My research has been largely motivated by the desire to understand and explore the connexions between logical and philosophical principles of inference, mathematical structures, and data-analysis.  Much of my work is a critical examination of the logical Foundations of Statistics , both from a general standpoint and in respect of particular schools of thought--especially Bayesian, but also Likelihood, Frequentist, Structural and Fiducial.  Early work on the Marginalization Paradox in improper Bayesian inference continues to stimulate debate.  I have developed and applied Bayesian decision-theoretic ideas to clarify the theory of Optimal  Experimental Design.  Other Bayesian interests include Coherent Combination of Expert Opinion, Matrix Distribution Theory and Bayesian Multivariate Analysis, Model Uncertainty, Maximum Entropy and Robust Bayesian  Decision Theory. 

I introduced the general axiomatic approach to Conditional Independence, and showed how, as well as being a valuable tool for formulating and solving a variety of conceptual and technical problems, it provides a unifying thread drawing together many scattered concepts of statistical inference.  I later developed a novel abstract mathematical framework, the Separoid , to support the application of these ideas to a wide range of other "irrelevance" concepts, of independent interest.  This work has been widely applied and extended, especially by workers in in Artificial Intelligence.

Another continuing interest has been in abstract geometrical descriptions of statistical models.  Early work showed the relevance and importance of the exponential and mixture connexions, and of Fisher information, thus laying the foundations for Information Geometry.  More recently, extending these ideas, I introduced a more general Decision Geometry.

A major research effort over many years has been devoted to Probabilistic Expert Systems, a multi-faceted topic which applies graph-theoretic representations to analyse, structure and manipulate complex multivariate distributions.  The theory of Conditional Independence has played an important role in this.  Together with Steffen Lauritzen, David Spiegelhalter and Robert Cowell, I developed fundamental theory and algorithms for computer implementation, and application to real problems.  This work was encapsulated in a monograph published by Springer, and has been implemented in software systems such as Hugin.  Further research has focused on the development and use of exact and approximate Bayesian methods for learning, from data, about both the quantitative and the qualitative structure of the system; on the identification and exploitation of the abstract structure common to many related computational algorithms; and on the integration of exact algorithms with Monte Carlo methods. 

An important philosophical attitude running through my work has been a "positivist'' emphasis on observables, and on the falsifiability of inferences.  This has guided my approach to the interpretation, assessment and validation of probabilistic statements and models.  I introduced the Calibration Criterion of validity for probability forecasts, and developed this into a new philosophical theory of Objective Probability .  The same attitude provided the impetus for my introduction of Prequential Analysis , a novel general approach to problems of statistical inference and data analysis which takes seriously the problem of making and criticising statistical forecasts.  On the theoretical front, it allows for very broad extension of classical ideas such as estimation efficiency, or consistent model selection, and has many pleasing properties.  Practically, it provides a more generally applicable and interpretable alternative to cross-validation.  Prequential Analysis has been successfully applied to the monitoring and improvement of the predictive performance of Probabilistic Expert Systems.  It has close and interesting links with Bayesian Inference, with the theory of Stochastic Complexity, with Algorithmic Complexity Theory, with Computational Learning Theory (COLT) and with Rational Learning in Games.  In work with Volodya Vovk, we developed a new general mathematical theory of Prequential Probability, based on game-theoretic rather than measure-theoretic foundations. 

Another fundamental research theme, extending de Finetti's ideas of exchangeability, has been in Symmetry Modelling, which uses symmetry judgements and group representation theory to guide the construction, interpretation and analysis of statistical models. 

Causal Inference, a major current interest, has links with both Symmetry Modelling and Probabilistic Expert Systems.  My positivist world view being antipathetic to popular explications of cause based on counterfactual variables, I have been investigating the extent to which sensible and meaningful causal inferences can be justified without recourse to counterfactuals.  I have developed decision-theroetic concepts and tools, based on conditional independence and associated graphical representations, to support such causal modelling and inference, and have used them to shed light on such problems as confounding, incomplete compliance, and seqential decision-making. 

I have been interested in the application of Probability and Statistics to a variety of subject areas, in particular to Medicine (especially medical diagnosis and decision-making), Crystallography, Reliability (especially Software Reliability) and, most recently, Legal Reasoning.  I have acted as expert advisor or witness in a number of legal cases involving DNA profiling.  This has led me to a thorough theoretical examination of the use of Probability and Statistics for Forensic Identification .  I headed an international research team focusing on the analysis of complex forensic DNA identification cases using Probabilistic Expert Systems.

These legally inspired investigations highlighted the many logical subtleties and pitfalls that beset evidential reasoning more generally.  To address these I established a multidisciplinary research programme on Evidence, Inference and Enquiry at University College London. This brought together researchers from a wide diversity of disciplinary backgrounds to seek out common ground, to advance understandings, and to improve the handling of evidence.

I am currently Director of the Cambridge Statistics Initiative, which is charged with developing novel statistical methodology of value in real applications.

Marginalization paradoxes in Bayesian and structural inference (with  M. Stone and  J. V. Zidek).  With discussion.  J. Roy. Statist. Soc. B 35 (1973), 189-233. 

Likelihood and Bayesian inference from selectively reported data (with  J. M.  Dickey).  J. Amer. Statist. Assoc.  72 (1977), 845-850. 

