Seminars (SCB)

There are models for which the evaluation of the likelihood is infeasible in practice. For these models the Metropolis–Hastings acceptance probability cannot be easily computed. This is the case, for instance, when only departure times from a G/G/1 queue are observed and inference on the arrival and service distributions are required. Indirect inference is a method to estimate a parameter theta in models whose likelihood function does not have an analytical closed form, but from which random samples can be drawn for fixed values of theta. First an auxiliary model is chosen whose parameter beta can be directly estimated. Next, the parameters in the auxiliary model are estimated for the original data, leading to an estimate betahat. The parameter beta is also estimated by using several sampled data sets, simulated from the original model for different values of the original parameter theta. Finally, the parameter theta which leads to the best match to betahat is chosen as the indirect inference estimate. We analyse which properties an auxiliary model should have to give satisfactory indirect inference. We look at the situation where the data are summarized in a vector statistic T, and the auxiliary model is chosen so that inference on â is drawn from T only. Under appropriate assumptions the asymptotic covariance matrix of the indirect estimators is proportional to the asymptotic covariance matrix of T and componentwise inversely proportional to the square of the derivative, with respect to theta, of the expected value of T. We discuss how these results can be used in selecting good estimating functions. We apply our findings to the queuing problem.

Related Links

• http://www.blackwell-synergy.com/links/doi/10.1111/j.1369-7412.2003.05341.x/abs/

We introduce a methodology to sample from a sequence of probability distributions and estimate their unknown normalizing constants. This problem is traditionally addressed using Sequential Monte Carlo (SMC) methods which rely on importance sampling/resampling ideas. We design here an alternative iterative algorithm. This algorithm is based on a sequence of interacting Markov chain Monte Carlo (MCMC) algorithms. We establish the convergence of this non-Markovian scheme and demonstrate this methodology on various examples arising in Bayesian inference.

(This is a joint work with Anthony E. Brockwell, Department of Statistics, Carnegie Mellon)

MCMC provides a powerful set of tools for inference in Bayesian models. However, for many applications to large scale problems, MCMC methods can be relatively inefficient compared to new deterministic approximations developed in the machine learning community. I will describe several modern and generally applicable deterministic algorithms for approximate inference, and mention their advantages and disadvantages compared to MCMC. To focus the talk, I will describe all algorithms in the context of inference in Dirichlet process mixtures (DPMs), a classical non-parametric Bayesian statistical model used to define mixture models with countably infinitely many components. In particular, I will cover the following algorithms for inference in DPMs: (1) the traditional Gibbs sampling MCMC algorithm, (2) Variational Bayesian (VB) approximations, (3) the Expectation Propagation (EP) algorithm, and (4) a new approximate inference method for DPMs based on Bayesian hierarchical clustering (BHC). All these algorithms provide different speed / accuracy tradeoffs for large-scale problems, and the underlying concepts can be applied to virtually any statistical model. My conclusion is the following: MCMC is but one of many ways of approaching intractable inference problems, and modern statistics is likely to benefit by broadening the toolbox to include novel inference methods arising from other communities.

Joint work with: Matthew Beal (U Buffalo), Katherine Heller (UCL), and Tom Minka (Microsoft).

Related Links

• http://learning.eng.cam.ac.uk/zoubin/ - Zoubin Ghahramani's website

Sequential Monte Carlo methods for static problems combine the advantages of importance sampling and Markov chain based methods. We demonstrate how to use these exciting new techniques to fit generalised linear mixed models. A normal approximation to the likelihood is used to generate an initial sample, then transition kernels, reweighting and resampling result in evolution to a sample from the full posterior distribution. Since the technique does not rely on any ergodicity properties of the transition kernels, we can modify these kernels adaptively, resulting in a more efficient sampler.

