Geometrical Methods for Statistical Inference and Decision

Lead Research Organisation: University College London
Department Name: Statistical Science

Abstract

The important problems of statistics concern what we can learn from empirical data, what might happen next, and what is the best course of action. Statistical inference is the process of extracting information about the underlying nature of the data, to allow us to make predictions about future events. Statistical decision theory searches for strategies that will lead to optimal outcomes, taking into account the intrinsic uncertainty in our predictions. For instance, data on the response of patients to a pharmaceutical drug enable us to infer the effectiveness of the drug in the general population. Given a desirable social goal, such as maximising health benefits while minimising adverse reactions, we may then decide what treatment allocation and dosage levels would be optimal.It is remarkable fact that such statistical questions can be reframed in the mathematical language of geometry. To be more precise, geometric descriptions of objects, involving e.g. distances between points, can be applied to statistical models. However, instead of thinking of an object as a collection of points in 3-dimensional space, the points are now the various different probability distributions that could generate the data. To take a simple example, optimal estimation becomes the process of finding the geometric point which represents the best fitting distribution, and of quantifying how close it is to the true distribution generating the data. More sophisticated applications utilise e.g. the geometric curvature of the statistical model to quantify the uncertainty in our inferential conclusions.The geometric approach to statistical inference has been intensively studied, but there has been little attempt to apply it to statistical decision theory. Building on theoretical foundations recently laid down by Dawid and Lauritzen, this project will develop new theory and applications of geometric decision analysis. In particular it will introduce geometric concepts and techniques originating in Physics and Cosmology to the study of problems of statistical inference.

Publications

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Dawid A (2012) Proper local scoring rules on discrete sample spaces in The Annals of Statistics

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Dawid A (2012) Proper local scoring rules on discrete sample spaces in The Annals of Statistics

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Dawid A (2012) Estimation of spatial processes using local scoring rules SPATIAL SPECIAL ISSUE in AStA Advances in Statistical Analysis

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Parry M (2012) Proper local scoring rules in The Annals of Statistics

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Parry M (2012) Proper local scoring rules in The Annals of Statistics

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Parry, M. (2013) Multidimensional local scoring rules in Proceedings 59th ISI World Statistics Congress

Related Projects

Project Reference Relationship Related To Start End Award Value
EP/E009670/1 01/10/2006 14/01/2008 £264,102
EP/E009670/2 Transfer EP/E009670/1 15/01/2008 14/10/2009 £153,467
 
Description The concept of a scoring rule is applicable to both statistical inference and statistical decsision theory. For this reason, scoring rules were a natural starting point for our work.



The results we obtained follow from one simple organising principle: that a scoring rule should depend only on the observed data and data that is, in a sense to be made precise, near to the observed data. We call such scoring rules "local". A remarkable fact, which is true for all local scoring rules, is that the quoted probability distribution need not be normalized. This is remarkable because the normalization is vital in many statistical applications, e.g. maximum likelihood estimation, and is the main quantity of interest in statistical physics.



For continous outcome spaces, we were able to characterize essentially all local scoring rules. Such scoring rules depend on the probability density and its derivatives. We found a connection to the time-independent Schroedinger equation and, as a consequence, a fascinating relationship to a recent approach to clustering in data mining. Another nice feature of these scoring rules is that they are invariant to invertible transformations of the outcome space. Scoring rules can also be made robust.



Locality on discrete outcome spaces is rather more flexible: nearness is defined by an arbitrary undirected graph on the outcomes. The specification of local scoring rules in this case then follows from a lovely connection to the factorization theorem for the joint probability distribution of random variables on a graph. Because the graph on the outcomes can be specified by the user, applications to missing data problems and sequential prediction are possible. Furthermore, the well known pseudolikelihood approach turns out to be an example of a local scoring rule.



Local scoring rules have a natural geometrical structure. We found the metric determines the Godambe efficiency of estimators derived from the scoring rule. Furthermore, the metric appears, in some cases, to share with the Fisher metric the property of being invariant under sufficient reduction of the data. The role of curvature is the subject of continuing investigation.
Exploitation Route Local scoring rules make statistical estimation possible in a number of statistical areas and hence are ripe for application. Their connection to information and decision geometries are currently being explored.
Sectors Agriculture, Food and Drink

 
Description Our findings have been used in the statistical and machine learning communities. From a theoretical perspective, they have been used to unify and justify a number of existing statistical techniques. From an applied perspective, they have been used in plant epidemiology and in time series analysis.
Sector Agriculture, Food and Drink
 
Description Invited conference speaker I 
Form Of Engagement Activity Scientific meeting (conference/symposium etc.)
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Other academic audiences (collaborators, peers etc.)
Results and Impact Invited conference speaker on scoring rules.

