Visions of a theory of Imprecise Probabilities VIP - Oxford Brookes ...

Cuzzolin Part B section1 (B1) VIP 

European Research Council 

ERC Starting Grant 

Research proposal (Part B section 1 (B1)) 1 

(to be evaluated in Step 1) 

Visions of a theory of Imprecise Probabilities 

Cover Page: 

- Fabio Cuzzolin 

- Oxford Brookes University 

- Visions of a theory of Imprecise Probabilities 

- VIP 

- 60 months 

VIP 

Decision making and estimation are central problems in most applied sciences, as people or machines need 

to make inferences about the state of the external world, and take consequent actions. Traditionally, the 

state of the world is described by a probability distribution over a set of alternative hypotheses. In many 

cases, however, such as extremely rare events (e.g., a volcanic eruption), few statistics are available to 

drive the estimation. Part of the observed data is often missing. Besides, under the law of large numbers, 

probability distributions are the outcome of an infinite process of evidence accumulation, while in all 

practical cases the available evidence only provides some constraints on the “true” probability governing 

the process. These issues have led to a widespread recognition of the need for a coherent mathematical 

theory of "imprecise probabilities". 

The theory of belief functions, with its simple rationale, its multiple semantic and mathematical 

interpretations, its generalization potential, is currently arguably the most developed such theory. Inference 

and decision making tools with belief functions have been developed. Their application to engineering, 

business, and artificial intelligence has been steadily growing in the last thirty years. However, the 

spreading of such applications and the interest of the statistical community seem to have reached a plateau. 

This is the signal of a crisis, possibly a blessing in disguise, that needs to be addressed. Belief calculus is 

not yet an entirely viable alternative to traditional probability theory, for instance in its Bayesian 

interpretation. 

A second effort is needed to address a number of crucial issues which still impair the formalism, in order 

to make the the theory of belief functions a fully fledge mathematical theory of uncertainty and spread its 

adoption in the manifold fields of applied science, with a potentially enormous methodological impact. 

This is the goal of the present proposal. 

1 Instructions for completing Part B section 1 (B1) can be found in the Guide for Applicants for the Starting Grant 2011 

Call. 

1


Section 1: The Principal Investigator (see Guide for Applicants for the Starting Grant 2011 Call) 

1(a) Scientific Leadership Potential 2 (max 1 page) 

Dr Fabio Cuzzolin has a broad international experience, as he worked as a researcher in six world-class 

institutions of four different countries on both sides of the Atlantic. He graduated in 1997 from the 

University of Padua (Universitas Studii Paduani, founded 1222, is the seventh most ancient university in the 

world) with a laurea magna cum laude in Computer Engineering. He received a Ph.D. degree from the same 

institution in 2001, for a thesis entitled “Visions of a generalized probability theory". During his Ph.D. he 

was Visiting Scholar in the ESSRL laboratory of the Washington University in St. Louis (12th in the US 

universities ranking, 2009). He was later appointed fixed-term Assistant Professor by Politecnico di Milano, 

Milan, Italy, consistently recognized as the top university in the country. He later moved as a Postdoctoral 

Fellow to the UCLA Vision Lab, University of California at Los Angeles (24th in the US ranking). In 2006 

he was awarded a Marie Curie Fellowship in partnership with INRIA Rhone-Alpes, France. Since September 

2008 he has been a Lecturer and Early Career Fellow in the Department of Computing of Oxford Brookes 

University, Oxford, U.K. Dr Cuzzolin is an integral part of the Oxford University - Oxford Brookes joint 

research group led by Professors Phil Torr and Andrew Zisserman. The group is well known as one of 

world's best and has won an incredible number of awards in the recent past, last ones the best paper awards 

at the British Machine Vision Conference 2010 and the European Conference on Computer Vision 2010. 

Dr Cuzzolin classified second in the 2007 Senior Researcher (CR1) national recruitment at INRIA, and had 

interviews with/offer from Oxford University, EPFL, Universitat Pompeu Fabra, UCSD, GeorgiaTech, U. 

Houston, Honeywell Labs, Riya. 

Given his career stage and his early achievements, Dr Cuzzolin can be considered a “consolidator”. 

Dr Cuzzolin's research interests span imprecise (non-additive) probabilities, with a focus on belief 

functions, computer vision, and machine learning. 

He is a recognized prominent expert in the field of belief functions and finite uncertainty measures, and a 

member of the Board of Directors of the Belief Functions and Applications Society. He formulated a 

geometric approach to uncertainty measures in which various uncertainty measures can be represented as 

points of a Cartesian space and there analyzed. Within computer vision, his work focuses on human motion 

analysis and action recognition, while in machine learning it mainly concerns large scale manifold learning. 

Dr Fabio Cuzzolin is author at the present time of 52 peer-reviewed publications. He is first or single author 

of 44 such publications. Of these, thirty have been published in the last three years alone. He is single author 

of 2 book chapters, 8 journal papers, and 10 chapters in book series. One of his papers won the best paper 

award at the PRICAI'08 symposium. Four more articles (all of them single-author) are currently under 

review. Twelve additional journal papers are in the process of being submitted in the very near future. 

Dr Cuzzolin is working on a monograph collecting his work on the geometry of uncertainty measures. 

He collaborates as a reviewer with several journals in both computer vision and imprecise 

probabilities (such as the IEEE Trans. SMC – B and C, the IEEE Trans. on Fuzzy Systems, the Int. J.l on 

Approximate Reasoning, Computer Vision and Image Understanding, Information Sciences, the Journal of 

Risk and Reliability, the Int. J. on Uncertainty, Fuzziness, and Knowledge-Based Systems, etc, see CV). 

He has been a guest editor for Information Fusion, has served in the Programme Committee of some 20 

international conferences, and serves as a reviewer in major vision conferences such as BMVC and ECCV. 