Fuelled by the proliferation of large and complex data sets, statistical modelling has become increasingly ambitious. In fact, the use of models parameterised by an infinite number of parameters or latent variables is increasingly common. This is particularly appealing when models are most naturally formulated within an infinite dimensional setting, for instance continuous time-series.

It is well-understood that to obtain widely applicable statistical methodology for flexible families of complex models, requires powerful computational techniques (such as MCMC, SMC, etc). Unfortunately, infinite dimensional simulation is not usually feasible, so that the use of these computational methods for infinite-dimensional models is often only possible by adopting some kind of finite dimensional approximation to the chosen model. This is unappealing since the impact of the approximation is often difficult to assess, and the procedure often involves essentially using an approximate finite-dimensional model.

The talk will discuss a general technique for simulation which attempts to work directly with infinite dimensional random variables without truncation nor approximation. The talk is illustrated by concrete examples from the simulation of diffusion processes and Bayesian inference for Dirichlet mixture models. One surprising feature of the methodology is that exact infinite-dimensional algorithms are commonly far more efficient than approximate models

The Cross-Entropy method is a new Monte Carlo paradigm pioneered by Rubinstein (1999) in Operation Research. Its primary applications are (i) the calculation of probability of rare events, and (ii) the optimisation of irregular, multi-modal functions. While these two objectives seem to have a little in common, the CE approach manages to express them in a similar framework. In this talk, we will explain how Statistics can benefit from the CE method, and how the CE method can also benefit in turn from Statistical methodology. We will discuss the following particular applications in Statistics: Monte-Carlo p-values, simulation of truncated distributions, variable selection, and mixture estimation. We will see that in each case CE provides significant improvements over current methods. Interestingly, we will see also vanilla CE rarely works directly, but tandard tools from Statistical Inference allow for developing more efficient algorithms. In particular, we will discuss a CE-EM algorithm for mixture estimation, which outperform any straight CE or EM algorithm, in terms, of finding higher modes of the likelihood function.

Sequential Monte Carlo methods provide reliable approximations of the conditional ditribution of a certain signal process given the data obtined from an associated observation process. The generic SMC method involves sampling from the prior distribution of the signal and then using a weighted bootstrap technique (or equivalent) with weights defined by the likelihood of the most recent observation data. If the number of updating stages becomes large, the repeated application of the the weighted bootstrap may lead to what the literature describes as "impoverished sample" or "sample attrition". That means that the sample being carried forward will have fewer and fewer distict values. In this talk, I propose a method that attempts to solve this problem for the continuous time filtering problem. The method replaces the sampling from the prior step with sampling from a distribution that depends on the entire (existing) sample and the most recent observation data and it does not contain the weighted bootstrap step. The method is motivated by recent advances in the area of McKean-Vlasov representations for solutions of stochastic PDEs and their application to solving the filtering problem in a continuous time framework.

In this talk I will give an overview of the problem of conducting Bayesian inference for the fixed parameters of nonlinear multivariate diffusion processes observed partially, discretely, and possibly with error. I will present a sequential strategy based on either SIR or MCMC-based filtering for approximate diffusion bridges, and a "global" MCMC algorithm that does not degenerate as the degree of data augmentation increases. The relationship of these techniques to methods of approximate Bayesian computation will be highlighted.

This talk is an exploration of the possible role for branching processes and related models in Monte Carlo simulation from a complex distribution, such as a Bayesian posterior. The motivation is that branching processes can support antithetic behaviour in a natural way by making offspring negatively correlated, and also that branching paths may assist in navigating past slowly-mixing parts of the state space. The basic theory of branching processes as used for sampling is established, including the appropriate analogue of global balance with respect to the target distribution, evaluation of moments, in particular asymptotic variances, and a start on the spectral theory. Although our model is a kind of 'population Monte Carlo', it should be noted that it has virtually nothing to do with particle filters, etc. Our target is not sequentially evolving, and we rely on ergodicity for convergence of functionals of the target distribution, rather than using importance sampling.

