It has been argued that imprecise probabilities are a natural and intuitive way of overcoming some of the issues with orthodox precise probabilities. Models of this type have a long pedigree, and interest in such models has been growing in recent years. This article introduces the theory of imprecise probabilities, discusses the motivations for their use and their possible advantages over the standard precise model. It then discusses some philosophical issues raised by this model. There is also a historical appendix which provides an overview of some important thinkers who appear sympathetic to imprecise probabilities.2. Motivations
2.1 Ellsberg decisions
2.2 Incompleteness and incomparability
2.3 Weight of evidence, balance of evidence
2.4 Suspending judgement
2.5 Unknown correlations
2.6 Nonprobabilistic chances
2.7 Group belief
3. Philosophical questions for IP
3.1 Dilation
3.2 Belief inertia
3.3 Decision making
3.4 Interpreting IP
3.4.1 What is a belief?
3.4.2 What is a belief in [ltr]X[/ltr]?
3.5 Regress
3.6 What makes a good imprecise belief?
4. Summary
Bibliography
Academic Tools
Other Internet Resources
Related Entries
[size=30]1. Introduction
Probability theory has been a remarkably fruitful theory, with applications in almost every branch of science. In philosophy, some important applications of probability theory go by the name Bayesianism; this has been an extremely successful program (see for example Howson and Urbach 2006; Bovens and Hartmann 2003; Talbott 2008). But probability theory seems to impute much richer and more determinate attitudes than seems warranted. What should your rational degree of belief be that global mean surface temperature will have risen by more than four degrees by 2080? Perhaps it should be 0.75? Why not 0.75001? Why not 0.7497? Is that event more or less likely than getting at least one head on two tosses of a fair coin? It seems there are many events about which we can (or perhaps should) take less precise attitudes than orthodox probability requires. Among the reasons to question the orthodoxy, it seems that the insistence that states of belief be represented by a single real-valued probability function is quite an unrealistic idealisation, and one that brings with it some rather awkward consequences that we shall discuss later. Indeed, it has long been recognised that probability theory offers only a rather idealised model of belief. As far back as the mid-nineteenth century, we find George Boole saying:
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- It would be unphilosophical to affirm that the strength of that expectation, viewed as an emotion of the mind, is capable of being referred to any numerical standard. (Boole 1958 [1854]: 244)
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For these, and many other reasons, there is growing interest in
Imprecise Probability (IP) models. Broadly construed, these are models of belief that go beyond the probabilistic orthodoxy in one way or another.
IP models are used in a number of fields including:
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- Statistics (Walley 1991; Augustin et al. 2014)
- Psychology of reasoning (Pfeifer and Kleiter 2007)
- Linguistic processing of uncertainty (Wallsten and Budescu 1995)
- Neurological response to ambiguity and conflict (Smithson and Pushkarskaya forthcoming)
- Philosophy (Levi 1980; Joyce 2011; Sturgeon 2008; Kaplan 1983; Kyburg 1983)
- Behavioural economics (Ellsberg 1961; Camerer and Weber 1992; Smithson and Campbell 2009)
- Mathematical economics (Gilboa 1987)
- Engineering (de Cooman and Troffaes 2004; Oberguggenberger 2014)
- Computer science (Cozman 2000; Cozman and Walley 2005)
- Physics (Suppes and Zanotti 1991; Hartmann and Suppes 2010; Frigg et al. 2014)
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This article identifies a variety of motivations for IP models; introduces various formal models that are broadly in this area; and discusses some open problems for these frameworks. The focus will be on formal models of belie[/size]
Imprecise Probabilities
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