Dr Cuzzolin has an extensive teaching experience at university level in three different countries, first 

as a teaching assistant (T.A) in Padua and Milan, later as a T.A. at UCLA, and eventually as the Module 

Leader for a number of honours courses at Oxford Brookes University. He has been co-supervisor of two 

Ph.D. students in France and the UK, and supervised several Master's students' final year dissertations. 

Concerning grants and awards,Dr Cuzzolin has recently submitted two Royal Society's Newton 

International Fellowshipapplications as UK sponsor. He has just submitted his EPSRC First Grant, and has 

applied for a Project Grant on “The total probability theorem for finite random sets" with the Leverhulme 

Trust. He is now finalizing as the coordinator a Short Proposal for a Future and Emerging Technology (FET- 

Open)FP7 EU grant on “Large scale hybrid manifold learning", with INRIA, Pompeu Fabra and Technion as 

partners. At European level he is also organizing with IDSIA, Universiteit Gent, and the Milan company 

March Networks a STREP on “Imprecise Markov models for gesture recognition" which will be submitted 

to Call 7 in January 2011. Finally, he is setting up an EPSRC project at UK level on inference and decision 

making under uncertainty in partnership with U. Bristol, Manchester Metropolitan, and U. Belfast. 

2 By default "Starters" being those awarded their PhD from 2 up to 7 years prior to the Starting Grant call publication, 

and "Consolidators" being normally those awarded their PhD over 7 years and up to 12 years prior to the Starting Grant 

call publication. 

2


1(b) Curriculum Vitae (max 2 pages) 

Personal information 

Nationality : Italian, Birthdate : October 5, 1971, Current residence : United Kingdom 

Education 

2001 Ph.D. Industrial Electronics and Computer Science, University of Padua (Italy) (date awarded: 

February 19, 2001) 

1997 M.S. Computer Engineering, University of Padua 

Professional experience 

9/2008 – present Lecturer and Early Career Fellow 

Department of Computing, Oxford Brookes University 

9/2006 – 9/2008 Marie-Curie fellow 

INRIA - Insitut National de Recherche en Informatique et en Automatique, Grenoble, France 

10/2004 – 4/2006 Post-doctoral researcher 

UCLA Vision Lab, Computer Science Department, University of California at Los Angeles 

1/2003 – 12/2004 Fixed-term Assistant Professor (Giovane Ricercatore) 

Department of Electronics and Information, Politecnico di Milano, Milan, Italy 

5/2001 – 5/2003 Post-doctoral researcher 

Autonomous Navigation and Computer Vision Lab, Dept. of Information Engineering., U. Padua 

6/2000 – 12/2000 Visiting student 

ESSRL Laboratory, Washington University in St. Louis 

3/1998 – 3/2001 Ph.D. Student - Research Assistant 

Autonomous Navigation and Computer Vision Lab, Dept. of Information Engineering., U. Padua 

Recent job offers and interviews 

• Dr Cuzzolin classified second in the final ranking for the 2007 Senior Researcher (Chargèe de 

Recherche de Premiere Classe) recruitment at INRIA – Rennes (May 2007); 

• He was short-listed for a faculty position by Robert Gordon University, Aberdeen (June 2008); 

• He was offered a four-year fellowship by: Pompeu Fabra, Barcelona - CILAB (May 2006); 

• He recently had interviews with/offers from: Oxford University, Georgia Institute of Technology; 

University of California at San Diego; EPFL - IDIAP - Martigny, Switzerland; University of Houston; 

• He had interviews for senior researcher positions with/offers from: Honeywell Labs - Minneapolis (June 

2006) Riya - Mountain View, CA (June 2006). 

Awards 

• Best Paper Award for the outstanding technical contribution: «Alternative formulations of the 

theory of evidence based on basic plausibility and commonality assignments», Pacific Rim 

International Conference on Artificial Intelligence, Hanoi, Vietnam, December 2008 

• Marie Curie fellowship, September 2006 - September 2008 

Membership of Boards of Societies 

• Member of the Board of Directors of the Belief Functions and Applications Society 

Membership of Editorial boards of international journals 

• Guest editor for Elsevier's Information Fusion Journal: CFP on "Competing Fusion Methods - Realworld 

Performances" 

Membership of Technical Program Committe of international conferences 

• ISIPTA 2009, ISIPTA 2011 –Int. Symposium on Imprecise Probabilities and Their Applications 

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• ECSQARU 2009, 2011 – European Conf. on Symbolic and Quantitative Approaches to Uncertainty 

• BELIEF 2010 - First International Workshop on the Theory of Belief Functions 

• IUM2010 - International Symposium on Integrated Uncertainty Management and Applications 

• FLAIRS'08 - FLAIRS'09 - FLAIRS'10 - FLAIRS'11 - Uncertain Reasoning track of the 21st, 22nd, 23rd, 

and 24th International Florida Artificial Intelligence Research Society Conferences 

• Session Chair for the 10th International Symposium on Artificial Intelligence and Mathematics (ISAIM 

2008), Fort Lauderdale, Florida, January 2-4 2008. 

• IMMERSCOM 2007 - 2nd International Conference on Immersive Telecommunications 

• VISAPP 2006, VISAPP 2007, VISAPP 2008, VISAPP 2009, VISAPP 2010, VISAPP 2011 - 

International Conference on Computer Vision Theory and Applications 

Reviewer for the following international journals on artificial intelligence, uncertainty theory, and vision: 

IEEE Trans. on Systems, Man, and Cybernetics - parts B and C, IEEE Trans. on Fuzzy Systems, IEEE Trans. 

on PAMI, Int. J. on Approximate Reasoning, Information Fusion, Computer Vision and Image 

Understanding, Int. J. on Uncertainty, Fuzziness, and Knowledge-Based Systems, Image and Vision 

Computing, Information Sciences, J. of Risk and Reliability, Annals of Operations Research. 

Also reviewer for major computer vision conferences such as ECCV and BMVC. 

