Research Interests
My main research interests concern the foundations of statistics and decision making under severe uncertainty, including:
- sequential decision processes: backward induction and dynamic programming
- uncertainty modelling: lower and upper previsions, set-valued choice functions, p-boxes, info-gap theory
- elicitation, representation, and aggregation of expert information
- optimal control
- bioinformatics, sequence alignment
Lower Previsions
In modelling a system, it often occurs that some of its aspects, or some of the influences acting on it, are not well known. The uncertainty this produces about the system's behaviour is usually modelled by a probability distribution, and treated using techniques from probability theory. Such a model will often not be adequate, simply because not enough information is available in order to identify a unique probability distribution. In that case, techniques from the theory of imprecise probabilities can be applied in order to represent and manipulate the really available knowledge about the system. The term imprecise probability theory actually covers a wide range of extensions of the classical theory of probability.
A behavioural approach to generalising probability theory, which is termed lower previsions and which can be seen as a generalisation of the work of de Finetti in the classical theory of probability, has been extensively studied and developed during the 1980's by Peter Walley, based on the work of Boole (1854), Smith (1961), Williams (1975), and many others. In Russia, Vladimir Kuznetsov developed, simultaneously and independently from Peter Walley, a theory which is mathematically very similar to Walley's lower previsions.
There are many reasons to prefer lower previsions above other well-known generalisations of probability theory (such as belief functions, possibility measures, fuzzy measures, credal sets, risk measures, Choquet capacities, comparative probability orderings, p-boxes, etc.). The most important reasons are:
- Mathematically, the theory has a unifying character: it unifies a large number of other generalisations of probability theory. Interestingly, imprecise probability also unifies the classical theory of probability with the classical theory of logic: see Boole's "An investigation of the laws of thought" (1854).
- The theory has a very clear behavioural interpretation in terms of buying (or, equivalently, selling) prices, similar to de Finetti's approach to probability theory.
- And last but not least, it leads naturally to a decision theory, essentially dropping Savage's completeness axiom, and whence admitting set-valued choice functions, which reflect more accurately how the available information leads to optimal choices (see discussion below).
There are of course also reasons not to prefer lower previsions. The most important drawbacks are:
- Computational complexity can become prohibitive. Indeed, computing with lower previsions essentially involves solving linear programming problems. Even though very large linear programming problems can be solved quite efficiently by computer, yet in many applications, especially when involving many variables, linear programming problems can simply become too large to solve practically.
- Lower previsions also require more complex mathematical tools, such as non-linear functionals and non-additive measures. This can be mitigated in part by considering particular models that yield simpler mathematical descriptions, possibly at the expense of generality, but gaining ease of use.
- Lower previsions rely heavily on the concept of a (precise) linear utility.
Set-Valued Choice Functions
One of the most profound results that arise from imprecise probability theory is found in its treating of optimality. It is intuitively clear that if only a scarce amount of information is available about a variable, then the optimal decision, whose gains and losses depend on that variable, cannot be completely determined.
Imprecise probability theory, and the theory of lower previsions in particular, grasps this aspect of the optimal decision problem in a rigorous manner, resulting in a set of possibly optimal decisions, that is, a set-valued choice function, rather than giving us, seemingly arbitrarily, only a single optimal decision from this set. From a decision making point of view, imprecise probability theory drops the completeness axiom. Imprecise probability theory is especially more desirable than probability theory in critical decision problems, that is, when gains and losses heavily depend on objects which are not completely known. One could say that imprecise probability theory allows for a more accurate description of our knowledge of reality. It therefore allows for a more accurate criterion of optimality too.
Challenges
However, greater generality usually results in greater complexity, and imprecise probability theory provides no exception. Whereas the theory of probability is essentially a sub-domain of linear mathematics (the expectation operator is a linear operator), imprecise probability theory has to rely on non-linear mathematics. Many mature concepts of probability theory, such as independence, characteristic functions, Markov processes, etc. do not carry over to imprecise probability theory in a straightforward manner.
Challenges in imprecise probability theory are (i) theoretical—the necessary tools need to be developed for describing and solving systems in which the amount of information is typically scarce; (ii) numerical—we must implement these theoretical tools in such a way that computers can solve real-life problems within a reasonable amount of time; and (iii) practical—imprecise probability theory needs to be applied in real-life problems, particularly where traditional methods are known to fail.
References
- George Boole. An investigation of the laws of thought on which are founded the mathematical theories of logic and probabilities. Walton and Maberly, London, 1854.
- Bruno de Finetti. Theory of Probability: A Critical Introductory Treatment. Wiley, New York, 1974-5. Two volumes.
- Cedric A. B. Smith. Consistency in statistical inference and decision. Journal of the Royal Statistical Society, B(23):1-37, 1961. [ http ]
- Peter M. Williams. Notes on conditional previsions. International Journal of Approximate Reasoning, 44(3):366-383, 2007. [ DOI ]
- Peter M. Williams. Notes on conditional previsions. Technical report, School of Math. and Phys. Sci., Univ. of Sussex, 1975.
- Vladimir P. Kuznetsov. Interval Statistical Models. Radio i Svyaz Publ., Moscow, 1991. In Russian.
- Peter Walley. Statistical Reasoning with Imprecise Probabilities. Chapman and Hall, London, 1991.