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

Cuzzolin Part B section1 (B1) VIP
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);
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