This is joint work with Antonietta Mira (University of Insubria, Varese, Italy).

Quasi-Monte Carlo (QMC) methods are numerical techniques for estimating large-dimensional integrals, usually over the unit hypercube. They can be applied, at least in principle, to any stochastic simulation whose aim is to estimate a mathematical expectation. This covers a wide range of applications. Practical error bounds are hard to obtain with QMC but randomized quasi-Monte Carlo (RQMC) permits one to compute an unbiased estimator of the integral, together with a confidence interval. RQMC can in fact be viewed as a variance-reduction technique.

In this talk, we review some key ideas of RQMC methods and provide concrete examples of their application to simulate systems modeled as Markov chains. We also present a new RQMC method, called array-RQMC, recently introduced to simulate Markov chains over a large number of steps. Our numerical illustrations indicate that RQMC can dramatically reduce the variance compared with standard Monte Carlo.

The analysis of mark-recapture data is undergoing a period of development and expansion.Here we contribute to that by presenting a model which include both births and immigration,as well as the usual deaths.Data come from a long-term study of the willow tit (Parus montanus ), where we can assume that all births are recorded,and hence immigrants can also be identified. We model the rates of immigration,birth rare per parent,and death rates of juveniles and adults. Using a hierarchical model allows us to incorporate annual variation in these parameters.The model is fitted to the data using MCMC,as a Bayesian analysis. In addition to the model fitting,we also check several aspects of the model fit,in particular whether survival varies with age or immigrant status,and whether capture probability is affected by previous capture history.The latter check is important,as independence of capture histories is a key assumption that simplifies the model considerably.Here we find that the capture probability depends strongly on whether the individual was captured in the previous year. Our work moves MRR modelling closer to a description of the dynamics of the whole population,with the obvious potential for prediction,and use in making decisions about population management.

Increasing pressures on the environment are generating an ever-increasing need to manage animal and plant populations sustainably, and to protect and rebuild endangered populations. Effective management requires reliable mathematical models, so that the consequences of management action can be predicted, and the uncertainty in these predictions quantified. These models must be able to predict the response of populations to anthropogenic change, while handling the major sources of uncertainty. We describe a simple ‘building block’ approach to formulating discrete-time models. These models may include demographic stochasticity, environmental variability through covariates or random effects, multi-species dynamics such as in predator-prey and competition models, movement such as in metapopulation models, non-linear effects such as density dependence, and mating models. We discuss methods for fitting such models to time series of data, and quantifying uncertainty in parameter estimates and population states, including model uncertainty, using computer-intensive Bayesian methods.

We examine the hypothesis that wandering albatrosses (Diomedea exulans) undergo Levy flights when roaming the skies in search of oceanic food sources. Levy flights are random walks whose step lengths are taken from a distribution with infinte variance, such as a power-law. The Levy flight consequently has no typical scale, and this has been interpreted as being an efficient way of searching for food on the ocean surface. We first re-analyse the original data that were used to infer Levy flights. These data come from wet/dry loggers that record the time periods for which the birds were airborne or on the ocean surface. We cast doubt as to whether these data are sufficient to conclude Levy flight behaviour. This prompts us to analyse recent data from birds fitted with much higher resolution wet/dry loggers. We find that the widely-held Levy flight hyopthesis can be refuted by the newer data. We will also briefly discuss other data sets and ecological questions arising from the unique Antarctic environment.

Related Links

• http://www.chebucto.ns.ca/~english - Lead author's homepage
• http://www.bas.ac.uk - British Antarctic Survey homepage

Not only have marked point processes received relatively little attention in the literature, but most analyses ignore the fact that in real life spatial structure often develops dynamically through time. We therefore develop a computationally fast and robust spatial-temporal process, based on stochastic immigration-death and deterministic growth-interaction. For this enables both single and multiple snap-shot marked point process data to be studied in considerable depth. Combining logistic and linear growth with (symmetric) disc-interaction and (asymmetric) area-interaction generates a wide variety of mark-point spatial structures. A maximum psuedo-likelihood approach is developed for parameter estimation at fixed times, and a least squares procedure for parameter estimation based on multiple time points.

