for Gibbs Sampling

SUMMARY We propose a method for rejection sampling from any univariate log-concave probability density function. The method is adaptive: as sampling proceeds, the rejection envelope and the squeezing function converge to the density function. The rejection envelope and squeezing function are piecewise exponential functions, the rejection envelope touching the density at previously sampled points, and the squeezing function forming arcs between those points of contact. The technique is intended for situations where evaluation of the density is computationally expensive, in particular for applications of Gibbs sampling to Bayesian models with non-conjugacy. We apply the technique to a Gibbs sampling analysis of monoclonal antibody reactivity.

Highly Conserved Non-Coding Sequences Are Associated with Vertebrate Development

In addition to protein coding sequence, the human genome contains a significant amount of regulatory DNA, the identification of which is proving somewhat recalcitrant to both in silico and functional methods. An approach that has been used with some success is comparative sequence analysis, whereby equivalent genomic regions from different organisms are compared in order to identify both similarities and differences. In general, similarities in sequence between highly divergent organisms imply functional constraint. We have used a whole-genome comparison between humans and the pufferfish, Fugu rubripes, to identify nearly 1,400 highly conserved non-coding sequences. Given the evolutionary divergence between these species, it is likely that these sequences are found in, and furthermore are essential to, all vertebrates. Most, and possibly all, of these sequences are located in and around genes that act as developmental regulators. Some of these sequences are over 90% identical across more than 500 bases, being more highly conserved than coding sequence between these two species. Despite this, we cannot find any similar sequences in invertebrate genomes. In order to begin to functionally test this set of sequences, we have used a rapid in vivo assay system using zebrafish embryos that allows tissue-specific enhancer activity to be identified. Functional data is presented for highly conserved non-coding sequences associated with four unrelated developmental regulators (SOX21, PAX6, HLXB9, and SHH), in order to demonstrate the suitability of this screen to a wide range of genes and expression patterns. Of 25 sequence elements tested around these four genes, 23 show significant enhancer activity in one or more tissues. We have identified a set of non-coding sequences that are highly conserved throughout vertebrates. They are found in clusters across the human genome, principally around genes that are implicated in the regulation of development, including many transcription factors. These highly conserved non-coding sequences are likely to form part of the genomic circuitry that uniquely defines vertebrate development.

Following a moving target—Monte Carlo inference for dynamic Bayesian models

Markov chain Monte Carlo (MCMC) sampling is a numerically intensive simulation technique which has greatly improved the practicality of Bayesian inference and prediction. However, MCMC sampling is too slow to be of practical use in problems involving a large number of posterior (target) distributions, as in dynamic modelling and predictive model selection. Alternative simulation techniques for tracking moving target distributions, known as particle filters, which combine importance sampling, importance resampling and MCMC sampling, tend to suffer from a progressive degeneration as the target sequence evolves. We propose a new technique, based on these same simulation methodologies, which does not suffer from this progressive degeneration.

Adaptive Markov Chain Monte Carlo through Regeneration

Abstract Markov chain Monte Carlo (MCMC) is used for evaluating expectations of functions of interest under a target distribution π. This is done by calculating averages over the sample path of a Markov chain having π as its stationary distribution. For computational efficiency, the Markov chain should be rapidly mixing. This sometimes can be achieved only by careful design of the transition kernel of the chain, on the basis of a detailed preliminary exploratory analysis of π. An alternative approach might be to allow the transition kernel to adapt whenever new features of π are encountered during the MCMC run. However, if such adaptation occurs infinitely often, then the stationary distribution of the chain may be disturbed. We describe a framework, based on the concept of Markov chain regeneration, which allows adaptation to occur infinitely often but does not disturb the stationary distribution of the chain or the consistency of sample path averages.

Dynamic conditional independence models and Markov chain Monte Carlo methods

Abstract In dynamic statistical modeling situations, observations arise sequentially, causing the model to expand by progressive incorporation of new data items and new unknown parameters. For example, in clinical monitoring, patients and data arrive sequentially, and new patient-specific parameters are introduced with each new patient. Markov chain Monte Carlo (MCMC) might be used for continuous updating of the evolving posterior distribution, but would need to be restarted from scratch at each expansion stage. Thus MCMC methods are often too slow for real-time inference in dynamic contexts. By combining MCMC with importance resampling, we show how real-time sequential updating of posterior distributions can be effected. The proposed dynamic sampling algorithms use posterior samples from previous updating stages and exploit conditional independence between groups of parameters to allow samples of parameters no longer of interest to be discarded, such as when a patient dies or is discharged. We apply the ...