Within-discipline interest
Year(s) Of Engagement Activity 2012
 
Description Invited conference speaker II 
Form Of Engagement Activity Scientific meeting (conference/symposium etc.)
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Other academic audiences (collaborators, peers etc.)
Results and Impact Invited conference speaker on information geometry.

Invited talk on "Decision Geometry" at the conference Geometric Aspects of Conditional Independence and Information at the Max Planck Institute for Mathematics, Leipzig, Germany

Further collaboration
Year(s) Of Engagement Activity 2008
 
Description Invited conference speaker III 
Form Of Engagement Activity Scientific meeting (conference/symposium etc.)
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Other academic audiences (collaborators, peers etc.)
Results and Impact Invited conference speaker on local scoring rules.

Invited talk on "Local proper scoring rules" at the conference Information Geometry and its Applications III at the Max Planck Institute for Mathematics in Leipzig, Germany

Interdisciplinary interest
Year(s) Of Engagement Activity 2010
 
Description Invited conference speaker IV 
Form Of Engagement Activity Scientific meeting (conference/symposium etc.)
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Other academic audiences (collaborators, peers etc.)
Results and Impact Invited conference speaker on scoring rules in multidimensional outcome spaces.

Invited conference speaker on scoring rules at the session "Probability Forecasting" at the 59th World Statistics Congress in Hong Kong, China

Further collaboration
Year(s) Of Engagement Activity 2013
 
Description Invited conference speaker V 
Form Of Engagement Activity Scientific meeting (conference/symposium etc.)
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Other academic audiences (collaborators, peers etc.)
Results and Impact Invited conference speaker on local scoring rules.

Invited conference speaker on scoring rules at the International Conference on Robust Statistics 2012, Cagliari, Italy

Further collaboration
Year(s) Of Engagement Activity 2012
 
Description Invited contribution I 
Form Of Engagement Activity Scientific meeting (conference/symposium etc.)
Part Of Official Scheme? No
Geographic Reach Local
Primary Audience Other academic audiences (collaborators, peers etc.)
Results and Impact Invited contribution to a research meeting.

Invited contribution on information geometry to a special one-day meeting of the Cambridge Statistics Initiative, University of Cambridge

Interdisciplinary interest
Year(s) Of Engagement Activity 2008
 
Description Invited contribution II 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Other academic audiences (collaborators, peers etc.)
Results and Impact Invited contribution to the discussion section of a paper.

Invited by the authors to contribute to the discussion section of "Riemann manifold Langevin and Hamiltonian Monte Carlo methods", M. Girolami & B. Calderhead, J.R.Statisi. Soc. B (2011) 73, Part 2, 123-214

Further collaboration
Year(s) Of Engagement Activity 2010
 
Description Invited guest 
Form Of Engagement Activity Scientific meeting (conference/symposium etc.)
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Other academic audiences (collaborators, peers etc.)
Results and Impact Invited guest and contributor to a one-day symposium on information geometry.

Invited guest of Prof. Shinto Eguchi at the Institute of Statistical Mathematics, Tokyo, Japan. Contributed talks to a one-day symposium on information geometry.

Further collaboration
Year(s) Of Engagement Activity 2009
 
Description Invited research seminar I 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Other academic audiences (collaborators, peers etc.)
Results and Impact Invited research seminar on local scoring rules.

Further collaboration
Year(s) Of Engagement Activity 2010
 
Description Invited research seminar II 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Other academic audiences (collaborators, peers etc.)
Results and Impact Invited research seminar on local scoring rules.

Further collaboration
Year(s) Of Engagement Activity 2011
 
Description Invited research seminar III 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Local
Primary Audience Other academic audiences (collaborators, peers etc.)
Results and Impact Invited research seminar on local scoring rules.

Further collaboration
Year(s) Of Engagement Activity 2010
 
Description Invited research seminar IV 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Regional
Primary Audience Other academic audiences (collaborators, peers etc.)
Results and Impact Invited research seminar on local scoring rules.

