What Is Imprecision?

Uncertainty is usually modelled by a probability distribution, and treated using techniques from probability theory. Such an uncertainty model will often be 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.

Similar concerns arise when dealing with utility. In making decisions, each reward is assigned a single real number, called utility, and rewards are accordingly ranked. However, in many practical cases, a complete ranking over all rewards is unrealistic. Imprecise utility aims to represent and reason with such incomplete preferences over rewards.

The main benefits of using imprecise probabilities and imprecise utilities, compared to classical statistical methods, are

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

Recent Advances in Statistics Using Imprecision

Currently, a popular approach to statistics using imprecision, is by use of Walley's generalised Bayes rule, which is close in nature to the robust Bayesian approach where a set of priors is used, and each prior in the set is updated to produce a set of posteriors. A particularly successful model, be it not without its critics, is the imprecise Dirichlet model, which has been applied for example in game theory, classification, Markov decision processes, aggregation, etc. Other statistical methods with promising potential for application include the bounded derivative model, and nonparametric predictive inference.

In classical statistics, limit theorems play a crucial role. Recently, generalised versions of such theorems have been presented, opening new possibilities from a frequentist perspective.

Imprecise probabilities and utilities also lead the way to generalised decision support methods. For example, generalisations of maximising expected utility lead to interesting research challenges, the solutions to some of which have been presented and addressed. A common theme in all of these generalised methods, is that they focus on realistic reflection of available information and preferences, and as such support informative decisions. This also poses challenges for elicitation and data collection.

Aim Of The Special Issue

With this special issue on imprecision we hope to promote new and recent techniques that employ imprecise methods in a useful way, and advance them to a wider audience. We especially hope to demonstrate the benefits of imprecise models over traditional statistical methods. In particular we are looking for (but not exclusively):

References