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Bayesian and computational statistics
Bayesian inference differs from classical procedures in that unknown parameters are treated as random variables. This requires the specification of a prior distribution for the parameters (this distribution represents our belief about the values of the parameters prior to seeing the data), which is then updated by the information in the data. Bayesian inference has advantages over classical inference in terms of more naturally representing uncertainty, dealing with nuisance parameters, performing prediction and analysing complex stochastic models.
Recent development of very powerful computational algorithms (collectively known as Markov chain Monte Carlo -- MCMC), has led to a great upsurge in popularity of the Bayesian paradigm. These computational methods make it possible to analyse data under complex and highly structured models that are suitable for real-life scenarios.
About the Bayesian and computational statistics group
At Lancaster, we have an internationally leading group that works in all aspects of Bayesian inference and the associated computational techniques. Group members are Paul Fearnhead, Chris Sherlock, Honsheng Dai, and Gareth Ridall, together with a variety of research students and research associates.
We organise an informal computational statistics seminar series where group members discuss and develop current research ideas.
Overview of current research activities
The group's research is on a variety of theoretical, methodological and applied topics. Much of the research is motivated through interdisciplinary collaborations, and particularly through genetics, engineering, financial, wildlife, biological and environmental applications. Below we give you a flavour of some of our research areas.
- Modelling of complex systems
- Making accurate inference in complex real-world settings requires the development of suitable stochastic models that capture the main features of the systems being studied. Stochastic processes are suitable for modelling systems that evolve over time, such as stock market prices, the development of an epidemic, the dynamics of wildlife populations, and natural phenomena such as river or sea levels, or temperature values. In other instances, our work in this area requires spatial models.
- MCMC theory and methods
- MCMC algorithms have convergence properties which vary dramatically in different models, and even for different datasets from the same model. Therefore to have reliable algorithms, it is important to have some understanding of the factors that affect convergence. This involves probabilistic analyses of the Markov chains produced by the algorithms, and this in turn poses challenging new problems in Markov chain theory. We work actively on the interface between the theory of Markov chains and MCMC. In many problems, existing "standard" algorithms are too slow, and new MCMC methodology needs to be developed to explore posterior distributions. We also work actively in developing new MCMC methodology; and in the theoretical analysis of MCMC algorithms, which is important for producing guidelines on how to implement MCMC.
- Perfect and direct simulation
- MCMC algorithms are based on theory which says that they will produce draws from the posterior distribution of interest if run forever! There has been recent interest in adapting MCMC algorithms so that they can produce a draw from the posterior distribution in a finite amount of time; a process called perfect simulation. We have developed perfect simulation methods which can be applied to diffusions, models in population genetics and mixture models amongst others. A related idea is to use filtering algorithms to draw directly from the posterior distribution, and we have developed such algorithms for changepoint and mixture models, and continuous time hidden Markov processes.
- Particle filtering
- For many applications it is necessary to perform inference "online" as the data is collected. For example in tracking problems, it is necessary to have current estimates of the target which are updated as soon as a new measurement arrives. We work on sequential Monte Carlo methods, known as particle filters, for such problems. Research focuses on developing new particle filter methods, analysing particle filters to obtain guidelines for implementing them, and applying particle filters to real-life problems.
Some ongoing applied projects
- Motor unit number estimation.
- Inference of fine-scale recombination rates.
- Analysis of Human Isochore structure.
Recent group publications
Ridall, P.G., Pettitt, A.N., Friel, N., McCombe, P.A. and Henderson, R.D. (2007) Motor unit number estimation using reversible jump MCMC methods (with discussion). Applied Statistics, 56, 235-269.
Beskos, A. Papaspiliopoulos, O. Roberts, G.O. and Fearnhead P. (2006) Exact and computationally efficient likelihood-based estimation for discretely observed diffusion processes (with discussion). J. Roy. Stat. Soc., B, 68, 333-382.