Further collaboration
Year(s) Of Engagement Activity 2010
 
Description Invited research seminar IX 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Other academic audiences (collaborators, peers etc.)
Results and Impact Invited research seminar on local scoring rules.

Further collaboration
Year(s) Of Engagement Activity 2012
 
Description Invited research seminar V 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Other academic audiences (collaborators, peers etc.)
Results and Impact Invited research seminar on local scoring rules.

Further collaboration
Year(s) Of Engagement Activity 2011
 
Description Invited research seminar VI 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Other academic audiences (collaborators, peers etc.)
Results and Impact Invited research seminar on local scoring rules.

Further collaboration
Year(s) Of Engagement Activity 2011
 
Description Invited research seminar VII 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Regional
Primary Audience Other academic audiences (collaborators, peers etc.)
Results and Impact Invited research seminar on local scoring rules.

Interdisciplinary interest
Year(s) Of Engagement Activity 2011
 
Description Invited research seminar VIII 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach Local
Primary Audience Other academic audiences (collaborators, peers etc.)
Results and Impact Invited research seminar on local scoring rules.

Further collaboration
Year(s) Of Engagement Activity 2012
 
Description Invited research seminar X 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Other academic audiences (collaborators, peers etc.)
Results and Impact Invited research seminar on local scoring rules.

Further collaboration
Year(s) Of Engagement Activity 2012
 
Description Invited research seminar XI 
Form Of Engagement Activity A talk or presentation
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Other academic audiences (collaborators, peers etc.)
Results and Impact Invited research seminar on local scoring rules.

Further collaboration
Year(s) Of Engagement Activity 2012
 
Description Invited research seminar XII 
Form Of Engagement Activity Scientific meeting (conference/symposium etc.)
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Other academic audiences (collaborators, peers etc.)
Results and Impact Invited research seminar on local scoring rules.

Within-discipline interest
Year(s) Of Engagement Activity 2013
 
Description Invited research seminar XIII 
Form Of Engagement Activity Scientific meeting (conference/symposium etc.)
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Other academic audiences (collaborators, peers etc.)
Results and Impact Invited research seminar on local scoring rules.

Within-discipline interest
Year(s) Of Engagement Activity 2013
 
Description Invited workshop presentation IV 
Form Of Engagement Activity Scientific meeting (conference/symposium etc.)
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Other academic audiences (collaborators, peers etc.)
Results and Impact Invited workshop speaker on scoring rules and information geometry.

Invited workshop speaker at The Seventh Workshop on Information Theoretic Methods in Science and Engineering (WITMSE), USA; 5-7 July 2014

Interdisciplinary interest
Year(s) Of Engagement Activity 2014
 
Description Invited workshop speaker I 
Form Of Engagement Activity Scientific meeting (conference/symposium etc.)
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Other academic audiences (collaborators, peers etc.)
Results and Impact Invited workshop speaker on scoring rules and information geometry.

Invited workshop speaker on "Local and discrete scoring rules" at the 1st Workshop on Geometric and Algebraic Statistics, Open University

Further collaboration
Year(s) Of Engagement Activity 2009
 
Description Invited workshop speaker II 
Form Of Engagement Activity Scientific meeting (conference/symposium etc.)
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Other academic audiences (collaborators, peers etc.)
Results and Impact Invited workshop speaker on local scoring rules and decision geometry.

Invited workshop speaker and working group convenor on decision geometry and geometry of Markov chain Monte Carlo methods at the 3rd Workshop on Geometric and Algebraic Statistics, Warwick University

Further collaboration
Year(s) Of Engagement Activity 2011
 
Description Invited workshop speaker III 
Form Of Engagement Activity Scientific meeting (conference/symposium etc.)
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Other academic audiences (collaborators, peers etc.)
Results and Impact Invited workshop speaker on information geometry.

Invited workshop speaker on "A geometric approach to Bayesian local sensitivity" at the 3rd Workshop on Geometric and Algebraic Statistics, Warwick University

Further collaboration
Year(s) Of Engagement Activity 2011
 
Description Local scoring rules on discrete outcome spaces 
Form Of Engagement Activity Scientific meeting (conference/symposium etc.)
Part Of Official Scheme? No
Geographic Reach International
Primary Audience Other academic audiences (collaborators, peers etc.)
Results and Impact Conference poster.

Poster submission to the International Society for Bayesian Analysis 2010 World Meeting in Valencia, Spain

Further collaboration
Year(s) Of Engagement Activity 2010