A related problem in spatial statistics and stochastic geometry concerns the modelling and statistical analysis of hard particle systems involving discs or spheres. For successively filling remaining empty structure leads to a limiting maximum packing pattern whose structure depends on the given characteristics of the particles. Using our process to develop such patterns extends current methods, since a newly arrived particle is not immediately rejected if it does not fit into a specific gap, but can change size to adapt to the interaction pressure placed on it.

In general, inference problems for disease outbreak data are complicated by the facts that (i) the data are inherently dependent and (ii) the data are usually incomplete in the sense that the actual process of infection is not observed. We adopt a Bayesian approach and apply Markov Chain Monte Carlo (MCMC) methods in order to make inference for the parameters of interest (such as infection and removal rates). We show that once the size of the data set ncreases, the standard methods perform poorly. Therefore, apart from centered reparameterisation we extend the Non-Centered and partially Non-Centered algorithms presented in Neal and Roberts (2005). Finally, we adopt a fully Bayesian approach to analyze the Foot-and-Mouth disease occurred in 2001 in the UK and also discus modelling approaches for a potential Avian Influenza outbreak in the poultry industry of the UK.

Inference and parameter estimation for stochastic epidemic models has been greatly facilitated by Bayesian methods and associated computational techniques such as Markov chain Monte Carlo. The question of how experiments should be designed ­ e.g. how populations should be sampled in space and time - to maximise the insights gained from these analyses is now being considered. This talk will describe how the Bayesian approach to experimental design, originally due to Muller, can be applied in the context of nonlinear stochastic epidemic models. In this approach, the design itself is treated as a random quantity. A distribution, which depends fundamentally on the utility of the design, is assigned to model parameters, experimental outcome and experimental design jointly. The design which is optimal, in terms of having the highest expected utility, corresponds to the mode of the design marginal distribution. We will demonstrate how, by using approximations to parameter likelihoods based on moment closure methods, it is computationally feasible to implement this approach to design experiments in practically relevant situations. In particular, we use the methods to explore possible designs for microcosm experiments on epidemics of fungal pathogens in plant communities.

A neutral model of community dynamics has been built using the hierarchical Bayesian framework and fitted with Markov Chain Monte Carlo methods to three community datasets. To fit the data well, the model would need parameter values that are impossible. This suggests that variation in species abundances cannot be explained solely by random drift between species as suggested by the neutral model.

We describe a discrete event simulation model of tuberculosis (TB) and HIV disease, parameterized to describe the dual epidemics in Harare, Zimbabwe. TB and HIV are the leading causes of death from infectious disease among adults worldwide and the number of TB cases has risen significantly since the start of the HIV epidemic, particu-larly in Sub-Saharan Africa, where the HIV epidemic is most severe. There is a need to devise new strategies for TB control in countries with a high prevalence of HIV. This model has been designed to investigate strategies for reducing TB transmission by more efficient TB case detection. The model structure and its validation are discussed.

We treat a panmictic host population interacting with a virus. The virus is transmitted both horizontally and vertically. We modify the Moran model to describe the stochastic dynamics of individual host and viral lineages. For a sample of individuals from the population, the model gives rise to a branching and coalescing graph that contains the combined host and viral genealogies as a subgraph. The associated diffusion process, obtained in the limit of large host population, is related to the Neuhauser-Krone selection graph process.

We consider two study populations: cougars infected with FIV and a UK cohort of HIV patients. We fit the joint host-virus process to viral sequence data and known host pedigrees (which are trivial in the human case). We use MCMC to average over the variable dimension parameter space of labeled graphs.