Other collaborations 

• Member of the Society for Imprecise Probabilities and Their Applications; 

• Member of the IEEE Systems, Man, and Cybernetics Society 

Tutored Master's and PhD students 

• Mostafa Kamali Tabrizi, Ph.D. student - Oxford Brookes University, UK; 

• Massimiliano Pierobon (Politecnico di Milano), Laurent Simon, Thomas Coville, Dan Williams, 

Deepak Sharma (Oxford Brookes University). 

Funding ID - Research grant proposals as Principal Investigator 

• Oxford Brookes University - Central university funding: Uncertainty in Computer Vision, 12 

months, 3600 pounds, awarded; 

• The Royal Society - Newton International Fellowships: The total probability theorem for finite 

random sets or belief functions, 24 months, 98000 pounds, pending; 

• Engineering and Physical Sciences Research Council (EPSRC) - First Grant Scheme: Tensorial 

modeling of dynamical systems for gait and activity recognition, 2 years, 122000 pounds, pending; 

• Leverhulme Trust - Research Projects Grants: The total probability theorem for belief functions, 

3 years, 121781 pounds, pending; 

• European Union FP7 – FET-Open: Large scale hybrid manifold learning, 3 years, ~2 million 

euros (partners: INRIA, U. Pompeu Fabra, Technion), in preparation; 

• European Research Council - Starting Grant: Visions of a theory of Imprecise Probabilities, 5 

years, 1.45 million euros, in preparation; 

• European Union FP7 - Call 7 - STREP: Imprecise Markov chains for gesture recognition, 3 years, 

~1.4 million euros (partners: IDSIA, U. Gent, March Networks), in preparation; 

• Engineering and Physical Sciences Research Council (EPSRC): A network on uncertainty theory 

(partners: U. Bristol, Manchester Metropolitan, U. Belfast), in preparation. 

Other research projects I partecipated in as a paid member of staff: 

• Politecnico di Milano - FIRB - VICOM, Virtual Immersive Communication, Italy Ministry of 

University and Scientific Research, 2003-5; 

• INRIA Rhone-alpes - VISIONTRAIN, network of excellence in computer vision. 

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1(c) Early Achievement-Track-Record (max 2 pages) 

Publications as main author, most relevant to this research project (selected out of 52 peer reviewed papers), 

which are used for citation in the body of the proposal. 

Publications as main author in peer-reviewed international journals and conference proceedings 

[1] F. Cuzzolin, Geometry of relative plausibility and relative belief of singletons, Annals of Mathematics 

and Artificial Intelligence, pp. 1-33, July 26 2010 

[2] F. Cuzzolin, The geometry of consonant belief functions: simplicial complexes of necessity measures, 

Fuzzy Sets and Systems, 161(10), pp. 1459-1479, 16 May 2010 

[3] F. Cuzzolin, Credal semantics of Bayesian transformations in terms of probability intervals, IEEE 

Transaction on Systems, Man, and Cybernetics - part B, Vol. 40(2), pp. 421-432, April 2010 

[4] F. Cuzzolin, Three alternative combinatorial formulations of the theory of evidence, Special Issue of 

PRICAI'08 Selected Papers, Intelligent Decision Analysis journal, 2010 

[5] F. Cuzzolin, A geometric approach to the theory of evidence, IEEE Transactions on Systems, Man, 

and Cybernetics part C, 38(4), pp. 522-534, July 2008 

[6] F. Cuzzolin, Two new Bayesian approximations of belief functions based on convex geometry, IEEE 

Transactions on Systems, Man, and Cybernetics part B, 37(4), pp. 993-1008, August 2007. 

[7] F. Cuzzolin, Algebraic structure of the families of compatible frames of discernment, Ann. Math. Artif. 

Intell. 45(1-2), pp. 241-274, 2005. 

[8] F. Cuzzolin, Geometry of Dempster's rule of combination, IEEE Transactions on Systems, Man, and 

Cybernetics part B, 34(2), pp. 961- 977, April 2004. 

[9] F. Cuzzolin, Geometric conditioning of belief functions, Proceedings of the First International Workshop 

on the Theory of Belief Functions (BELIEF 2010), Brest, April 2010 

[10] F. Cuzzolin, Consistent approximations of belief functions, Proceedings of the International Symposium 

on Imprecise Probabilities and Their Applications (ISIPTA'09), Durham, UK, July 14-18 2009; 

[11] F. Cuzzolin, Credal semantics of Bayesian approximations, Proc. of ISIPTA'09, Durham, July 2009; 

[12] F. Cuzzolin, Learning pullback metrics for linear models, European Conference on Computer Vision 

(ECCV'08), First workshop on Machine Learning for Vision-based Motion Analysis, Marseille, 2008 

[13] F. Cuzzolin, D. Mateus, D. Knossow, E. Boyer, R. Horaud, “Coherent Laplacian 3-D protrusion 

segmentation”, Proc. of CVPR'08 (Computer Vision and Pattern Recognition), pp.1 -8, Anchorage, 2008. 

[14] F. Cuzzolin, Boolean and matroidal independence in uncertainty theory, Proc. of the 10th 

International Symposium on Mathematics and Artificial Intelligence ISAIM'08, Fort Lauderdale, 

January 2-4 2008. 

[15] F. Cuzzolin, An interpretation of consistent belief functions in terms of simplicial complexes, 

Proceedings of ISAIM'08, Fort Lauderdale, Florida, January 2-4 2008. 

[16] F. Cuzzolin, Using Bilinear Models for View-invariant Action and Identity Recognition, Proc. of 

CVPR'06, pp. 1701-1708, New York, June 18-22 2006. 

[17] F. Cuzzolin, The geometry of relative plausibilities, Proc of the 11th International Conference on 

Information Processing and Management of Uncertainty IPMU'06, Paris, July 2006. 

[18] F. Cuzzolin, On the properties of relative plausibilities, Proc. of the International Conference of the 

IEEE Systems, Man, and Cybernetics Society (SMC'05), vol.1, pp. 594-599, Hawaii, October 2005. 