A novel algorithm and web-based tool for comparing two alternative phylogenetic trees

SUMMARY We describe an algorithm and software tool for comparing alternative phylogenetic trees. The main application of the software is to compare phylogenies obtained using different phylogenetic methods for some fixed set of species or obtained using different gene sequences from those species. The algorithm pairs up each branch in one phylogeny with a matching branch in the second phylogeny and finds the optimum 1-to-1 map between branches in the two trees in terms of a topological score. The software enables the user to explore the corresponding mapping between the phylogenies interactively, and clearly highlights those parts of the trees that differ, both in terms of topology and branch length. AVAILABILITY The software is implemented as a Java applet at http://www.mrc-bsu.cam.ac.uk/personal/thomas/phylo_comparison/comparison_page.html. It is also available on request from the authors.

Disease mapping with errors in covariates.

We describe Bayesian hierarchical-spatial models for disease mapping with imprecisely observed ecological covariates. We posit smoothing priors for both the disease submodel and the covariate submodel. We apply the models to an analysis of insulin Dependent Diabetes Mellitus incidence in Sardinia, with malaria prevalence as a covariate.

Statistical analysis of domains in interacting protein pairs

MOTIVATION Several methods have recently been developed to analyse large-scale sets of physical interactions between proteins in terms of physical contacts between the constituent domains, often with a view to predicting new pairwise interactions. Our aim is to combine genomic interaction data, in which domain-domain contacts are not explicitly reported, with the domain-level structure of individual proteins, in order to learn about the structure of interacting protein pairs. Our approach is driven by the need to assess the evidence for physical contacts between domains in a statistically rigorous way. RESULTS We develop a statistical approach that assigns p-values to pairs of domain superfamilies, measuring the strength of evidence within a set of protein interactions that domains from these superfamilies form contacts. A set of p-values is calculated for SCOP superfamily pairs, based on a pooled data set of interactions from yeast. These p-values can be used to predict which domains come into contact in an interacting protein pair. This predictive scheme is tested against protein complexes in the Protein Quaternary Structure (PQS) database, and is used to predict domain-domain contacts within 705 interacting protein pairs taken from our pooled data set.

Substantial benefits of tissue matching in renal transplantation

The purpose of this study was to perform a rigorous statistical analysis of the benefits of HLA-A,B, and DR matching in renal transplantation. Graft survival in 2282 first cadaver kidney transplants, recorded and followed up by the United Kingdom Transplant Service (UKTS), was analyzed using the piecewise proportional hazards regression method. The results show that substantial improvements in graft survival are obtained when there is DR compatibility and at most one A or B mismatch, but that there is little advantage in tissue matching unless this degree of matching can be attained. So far, few graft recipients have benefited substantially through tissue matching (24% of kidneys exchanged through UKTS in 1984). This is partly attributable to unresolved technical problems in DR typing. However simulations show that under ideal conditions, with a pool of 3000 patients awaiting transplantation, considerable improvements in graft survival can be obtained in over 60% of recipients.

RESAMPLE-MOVE Filtering with Cross-Model Jumps

In standard sequential imputation, repeated resampling stages progressively impoverish the set of particles, by decreasing the number of distinct values represented in that set. A possible remedy is Rao-Blackwellisation (Liu and Chen 1998). Another remedy, which we discuss in this chapter, is to adopt a hybrid particle filter, which combines importance sampling/resampling (Rubin 1988, Smith and Gelfand 1992) and Markov chain iterations. An example of this class of particle filters is the RESAMPLEMOVE algorithm described in (Gilks and Berzuini 1999), in which the swarm of particles is adapted to an evolving target distribution by periodical resampling steps and through occasional Markov chain moves that lead each individual particle from its current position to a new point of the parameter space. These moves increase particle diversity. Markov chain moves had previously been introduced in particle filters (for example, (Berzuini, Best, Gilks and Larizza 1997, Liu and Chen 1998)), but rarely with the possibility of moving particles at any stage of the evolution process along any direction of the parameter space; this is, indeed, an important and innovative feature of RESAMPLE—MOVE. This allows, in particular, to prevent particle depletion along directions of the parameter space corresponding to static parameters, for example when the model contains unknown hyper-parameters, a situation which is not addressed by the usual state filtering algorithms.