Conditional independence in statistical theory.  With discussion. J. Roy. Statist. Soc. B  41 (1979), 1-31. 

Some matrix-variate distribution theory: Notational considerations and a Bayesian application.  Biometrika  68 (1981), 265-274. 

The well-calibrated Bayesian.  With discussion.  J. Amer. Statist. Ass. 77 (1982), 604-613. 

Statistical theory: The prequential approach.  With discussion. J. Roy. Statist. Soc. A  147 (1984), 278-292. 

Probability, symmetry and frequency.  Brit. J. Phil. Sci. 36 (1985), 107-128. 

Calibration-based empirical probability.  With discussion. Ann. Statist.13 (1985), 1251-1285. 

Symmetry models and hypotheses for structured data layouts.  With discussion.  J. Roy. Statist. Soc. B  50 (1988), 1-34. 

Independence properties of directed Markov fields (with  S. L. Lauritzen, B. N. Larsen and H. G. Leimer).  Networks 20 (1990), 491-505. 

Fisherian inference in likelihood and prequential frames of reference. With discussion.  J. Roy. Statist. Soc. B  53 (1991), 79-109. 

Applications of a general propagation algorithm for probabilistic expert systems.  Statistics and Computing  2 (1992), 25-36. 

Prequential analysis, stochastic complexity and Bayesian inference. With discussion.  Bayesian Statistics 4 (1992), edited by J. M. Bernardo, J. O. Berger, A. P. Dawid and A. F. M. Smith.  Oxford University Press, 109-125. 

On testing the validity of sequential probability forecasts (with F. Seillier-Moiseiwitsch).  J. Amer. Statist. Ass.  88 (1993), 355-359. 

Hyper Markov laws in the statistical analysis of decomposable graphical models (with S. L. Lauritzen).  Ann. Statist.  21 (1993), 1272-1317. 

Coherent combination of experts' opinions (with M. H. DeGroot and J. Mortera).  TEST  4 (1995), 263-313. 

Coherent analysis of forensic identification evidence (with J. Mortera). J. Roy. Statist. Soc. B  58 (1996), 425-443. 

Prequential analysis.  Encyclopedia of Statistical Sciences , Update Volume 1, edited by S. Kotz, C. B. Read and D. L. Banks.  Wiley-Interscience (1997), 464-470. 

Conditional independence.  Encyclopedia of Statistical Sciences , Update Volume 2, edited by S. Kotz, C. B. Read and D. L. Banks.  Wiley-Interscience (1998), 146-155. 

Prequential probability: Principles and properties (with V. G. Vovk). Bernoulli 5 (1999), 125-162. 

Probabilistic Networks and Expert Systems (with R. G. Cowell, S. L. Lauritzen and D. J. Spiegelhalter.  Springer (1999), xii + 321 pp. 

Coherent dispersion criteria for optimal experimental design (with P. Sebastiani).  Ann. Statist. 27 (1999), 65-81. 

Causal inference without counterfactuals.  With discussion. J. Amer. Statist. Ass. 95 (2000), 407-448. 

Separoids: A mathematical framework for conditional independence and irrelevance.  Ann. Math. Artificial Intelligence 32 (2001), 335-372. 

Compatible prior distributions (with S. L. Lauritzen).  Bayesian Methods with Applications to Science, Policy and Official Statistics (2001), edited by  E. I. George.  Monographs of Official Statistics, Eurostat, 109-118. 

Bayes's theorem and weighing evidence by juries.  In Bayes's Theorem, edited by Richard Swinburne.  Proc. Brit. Acad. 113 (2002), 71-90. 

Influence diagrams for causal  modelling and inference.  Intern. Statist. Rev. 70 (2002), 161-189. 

Probabilistic expert systems for forensic inference from genetic markers (with J. Mortera, V. L. Pascali and D. van Boxel).  Scand. J. Statist. 29 (2002), 577-595.

Game theory, maximum entropy, minimum discrepancy, and robust Bayesian decision theory (with P. D. Grunwald).  Ann. Statist. 32 (2004), 1367-1433.

Probability, causality and the empirical world: A Bayes-de Finetti-Popper-Borel synthesis.  Statistical Science 19 (2004), 44-57.

Counterfactuals, hypotheticals and potential responses: a philosophical examination of statistical causality. In Causality and Probability in the Sciences, edited by F. Russo and J. Williamson.  London: College Publications, Texts In Philosophy Series Vol. 5 (2007), 503–32.

The geometry of proper scoring rules. Annals of the Institute of Statistical Mathematics 59 (2007), 77–93. doi:10.1007/s10463-006-0099-8

Object-oriented Bayesian networks for complex forensic DNA profiling problems (with J. Mortera and P. Vicard).  Forensic Science International 169 (2007), 195–205.  doi:10.1016/j.forsciint.2006.08.028

Object-oriented graphical representations of complex patterns of evidence (with A. B. Hepler and V. Leucari).  Law, Probability & Risk 6 (2007), 275–293.  doi:10.1093/lpr/mgm005

Statistics and the law.  In Evidence, edited by Andrew Bell, John Swenson-Wright and Karin Tybjerg.  Cambridge University Press (2008), 119–148. doi:10.2277/0521839610