We consider the problem of Bayesian inference for infection rates in a multi-type stochastic epidemic model in which the population has a given structure, given data on final outcome. For such data, a likelihood is both analytically and numerically intractable. This problem can be overcome by imputation of suitable latent variables. We describe two such approaches based on different representations of the epidemic model. We also consider extentions to the methodology for the situation where the observed data are a fraction of the entire population. The methods are illustrated with data on influenza outbreaks.

This talk is concerned with a stochastic model for the spread of an SIR (susceptible $\to$ infective $\to$ removed) epidemic among a closed, finite population that contains several types of individual and is partitioned into households. A pseudolikelihood framework is presented for making statistical inference about the parameters governing such epidemics from final outcome data, when possibly only some of the households in the population are observed. The framework includes parameter estimation, hypothesis tests and goodness-of-fit. Asymptotic properties of the procedures are derived when the number of households in both the sample and the population are large, which correctly account for dependencies between households. The methodology is illustrated by applications to data on a \emph{variola minor} outbreak in Sao Paulo and to data on influenza outbreaks in Tecumseh, Michigan.

We consider methodology for Bayesian model-based clustering of gene expression profiles, that is, measurements of expression levels of a large number of genes, typically from microarray assays, across a number of different experimental conditions and/or biological subjects. We follow a familiar approach using Dirichlet-process-based models to cluster the genes implicitly, but depart from standard practice in several ways. First, we incorporate regression on covariate information at the condition/subject level by modelling regression coefficients, not the expectations of the data directly. More importantly, we replace the Dirichlet process by one of a richer family of models, generated from a stick-colouring-and-breaking construction, under which cluster identities are not exchangeable: this allows modelling a 'background' cluster, for example. Finally, we follow a formal decision-theoretic approach to point estimation of the clustering, using a pairwise coincidence loss function. This is joint work with John Lau at Bristol.

Understanding the relationship between protein structure and biological function has long been a major goal of structural biology. With the advent of many structural genomics projects, there is a practical need for tools to analyse and characterise the possible functional attributes for a new structure.

One of the major challenges in assigning function is to recognise a cognate ligand which may be a small molecule or a large macromolecule. At EBI we have been developing a range of methods which seek to annotate a functional site. These methods include:

• Using sequence data and global and local structure comparisons to recognise distant relatives or short sequence patterns that are characteristic of binding sites. • Using 3-dimensional templates for functional sites defined from proteins of known structure and function which can identify similarities between the query protein and other proteins in the PDB. • Using spherical harmonics to define the shape of a binding site and to compare this shape with all known binding sites in the PDB and with the small molecule metabolome.

These methods have some success dependent upon the shape and flexibility of the binding site. In this presentation I will review our progress in this area and describe application to the sulpher transferases. Some of this work has been integrated into the ProFunc pipeline (coordinated by R.A. Laskowski) which is a web server which can provide automated annotation for a new protein structure. (http://www.ebi.ac.uk/thornton-srv/databases/ProFunc/).

References:

Laskowski, R.A., Watson, J.D. & Thornton, J.M. (2005). ProFunc: a server for predicting protein function from 3D structure. Nucleic Acids Res., 33, W89-W93. Laskowski, R.A., Watson, J.D. & Thornton, J.M. (2005) Protein function prediction using local 3D templates. J. Mol Biol., 351, 614-626. Morris, R.J., Kahraman, A. & Thornton, J.M. (2005) Binding pocket shape analysis for protein function prediction. Acta. Cryst. D61, C156-C157.

Microarray experiments and gene expression data have a number of characteristics that make them attractive but challenging for Bayesian analysis. There are many sources of variability, the variability is structured at different levels array specific, gene specific, ....) and the ratio of signal to noise is low. Typical experiments involve few samples but a large number of genes, so that borrowing information, e.g. across genes, to improve inference becomes essential. Hence embedding the inference in a hierarchical model formulation is natural.

