Imprecise Probability - Introduction

Introduction

Quantification of uncertainty is often done using a probability distribution, usually satisfying Kolmogorov's axioms (or a variation thereof), as argued by Laplace, de Finetti, Ramsey, Cox, Lindley, and many others. However, this has not been unanimously accepted by scientists, statisticians, and probabilists: it has been argued that some modification or broadening of probability theory is required, because one may not always be able to provide a probability for every event, particularly when only little information or data is available—an early example of such criticism is Boole's critique of Laplace's work—, or when we wish to model probabilities that a group agrees with, rather than those of a single individual.

Perhaps the most straightforward generalization is to replace a single probability specification with an interval specification. Lower and upper probabilities, denoted by and, or more generally, lower and upper expectations (previsions), aim to fill this gap:

• the special case with for all events provides precise probability, whilst
• and represents no constraint at all on the specification of ,

with a flexible continuum in between.

Some approaches, summarized under the name nonadditive probabilities, directly use one of these set functions, assuming the other one to be naturally defined such that, with the complement of . Other related concepts understand the corresponding intervals for all events as the basic entity.