[19] F. Cuzzolin, R. Frezza, Evidential modeling for pose estimation, Proc. of the 4th International 

Symposium on Imprecise Probabilities and Their Applications (ISIPTA'05), Pittsburgh, June 2005. 

[20] F. Cuzzolin, A. Sarti, S. Tubaro, Invariant action classification with volumetric data, International 

Workshop on Multimedia Signal Processing, pp. 395-398, Siena, Italy, 9/29-10/1 2004. 

[21] F. Cuzzolin, A. Sarti, S. Tubaro, Action modeling with volumetric data, Proc. of the 2004 International 

Conference on Image Processing (ICIP'04), vol. 2, pp. 881- 884, Singapore, October 24-27, 2004. 

[22] F. Cuzzolin, Simplicial complexes of finite fuzzy sets, Proc. of the International Conference on 

Information Processing and Management of Uncertainty (IPMU'04), pp. 1733-1740 Perugia, July 2004. 

[23] F. Cuzzolin, Geometry of upper probabilities, Proceedings of the 3rd International Symposium on 

Imprecise Probabilities and Their Applications (ISIPTA'03), Lugano, Switzerland, July 14-17, 2003. 

[24] F. Cuzzolin, R. Frezza, A. Bissacco, S. Soatto, Towards unsupervised detection of actions in clutter, 

Proc. of the 2002 Asilomar Conference on Signals, Systems, and Computers, vol.1, pp. 463-467, 2002. 

[25] F. Cuzzolin, Geometry of Dempster's rule, 2002 International Conference on Fuzzy Systems and 

Knowledge Discovery (FSKD'02), Special Session on Knowledge Representation under Vagueness and 

Uncertainty, Singapore, November 18-22, 2002. 

5


[26] F. Cuzzolin, R. Frezza, Geometric analysis of belief space and conditional subspaces, Proc. of the 2nd 

Int. Symposium on Imprecise Probabilities and Their Applications (ISIPTA'01), Ithaca, June 2001. 

[27] F. Cuzzolin, R. Frezza, Lattice structure of the families of compatible frames of discernment, Proc. of 

ISIPTA'01, Ithaca, NY, June 26-29, 2001. 

[28] F. Cuzzolin, R. Frezza, Integrating feature spaces for object tracking, Proc. of the International 

Symposium on the Mathematical Theory of Networks and Systems (MTNS2000), Perpignan, June 2000. 

[29] F. Cuzzolin, R. Frezza, An evidential reasoning framework for object tracking, Proc. of SPIE - 

Photonics East 99 - Telemanipulator and Telepresence Technologies VI, vol. 3840, pp. 13-24, 1999. 

Research monographs and chapters in collective volumes 

[30] F. Cuzzolin, Visions of a generalized probability theory, Ph.D. dissertation, Università degli Studi di 

Padova, Italy, March 2001. 

[31] F. Cuzzolin, Multilinear modeling for robust identity recognition from gait, in “Behavioral Biometrics 

for Human Identification: Intelligent Applications", L. Wang and X. Geng (Eds.), IGI Publishing, 2009 

[32] F. Cuzzolin, Manifold learning for multi-dimensional auto-regressive dynamical models, in “Machine 

Learning for Vision-based Motion Analysis", L. Wang, G. Zhao, L. Cheng, M. Pietikaaine (Eds.), 

Springer, 2010 

[33] F. Cuzzolin, Complexes of outer consonant approximations, in “Symbolic and Quantitative Approaches 

to Reasoning with Uncertainty" (ECSQARU'09), Lecture Notes in Computer Science, Vol. 5590/2009, 

pages 275-286, Springer Berlin / Heidelberg, 2009 

[34] F. Cuzzolin, The intersection probability and its properties, in “Symbolic and Quantitative Approaches 

to Reasoning with Uncertainty" (ECSQARU'09), Lecture Notes in Computer Science, Vol. 5590/2009, 

pages 287-298, Springer Berlin / Heidelberg, 2009 

[35] F. Cuzzolin, Alternative formulations of the theory of evidence based on basic plausibility and 

commonality assignments, in “PRICAI 2008: Trends in Artificial Intelligence", Lecture Notes in Computer 

Science, Vol. 5351/2008, pages 91-102, Springer Berlin / Heidelberg, 2009 

[36] F. Cuzzolin, Dual properties of the relative belief of singletons, in “PRICAI 2008: Trends in Artificial 

Intelligence", Lecture Notes in Computer Science, Vol. 5351/2008, pp. 78-90, Springer Berlin, 2009 

[37] F. Cuzzolin, On the credal structure of consistent probabilities, in Logics in Artificial Intelligence, 

Lecture Notes in Computer Science, Vol. 5293/2008, pp. 126-139, Springer Berlin / Heidelberg, 2008. 

[38] F. Cuzzolin, A lattice-theoretic interpretation of independence on frames, in «Interval / Probabilistic 

Uncertainty and Non-classical Logics», Advances in Soft Computing, Vol. 46, Huynh, V.-N.; Nakamori, 

Y.; Ono, H.; Lawry, J.; Kreinovich, V.; Nguyen, H.T. (Eds.), Springer-Verlag, Berlin, 2008. 

[39] F. Cuzzolin, Semantics of the relative belief of singletons, in «Interval / Probabilistic Uncertainty and 

Non-classical Logics», Advances in Soft Computing, Vol. 46, Springer-Verlag, Berlin, 2008. 

[40] F. Cuzzolin, D. Mateus, E. Boyer, R. Horaud, Robust spectral 3D-bodypart segmentation in time, in 

«Human Motion - Understanding, Modeling, Capture and Animation», LNCS, Vol. 4814/2007, pp. 196-211, 

Seidel, H.-P.; Wang, Y.; Rosenhahn, B.; Mori, G. (Eds.), Springer Berlin / Heidelberg, 2007. 