Bayesian models adapted to the level of information processed have been developed to address some of the questions raised that range from modelling the signal to synthesising gene lists across different experiments. In this talk, I shall illustrate their use in variety of contexts: probe level models attempting to quantify uncertainty of the signal, differential expression mixture models and gene list synthesis. Cutting across these developments are important issues of MCMC performance and model checking. These issues will be illustrated on case studies.

This is joint work with colleagues on the BGX project : Marta Blangiardo, Natalia Bochkina, Anne Mette Hein (now at Aarhus), Alex Lewin, Ernest Turro and Peter Green (Bristol).

Related Links

• http://www.bgx.org.uk - Related papers and technical reports

Admixed populations, such as African Americans, are an important resource for identifying mutations contributing to human disease. Until recently, disease mapping in such populations has used so called “admixture linkage disequilibrium" to map loci for diseases whose incidence varies markedly between populations. These analysis methods typically require unlinked markers spaced at wide intervals, while data now becoming available provide genotype information at much higher densities. High density data provides exquisite information about admixture, and association information, but presents methodological challenges. We have developed and implemented an approach to utilize such data to probabilistically infer admixture segments, and to perform disease mapping. The approach uses the HapMap data as a framework, and employs a fully Bayesian methodology, providing a natural weighting of both broad-scale admixture linkage disequilibrium and fine-scale association information. The software is currently being applied to data from a prostate cancer association study examining African American cases and controls. Future population genetics based directions are briefly discussed.

Finding the ancestral recombination graph (ARG) that explains a data set with the least number of recombination is the parsimony-ARG analogue to finding parsimonious phylogenies. This is a hard computational problem and two main approaches will be discussed. Firstly, a "scan along sequences” dynamic programming approach that works up to 10 sequences of any length. Secondly, a "trace history back in time" branch and bound approach that can work very fast for much larger number of sequences, but can also fail totally dependent on data". The second approach can also be extended to include gene conversion. Finally, the number of ancestral states that could be encountered for a given data set it counted for small number of sequences and segregating sites. It is also illustrated how likelihood calculations can be done on a restricted graph that contains close to minimal histories of a set of sequences

Allen, B. and Steel, M., Subtree transfer operations and their induced metrics on evolutionary trees,Annals of Combinatorics 5, 1-13 (2001)

Baroni, M., Grunewald, S., Moulton, V., and Semple, C. Bounding the number of hybridisation events for a consistent evolutionary history. Journal of Mathematical Biology 51 (2005), 171-182

Bordewich, M. and Semple, C. On the computational complexity of the rooted subtree prune and regraft distance. Annals of Combintorics 8 (2004), 409-423

Hein,J.J., T.Jiang, L.Wang & K.Zhang (1996): "On the complexity of comparing evolutionary trees" Discrete Applied Mathematics 71.153-169.

Hein, J., Schierup, M. & Wiuf, C. (2004) Gene Genealogies, Variation and Evolution, Oxford University Press

Lyngsø, R.B., Song, Y.S. & Hein, J. (2005) Minimum Recombination Histories by Branch and Bound. Lecture Notes in Bioinformatics: Proceedings of WABI 2005 3692: 239–250.

Myers, S. R. and Griffiths, R. C. (2003). Bounds on the minimum number of recombination events in a sample history. Genetics 163, 375-394.

Song, Y.S., Lyngsø, R.B. & Hein, J. (2005) Counting Ancestral States in Population Genetics. Submitted.

Song, Y.S. & Hein, J. (2005) Constructing Minimal Ancestral Recombination Graphs. J. Comp. Biol., 12:147–169

Song, Y.S. & Hein, J. (2004) On the minimum number of recombination events in the evolutionary history of DNA sequences. J. Math. Biol., 48:160–186.

Song, Y.S. & Hein, J. (2003) Parsimonious reconstruction of sequence evolution and haplotype blocks: finding the minimum number of recombination events, Lecture Notes in Bioinformatics, Proceedings of WABI'03, 2812:287–302.

