Imprecise probability

Imprecise probability generalizes probability theory to allow for partial probability specifications, and is applicable when information is scarce, vague, or conflicting, in which case a unique probability distribution may be hard to identify. Thereby, the theory aims to represent the available knowledge more accurately. Imprecision is useful for dealing with expert elicitation, because:

• People have a limited ability to determine their own subjective probabilities and might find that they can only provide an interval.
• As an interval is compatible with a range of opinions, the analysis ought to be more convincing to a range of different people.

Introduction

Uncertainty is traditionally modelled by a

Perhaps the most common generalization is to replace a single probability specification with an interval specification. Lower and upper probabilities, denoted by ${\displaystyle {\underline {P}}(A)}$ and ${\displaystyle {\overline {P}}(A)}$, or more generally, lower and upper expectations (previsions),^[4][5][6][7] aim to fill this gap. A lower probability function is superadditive but not necessarily additive, whereas an upper probability is subadditive. To get a general understanding of the theory, consider:

• the special case with ${\displaystyle {\underline {P}}(A)={\overline {P}}(A)}$ for all events ${\displaystyle A}$ is equivalent to a precise probability
• ${\displaystyle {\underline {P}}(A)=0}$ and ${\displaystyle {\overline {P}}(A)=1}$ for all non-trivial events represents no constraint at all on the specification of ${\displaystyle P(A)}$

We then have a flexible continuum of more or less precise models in between.

Some approaches, summarized under the name nonadditive probabilities,^[8] directly use one of these set functions, assuming the other one to be naturally defined such that ${\displaystyle {\underline {P}}(A^{c})=1-{\overline {P}}(A)}$, with ${\displaystyle A^{c}}$ the complement of ${\displaystyle A}$. Other related concepts understand the corresponding intervals ${\displaystyle [{\underline {P}}(A),{\overline {P}}(A)]}$ for all events as the basic entity.^[9][10]

History

The idea to use imprecise probability has a long history. The first formal treatment dates back at least to the middle of the nineteenth century, by George Boole,^[3] who aimed to reconcile the theories of logic and probability. In the 1920s, in A Treatise on Probability, Keynes^[11] formulated and applied an explicit interval estimate approach to probability. Work on imprecise probability models proceeded fitfully throughout the 20th century, with important contributions by

Standard consistency conditions relate upper and lower probability assignments to non-empty closed convex sets of probability distributions. Therefore, as a welcome by-product, the theory also provides a formal framework for models used in

The term "imprecise probability" is somewhat misleading in that precision is often mistaken for accuracy, whereas an imprecise representation may be more accurate than a spuriously precise representation. In any case, the term appears to have become established in the 1990s, and covers a wide range of extensions of the theory of probability, including:

• credal sets, or sets of probability distributions
• previsions^[2]
• Random set theory
• Dempster–Shafer evidence theory
• lower and upper probabilities, or interval probabilities^[3][9][11]
• belief functions^[18][19]
• possibility and necessity measures^[22][23][24]
• lower and upper previsions^{[5][6][7][25]}
• comparative probability orderings^{[11][26][27][28]}
• partial preference orderings
• sets of desirable gambles^[5][6][7]
• p-boxes^[29]
• robust Bayes methods^[30]

A more practical issue is what kind of decision theory can make use of imprecise probabilities.[31] For fuzzy measures, there is the work of Ronald R. Yager.^[32] For convex sets of distributions, Levi's works are instructive.^[33] Another approach asks whether the threshold controlling the boldness of the interval matters more to a decision than simply taking the average or using a Hurwicz decision rule.^[34] Other approaches appear in the literature.^{[35][36][37][38]}

See also

• Ambiguity aversion
• Robust decision making
• Imprecise Dirichlet process

References