[41] F. Cuzzolin, On the orthogonal projection of a belief function, in «Symbolic and Quantitative 

Approaches to Reasoning with Uncertainty», Lecture Notes in Computer Science, Vol. 4724/2007, pages 

356-367, Springer Berlin / Heidelberg, 2007. 

Invited presentations 

Awards 

• “Randomized Trees for Human Pose Detection”, British Machine Vision Association, 23/10/2009 

• Best Paper Award for the outstanding technical contribution: «Alternative formulations of the 

theory of evidence based on basic plausibility and commonality assignments», Pacific Rim 

International Conference on Artificial Intelligence, Hanoi, Vietnam December 2008 

• Marie Curie fellowship, September 2006 - September 2008 

Additional publications as a co-author are listed in Part B2 of the current proposal. 

6


Section 1d: Extended Synopsis of the project proposal (max 5 pages) 

(see Guide for Applicants for the Starting Grant 2011 Call) 

State of the art. Decision making and estimation are central problems in most applied sciences, as people or 

machines need to make inferences about the state of the external world, and take appropriate actions. 

Traditionally, the (uncertain) state of the world is assumed to be described by a probability distribution over 

a set of alternative, disjoint hypotheses. Making appropriate decisions or assessing quantities of interest 

requires therefore estimating such a distribution from the available data. Uncertainty is normally handled in 

the literature within the Bayesian framework, which is indeed quite intuitive and easy to use, and capable of 

providing a number of “off the shelf” tools to make inferences or compute estimates from time series. 

Sometimes, however, such as in the case of extremely rare events (e.g., a volcanic eruption), few statistics 

are available to drive the estimation [Falk04]. Part of the data can be missing [Zaffalon02]. Furthermore, 

under the law of large numbers, probability distributions are the outcome of an infinite process of evidence 

accumulation, while in all practical cases the available evidence only provides some sort of constraint on the 

unknown, “true” probability governing the process. These issues and many others have led to the recognition 

by many brilliant scientists of the need for a coherent mathematical theory of uncertainty. 

An extensive battery of different uncertainty theories has therefore been developed in the last half 

century or so, starting from De Finetti's pioneering work on subjective probability [DeFinetti74]. The most 

complete and successful frameworks are, arguably, the theory of belief functions or “theory of evidence” 

[Dempster67, Shafer76], possibility-fuzzy set theory [Zadeh78, Dubois90a], the theory of random sets 

[Matheron75, Nguyen78], the theory of imprecise probabilities [Walley91]. Other approaches to uncertainty 

modeling have also been proposed, including monotone capacities, Choquet integrals, rough sets, credal sets, 

hints [Kohlas95], lower and upper previsions, game theory [Vovk01]. Also referred to as imprecise 

probabilities (as most of them comprise classical probabilities as a special case) they form in fact an entire 

hierarchy of encapsulated formalisms. 

Most of them share the same rationale, the need to cope with some serious issues about the way uncertainty 

is traditionally handled in probability theory. We briefly sketch the most important ones here. 

Infinite accumulation of evidence (or lack thereof): Even if we assume that the ideal description of the 

phenomenon we study is a probability distribution, under the law of large numbers probability distributions 

are the outcome of an infinite process of evidence accumulation, drawn from an infinite series of samples. 

Quite the opposite is true for statistically rare events [Falk04], when very few statistics are available to drive 

the estimation. Think of the probability of volcanic eruptions or of critical accidents in a nuclear power plant. 

In all practical cases, indeed, the available evidence only provides some sort of partial belief [Shafer82]. 

Representation of ignorance: Ignorance is represented in classical probability theory as a uniform prior, 

which is quite a precise model of what is going on. This (often neglected) fact is indeed crucial, as it has 

been shown that, due to the nature of Bayes rule, an inadequate choice of the prior can be overturned by the 

available evidence only very slowly and with great effort. Moreover, it can be shown that when dealing with 

different but compatible decision spaces or “frames”, “uninformative” uniform priors on different domains 

are not compatible! [Shafer76] 

Choice of a model: In the Bayesian framework epistemic uncertainty is typically addressed by imposing 

additional assumptions in order to pick a specific a-priori distribution: in other words, people choose a 

specific “model”. The subsequent series of mathematical inferences is of course rigorous and self-contained, 

but, as it should be obvious, it only relates to the chosen “model” and has possibly nothing to do with the 

original phenomenon! Imprecise probabilities are more cautious, more “respectful” of the actual (partial) 

evidence at hand. They tend to avoid the introduction of specific prior distributions, while describing 

ignorance in a natural way. 

Missing data: It can be shown that [Zaffalon02], when part of the data used to estimate the desired 

probability distribution is missing, the resulting constraint is a credal set [Levi80] of the type associated with 

a belief function. There is no need to “fill in” the missing bit of data, as this is naturally coped with by 

imprecise probabilities. 

Different kinds of constraints are associated with different classes of imprecise probabilities. The simplest 

way to constrain a probability distribution is to put upper u(x) and lower l(x) bounds to its values on all the 

elements (x,y,z etc) of the domain: we get an interval probability [Moral94]. A more sophisticated constraint 

requires the (unknown) probability to fall inside a convex set of probability distributions or “credal set” 

[Levi80]. Convexity (as a mathematical requirement) is a natural consequence, in these theories, of 

rationality axioms such as coherence [Walley91]. 

The notion of belief function originally derives from a series of Dempster's works on upper and lower 

7


probabilities induced by multi-valued mappings. Given a probability distribution p on a given domain, and a 

one-to-many map x → Γ(x) to another domain, the original probability induces a probability distribution on 

the power set of the second collection [Dempster67], i.e., a “random set” [Matheron75, Nguyen78]. The term 

“belief function" (BF) was coined when Glenn Shafer [Shafer76, Shafer79] adopted these mathematical 

objects to represent evidence in the framework of subjective probability, and gave an axiomatic definition of 

them as non-additive (indeed, super-additive) probability measures. Belief functions can be also seen as 

“inner measures” [Fagin89], or as “sum functions” on the power set. In a rather controversial interpretation, 

rejected by many (including Shafer), belief functions can also be seen as a special case of credal set: as they 

determine a lower and an upper bound to the probability of each event A, they are naturally associated with 

the convex set of probabilities which “dominate” them: {P : P(A) ≥ Bel(A) ∀A}. The main (but not the only) 

problem with this interpretation is that it is not compatible with Shafer's original proposal (“Dempster's 

rule”) for the combination of BFs generated by different pieces of evidence. 