Suprisingly little is known about regulatory processes in prokaryotes outside a small group of model species such as Escherichia coli. Probabilistic models can help to combine the comparatively sparse direct experimental evidence for regulation in less well known organisms such as Mycobacterium tuberculosis with gene expression data and results from the application of bioinformatics and genomic tools. I will discuss the challenges of such a project and some of the statistical concepts that might be useful for tackling them.

The estimation of empirical codon models sheds new light on recently discussed questions about biological pressures and processes acting during codon sequence evolution (Averof et al., Science 287:1283 (2000), Bazykin et al., Nature 429:558 (2004), Friedman and Hughes, MBE 22:1285 (2005), Keightley et al., PLoS Biol 3:282 (2005)).

My results show that modelling the evolutionary process is improved by allowing for single, double and triple nucleotide changes; the affiliation between DNA triplets and the amino acid they encode is a main factor driving evolution; and the nonsynonymous-synonymous rate ratio is a suitable measure to classify substitution patterns observed for different proteins. However, comparing models estimated from genomic data and polymorphism data indicates that double and triple changes are not instantaneous.

This new view of how codon evolution proceeds leads to consequences for selection studies. I will discuss that under the new empirical codon model purifying selection is less purifying and that cases of positive selection are observed weaker than under the standard condon models (Yang et al.,Genetics 155: 431-449 (2000)).

When analyzing genetic data, one often wishes to determine if the samples are from a population that has structure. Can the samples be regarded as randomly chosen from a homogeneous population, or does the data imply that the population is not genetically homogeneous? We show that an old method (principal components) together with modern statistics (Tracy-Widom theory) can be combined to yield a fast and effective answer to this question. The technique is simple and practical on the largest datasets, and can be applied both to genetic markers that are biallelic or to markers that are highly polymorphic such as microsatellites. The theory also allows us to estimate the data size needed to detect structure if our samples are in fact from two populations that have a given, but small level of differentiation.

Testing one SNP at a time does not fully realise the potential of genome-wide association studies to identify multiple causal variants of small effect, which is a plausible scenario for many complex diseases. Moreover, many simulation studies assume a single causal variant and so more complex realities are ignored. Analysing large numbers of variants simultaneously is now becoming feasible, thanks to developments in Bayesian stochastic search methods. We combine Bayesian shrinkage methods together with a local stochastic model search to identify complex interactions, both local and distal. Our approach can analyse up to 10,000 SNPs simultaneously, and finds multiple potential disease models each with an associated probability. We illustrate its power in comparison with a range of alternative methods, in simulations that incorporate multiple causal loci, acting singly or in interacting pairs, among 4,000 SNPs in a 20Mb region. We argue that, implemented in a two-stage procedure, our hybrid Bayesian analysis can provide a powerful solution to the problem of extracting maximal information from genome-wide association studies.

Approximate Bayesian Computation (ABC) is a recent developed Bayesian technique that can be used to extract information from DNA data. This method has been firstly introduced to Population Genetics in 1997 by Pritchard.

Since 2002, with Beaumont’s paper on the subject, its usage has been strongly increased. This Bayesian approach is used to estimate several demographic history parameters, from populations, using DNA data. Its main advantages are the decrease on computation time demanding and the increase on efficiency and flexibility when dealing with multiparameter models.

In this project it has been studied a particular ABC method similar to the one used by Beaumont in 2006, against a commonly used Markov Chain Monte Carlo (MCMC) method (Hey and Nielsen, 2004) to infer about the accuracy of the first method. It was also explored the use of this method with more complex demographic models. These two approaches use DNA sequence data to extract demographic information (e.g. population sizes, time of splitting events, migration rates).

The study confirms the competitiveness of this method when compared to an MCMC approach as well as its potential role on researches with more complex, therefore more realistic, models.

Related Links

• http://www.rdg.ac.uk/~sar05sal - home page of the author