Appearance of the imprecision in uncertainty theories

The interpretation of probability is at the heart of the recent incorporation of imprecision in uncertainty. Two schools are competing constructively: the subjectivists and the objectivists, frequentists. This competition is constructive since people exist who impartially study both interpretations, like James Berger, Frank Coolen, Glenn Shafer or Didier Dubois and advancements in one school can be extended to the other. The frequentist interpretation of probability is based on the World, whose complexity is infinite. The assessment of probability depends on an infinite number of parameters, which can be more or less identified. The subjectivist interpretation of probability is based on people mind or behavior, whose complexity is infinite. The assessment of probability depends on an infinite number of parameters, which can hardly be identified.

The parameters or assessment rules are fuzzy or invisible when the human brain is considered. Perhaps attempts of simplifying our reasonning like the Critique of Pure Reason of Kant (I did not fully read it) could help subjectivists to model the uncertainty assessment rules of our reasoning. As far as the frequentist interpretation is concerned, the assessment rules are better identified, they are often visible. They are facts that rule the world.

So, it was harder to accept that there is imprecision in the objective interpretation of probability, since we have the illusion that we know the facts and the rules to assess an objective probability. Whereas in the subjective interpretation of probability, it is easier to admit ignorance: we are aware of the defects of our reasoning. Actually, it is easier to accept imperfection in the subjective knowledge than in the objective knowledge. In the latter, incompleteness (due to a lack of data) can be advanced, but it was to formalize and to incorporate in a general imprecise theory of uncertainty. Many attempts for taking incompleteness in the frequentist interpretation of probability havve been made but it was made on a case by case basis. It is interesting to note that it is the (to my opinion) well founded subjective theory of De Finetti that permitted an easy (sorry Mr Walley) incorporation of imprecision in uncertainty theories.

This remark is an attempt to explain the recent development of the imprecision in probability. History is also interesting. Kolmogorov, in 1933, exposed the objective measure theoretic framework for probabilty. The Bruno De Finetti subjective theory of probability was a fundamental step in the history of uncertainty theories. It took out probability from the measure-theoretic framework and exposed a new purely and radical subjective interpretation of probability. A probability of an event is the price that the assessor is ready to pay for a bet on this event. His work inspired the works of Peter Walley who introduced the imprecision in the precise probability theory of De Finetti.

Imprecision and objectivism

In my initial aim of imprecise sensor modeling, a justification of the introduction of the imprecision in objective probability seems necessary, since all the techniques involved are based on observations of the sensor behaviors: this is the sensor calibration. A sketch of justification is that we lack information about the sensor behavior. Precise probability theory is often inadequate in cases where insufficient information is available to identify a unique probability distribution. In that case, imprecise probabilities aim to represent and manipulate the really available knowledge about the system.

Another point is that the precise frequentist model links the probability to the World with a weakness: only very small probability can be interpreted in the World. Indeed you can say that if you try an experiment once, the small probability event will impossibly occur, but as for the other probability magnitudes (i.e. neither close to 0 nor to 1), they can just be interpreted as frequency limits, then with a huge number of repetitions of the experiment. As far as the imprecise probability interpretation is concerned, the upper probability can be interpreted as the maximum frequency of occurrence of the event, that we could have with n repetitions of the experiment and the lower probability as the minimum frequency of occurrence of the event, with n repetitions of the experiment. The probability is no more interpreted as a limit, but as bounds.

Imprecise models

The term imprecision actually covers a very wide range of extensions of the classical theory of probability. To sum just a few, they include

• lower and upper previsions (Walley, 1991)
• belief functions (Dempster, 1967; Shafer, 1976), theory of hints (Kohlas and Monney, 1995), transferable belief model (Smets, 1992)
• possibility measures (Dubois, 1985, 1988)
• non-additive measures (Denneberg, 1994)
• credal sets and sets of probabilities and utilities (Levi, 1980)
• risk measures (Artzner et al., 1999)
• 2- and n-monotone set functions, Choquet capacities (Choquet, 1953)
• comparative probability orderings (Keynes, 1921; De Finetti, 1931; Fine, 1973; also see overview by Fishburn, 1986)
• robust Bayes methods (Berger, 1984)
• sets of desirable gambles (Walley, 1991)
• p-boxes (Ferson et al., 2003)
• lower and upper envelopes/collectives (Papamarcou and Fine, 1991)
• interval probability (Weichselberger 2000, 2001)
• capacities (Huber, 1965; Huber and Strassen, 1973)
• ambiguity (Ellsberg, 1961)
• logical/fiducial probabilities (Hampel, 1993; Weichselberger, 2005)
• linear partial information (Kofler and Menges, 1976)
• multiple priors (Gilboa, 1989)
• partial identification of probability distributions (Manski, 2003)

The main question that had to be answered by this survey was: which model should we use as an alternative to the summative kernels, taking imprecision into account? Our answer is Possibility theory. Actually maxitive kernels are possibility distributions.