The theory of belief functions is appealing because it addresses all the above mentioned issues with 

the handling of uncertainty: it does not assume an infinite amount of evidence to model imprecision, but uses 

all the available partial evidence; it represents ignorance in a natural way, by means of the mass assigned to 

the whole decision space or “frame”, and deals with the problem of having to represent uncertainty on 

different but compatible domains; it does not need to resort to any specific “model” to enable us to make 

deductions on the observed phenomenon, but preserves the appropriate level of uncertainty at all stages of 

the calculations; it copes with missing data in the most natural of ways. 

Furthermore: its rationale is rather neat and simple; it is a straightforward generalization of probability 

theory, and it does not require to abandon the notion of event (like Walley's imprecise probability theory 

[Walley91]); it contains as special cases fuzzy set theory and possibility theory. 

Goals of the project. The rationale for a mathematical theory of imprecise probabilities is therefore 

quite apparent. The theory of belief functions, with its relative simplicity, its multiple semantic and 

mathematical interpretations, its generalization potential, is arguably one of the most developed such 

theories, and a most serious candidate for the job. 

Tools for inferring belief functions from the data or making decisions with them have been developed. The 

number of applications of the theory of evidence to engineering, business, artificial intelligence or robotics 

have been steadily growing in the last thirty years. 

However, the diffusion of such applications and even the interest of the statistical community in the related 

techniques seem to have reached a plateau. This is the signal of a crisis, possibly, as we claim in the abstract, 

a blessing in disguise, that needs to be addressed. Indeed, the theory of belief functions is not yet entirely a 

viable alternative to traditional probability theory in the Bayesian framework. The existing community has a 

tendency to focus on very interesting but somehow self-indulgent new theoretical results. Feedback from 

real-world applications is often missing. Many of what are, in our opinion, crucial questions that need to be 

addressed in order for practitioners to find the framework easy and convenient to use are being somehow 

neglected. Others have been well known for a long time, but are still far from a satisfactory resolution, often 

due to an over-abundance of proposal solutions rather than the opposite. 

A second effort is needed to address a number of crucial issues which currently impair the formalism and its 

practical application, in order to make the theory of evidence a fully fledged mathematical theory of 

uncertainty and spread its adoption in the manifold fields of applied science. We list them below. 

1- some of what are the crucial steps in any inference chain have been hotly debated in the past, and a 

settled resolution of many critical steps of belief calculus is not yet in sight; 

2- the theory is still incomplete under several points of view: fundamental mathematical tools, such as the 

total probability theorem and related issues are just not there; 

3- when compared with the Bayesian formalism, belief calculus lacks a number of techniques which, even 

though they cannot be classified as “fundamental”, are a sign of the significant development the Bayesian 

framework has gone through, and a valuable array of tools for practitioners: examples are Maximum A 

Posteriori, Maximum Likelihood, Expectation-Maximization estimation, Monte-Carlo methods; 

4- field testing of the different proposals has been neglected, leading to a certain degree of disconnection 

between theoretical debates and the reality of applications; 

5- at the opposite end of the spectrum, the mathematical properties of belief functions in their finite and 

continuous formulations are not yet entirely understood; the same holds for important links with many other 

fields of mathematics, such as linear algebra, combinatorics, geometry, discrete mathematics. 

The goals of the proposed research project reflect all such questions, and can be outlined as follows: 

1- a contribution to the resolution of the current ongoing debates on several crucial stages in the inference 

chain (illustrated below); 

8


2- an effort finalized at groundbreaking progress towards a completion of the theory of belief functions, to 

be achieved by tackling fundamental missing elements in the theory; 

3- with the aim of pushing towards a greater diffusion of the framework in ever more application fields, the 

design of some off-the-shelf inference tools to be made available to practitioners; 

4- the analysis and testing of the above tools and issues on real-world test-beds provided by classical 

artificial intelligence problems, with a focus on computer vision applications; 

5- an advancement of the understanding of the mathematical properties of belief functions as mathematical 

objects, and of their relationships with other fields of pure and applied mathematics. 

ii. Methodology 

Specific research objectives/workpackages (WP) 

Let us refine our research plan into a number of more specific activities (workpackages). The scope 

and ambition of the project are reflected in its articulation into several distinct but related workpackages, in 

turn clearly identifiable as stages of the inference process depicted in the diagram below. 

INFERENCE 

(WP1) 

BELIEF 

UPDATE OR 

REVISION 

(WP2) 

EXTENSION AND 

COMPLETION 

(WP5) 

TOTAL 

BELIEF 

THEOREM 

(WP3) 

MATHEMATICAL 

PROPERTIES 

(WP8) 

WP1 – The inference problem, i.e., how to build a belief function from input data. Data can be of different 

nature: statistical [Seidenfeld78], logical, expressed in terms of mere preferences, subjective. Different 

approaches to the inference problem produce different belief functions from the same statistical data. The 

problem has been studied by scholars of the caliber of Shafer, Seidenfeld, Walley, and others. Instead of 

trying and address the problem in its generality, we will focus on an interesting novel approach [Zhang10]. 

WP2 – Update/conditioning. The incorporation of new evidence in an existing state of belief is called 

“belief update”, while the act of merging the states of belief of different experts/sources is referred to as 

“belief revision”. Several different operators for the update, fusion or revision of a state of belief have been 

proposed [e.g. Jaffray92, Denneberg94]. This project will contribute to the ongoing debate about the range of 

possible operators, by investigating the introduction of an uniquely determined operator for belief functions, 

inspired by Walley's “natural extension” [Walley91], exploring a unified description of combination rules in 

terms of geometric conditioning [9], comparing and testing different approaches on real-world test-beds. 

WP3 – The total belief problem. A central question in the theory of BFs which has not been dealt with yet 

in its entirety and complexity is the generalization of the total probability theorem to belief functions. 

According to the choice of the conditioning operator a different “total belief theorem” has to be formulated 

[30]. Its general solution is one of the major goals of this project. Its relationship with marginal extension of 

lower previsions [Miranda07], its links with matroid theory [Oxley92] and positive systems will be explored. 

WP4 – Efficient implementation of belief calculus. The greatest obstacle to a naïve application of the 

theory of evidence is its (exponential) computational complexity. This issue has been recognized since its 

early days [Tessem93]. Sophisticated MonteCarlo algorithms have been proposed [Moral96]. Novel 

solutions to this problem based on transforming belief functions into less complex objects such as 

probabilities [1,3,6], possibilities [2], consistent [10] and k-additive BFs will be here brought forward. Also, 

lower probability approximations in credal networks for efficient reasoning will be proposed. 

WP5 – Extension and completion of the theory. The theory of belief functions has originally been 

formulated on finite domains (frames) [Shafer76], as a result of its focus on subjective probability. Several 

possible extensions of the theory to continuous domains have subsequently been studied [Shafer79, 

Nguyen78, Kohlas95]. In this more high-risk part of the project we will seek to contribute to the resolution 

of this issue, focussing in particular on: the completion of the theory of continuous BFs on closed intervals, 

9 

ESTIMATION 

TOOLS 

(WP5) 

DECISION 

MAKING 

(WP6) 

EFFICIENT 

IMPLEMENTATION 

(WP4) 

APPLICATIONS 

(WP7)


and the search for combination rules for random sets. In parallel, as easy-to-use estimation tools are 

absolutely crucial to widen scope and ambition of the theory of evidence, we will focus on a futher 

development of EM-like algorithms for BFs in imprecise hidden Markov models [DeCooman09]. 

WP6 – Decision making using belief functions and probability intervals. Several frameworks for decision 

making with BFs have been proposed in the past [e.g., Strat90, Smets05]. We will focus on the approach to 

decision making based on transforming belief functions into less complex measures. We will propose 

alternative decision frameworks based on different probability transformations of belief functions [3]; 

advance a similar framework for interval probabilities [34]; investigate the relationship between orderings 

and the “least commitment principle” for decision making; explore the use of the classes of consistent and kadditive 

belief functions for efficient decision making. 

WP7 – Testing and comparison on benchmark applications. A number of the above questions can only 

benefit from testing and comparing the different proposals and approaches between themselves and against 

more classical Bayesian methodologies on real-world applications. Given the proposer's unique 

interdisciplinary expertise three such problems will be selected from computer vision/machine learning: data 

association, pose estimation [19], action and gesture recognition [20,21]. 

WP8 – Mathematical properties. Given their inherent mathematical sophistication, belief functions exhibit 

a number of links with manifold fields of mathematics. We propose to investigate three of the most relevant 

of them: that with lattice theory, in particular the notion of independence [38] and the development of BFs on 

lattices [Grabish09] to represent mere “preferences”; combinatorics and the foundation of the theory of belief 

functions as sum functions [4,35]; geometry, with a focus on the extension of the proposer's geometric 

approach to the continuous case, and the link with geometric probability and the geometry of convex bodies. 

Ground-breaking nature and potential impact 

The ERC Starting Grant scheme has been designed on purpose to stimulate early career researchers 

to think big, focussing on what they need to accomplish in order to maximize the impact of their work, and 

allow them to makes plans for the medium term. This project, we believe, meets all these criteria. 

The proposed work concerns a fundamental methodology which is used as a tool for handling uncertainty in 

dozens of different application fields by thousands of practitioners already, many of them in the fast growing 

Asian scientific and technological community. 

The potential for an even more widespread application will be greatly increased by a success of this project, 

since the latter is designed to surgically target what we believe are currently weak spots of the formalism. 

Of particular ground-breaking nature are the tasks involving the total belief theorem (WP3), the continuous 

formulation of the theory (WP5), the inference problem (WP1), whose outcomes have the potential of 

changing the way people see the theory of evidence and its very position in the wider picture of uncertainty 

and probability theory. A success of the proposed project will help breaking an “impasse” in the 

development of an epistemically satisfactory mathematical theory of uncertainty, at the same time aiming at 

making it powerful and straightforward to use, and helping a better understanding of the important links 

between probability theory and the rest of mainstream mathematics. The dissemination of the outcomes of 

the project will be greatly helped by proposer's position at the crossroad between the communities of 

imprecise probabilities, belief functions, computer vision, and machine learning, possibly facilitating the 

introduction of the methodology to fast-growing communities such as vision and machine learning. 

Feasibility 

While admittedly ambitious, the proposed research (as detailed above) is realistically feasible given 

the long time span (5 years), the manpower available to complete it (if of course all the resources we ask for 

this purpose will be granted) and, most of all, the expertise and background of the proposer. 

Dr Cuzzolin is a recognized leader in the field of belief functions and uncertainty theory, recently elected a 

member of the Board of Directors of the newly established Belief Functions and Application Society, a 

member of the Society for Imprecise Probabilities, and an organizer for all the relevant major conferences. 

He enjoys a twelve-year experience as an active researcher in the field. His productivity has been judged 

impressive by his colleagues, and his original views on the mathematics of uncertainty have attracted 

growing interest, especially in the last four years, questioning more traditional approaches to the problem. 

As it can be seen from the detailed discussion of the different tasks involved by such a wide-scope project, 

he already possesses a significant expertise in all the research topics associated with the different stages of 

the inference chain depicted in the diagram, which are reflected in the proposed workpackages. 

Even a medium-term project of this sort cannot hope of addressing all the difficult questions involved by 

such an inference process, which have kept occupied many brilliant minds for a number of years. For this 

reason, at each stage we will focus on a subsets of more specific tasks or approaches, most of the time 

10


closely related to the proposer's expertise, with the objective of increasing the overall feasibility of the 

proposed research, without compromising on its ambition and impact. 

The proposer enjoys connections with all major research groups in the wider field of imprecise probabilities 

at world level (e.g. Durham, Lugano, Toulouse, Kansas, Rutgers, Cornell), which will allow him to recruit 

without difficulties the skilled young researchers which are crucial for the success of the project. Some of 

them already applied for international fellowships (e.g. the Newton scheme) with his support. 

Finally, the infrastructure available within the Oxford Brookes vision group, one of the top such groups in 

the world, winner of an astounding number of awards in recent years, will enormously facilitate the tests that 

will constitute a healthy reality check for the techniques developed in the course of the project. 

A selection of relevant references - Preference has been given to seminal and foundational works. Relevant 

papers by the authors are denoted by integer numbers and listed in Section 1c). 

[Dempster67] A.P. Dempster, Upper and lower probabilities induced by a multivaried mapping, 

Annals of Mathematical Statistics, Vol. 38(2), 325-339, 1967 

[DeFinetti74] B. De Finetti, Theory of probability, 1974 

[Matheron75] G. Matheron, Random sets and integral geometry, Wiley, NY, 1975 

[Shafer76] G. Shafer, A mathematical theory of evidence, Princeton University Press, 1976 

[Suppes77] P. Suppes and M. Zanotti, On using random relations to generate upper and lower 

probabilities, Synthese, Vol. 36(4), pp. 427–440, 1977 

[Nguyen78] H.T. Nguyen, On random sets and belief functions, Journal of Mathematical Analysis and 

Applications, Vol. 65, pp. 531-542, 1978 

[Zadeh78] L.A. Zadeh, Fuzzy sets as a basis for a theory of possibility, FSS, Vol. 1, pp. 3-28, 1978 

[Seidenfeld78] T. Seidenfeld, Statistical evidence and belief functions, Proc. of the Biennial Meeting of the 

Philosophy of Science Association, Vol. 1978, pp. 478-489 

[Shafer79] G. Shafer, Allocations of probability, Annals of Probability, Vol. 7(5), pp. 827-839, 1979 

[Levi80] I. Levi, The enterprise of knowledge, MIT Press, 1980 

[Kyburg87] H. Kyburg, Bayesian and non-Bayesian evidential updating, AIJ, Vol. 31, pp. 271–293, 1987 

[Tessem93] B. Tessem, Approximations for efficient computation in the theory of evidence, AIJ, Vol. 

61(2), pp. 315-329, 1993 

[Fagin89] R. Fagin and J.Y. Halpern, Uncertainty, belief, and probability, AAAI'89, pp. 1161-1167, 1989 

[Dubois90a] D. Dubois and H. Prade, Possibility theory, Plenum Press, New York, 1990 

[Strat90] T. M. Strat, Decision analysis using belief functions, IJAR, Vol. 4, pp. 391-417, 1990 

[Walley91] P. Walley, Statistical reasoning with imprecise probabilities, Chapman and Hall, 1991 

[Jaffray92] J.Y. Jaffray, Bayesian updating and belief functions, IEEE Transactions on Systems, Man and 

Cybernetics, Vol. 22, pp. 1144–1152, 1992 

[Oxley92] J. Oxley, Matroid theory, Oxford University Press, 1992 

[Moral94] L. de Campos, J. Huete, and S. Moral, Probability intervals: a tool for uncertain reasoning, 

Int. J. Uncertainty Fuzziness Knowledge-Based Syst., Vol. 1, pp. 167–196, 1994 

[Denneberg94] D. Denneberg, Conditioning (updating) non-additive probabilities, Ann. Operations Res., 

Vol. 52, pp. 21–42, 1994 

[Kohlas95] J. Kohlas and P.-A. Monney, Representation of Evidence by Hints, In “Classic Works of the 

Dempster-Shafer Theory of Belief Functions”, pp. 665-681, 2008 

[Moral96] S. Moral and N. Wilson, Importance Sampling Monte-Carlo Algorithms for the Calculation 

of Dempster-Shafer Belief, Proc. of IPMU'96, 1996 

[Vovk01] G. Shafer and V. Vovk, Probability and finance: It's only a game!, 2001 

[Zaffalon02] M. Zaffalon, Exact credal treatment of missing data, Journal of Statistical Planning and 

Inference, Vol. 105(1), pp. 105-122, 2002 

[Falk04] M. Falk, J. Husler and R-D. Reiss, Laws of Small Numbers: Extremes and Rare Events, 2004 

[Smets05] Ph. Smets, Decision making in the TBM: the necessity of the pignistic transformation, 

Journal of Approximate Reasoning, Vol. 38(2), pp. 133-147, 2005 

[Troffaes07] M.C.M. Troffaes, Decision making under uncertainty using imprecise probabilities, Int. J. 

Approx. Reasoning, Vol. 45(1), pp. 17-29, 2007 

[Miranda07] E. Miranda and G. de Cooman, Marginal extension in the theory of coherent lower 

previsions, International Journal of Approximate Reasoning, Vol. 46(1), pp. 188-225, 2007 

[Grabish09] M. Grabish, Belief functions on lattices, Int. J. of Intelligent Systems, pp. 1-20, 2009 

[DeCooman09] G. de Cooman, F. Hermans and E. Quaeghebeur, Imprecise Markov chains and their 

limit behavior, Tech. Report, Universiteit Gent, 2009 

[Zhang10] J. Zhang and C. Liu, Dempster-Shafer inference with weak beliefs, Statistica Sinica, 2010 

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