775:. In simple terms, a decision maker's lower prevision is the highest price at which the decision maker is sure he or she would buy a gamble, and the upper prevision is the lowest price at which the decision maker is sure he or she would buy the opposite of the gamble (which is equivalent to selling the original gamble). If the upper and lower previsions are equal, then they jointly represent the decision maker's
78:, 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
816:, or threshold of acceptance. This is not as much of a problem for intervals that are lower and upper bounds derived from a set of probability distributions, e.g., a set of priors followed by conditionalization on each member of the set. However, it can lead to the question why some distributions are included in the set of priors and some are not.
819:
Another issue is why one can be precise about two numbers, a lower bound and an upper bound, rather than a single number, a point probability. This issue may be merely rhetorical, as the robustness of a model with intervals is inherently greater than that of a model with point-valued probabilities.
688:
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
643:
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
811:
One issue with imprecise probabilities is that there is often an independent degree of caution or boldness inherent in the use of one interval, rather than a wider or narrower one. This may be a degree of confidence, degree of
827:. For convex sets of distributions, Levi's works are instructive. 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
782:
The allowance for imprecision, or a gap between a decision maker's upper and lower previsions, is the primary difference between precise and imprecise probability theories. Such gaps arise naturally in
449:
229:
763:
A unification of many of the above-mentioned imprecise probability theories was proposed by Walley, although this is in no way the first attempt to formalize imprecise probabilities. In terms of
588:
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
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for the gamble, the price at which the decision maker is willing to take either side of the gamble. The existence of a fair price leads to precise probabilities.
636:. Walley's theory extends the traditional subjective probability theory via buying and selling prices for gambles, whereas Weichselberger's approach generalizes
632:(which is also where the term "imprecise probability" originates). The 1990s also saw important works by Kuznetsov, and by Weichselberger, who both use the term
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249:
560:
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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
1849:
929:
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of probability. Walley defines upper and lower probabilities as special cases of upper and lower previsions and the gambling framework advanced by
30:
may be hard to identify. Thereby, the theory aims to represent the available knowledge more accurately. Imprecision is useful for dealing with
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1512:
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26:
to allow for partial probability specifications, and is applicable when information is scarce, vague, or conflicting, in which case a unique
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A more practical issue is what kind of decision theory can make use of imprecise probabilities. For fuzzy measures, there is the work of
2007:
1796:
1385:
38:
People have a limited ability to determine their own subjective probabilities and might find that they can only provide an interval.
169:
but not necessarily additive, whereas an upper probability is subadditive. To get a general understanding of the theory, consider:
383:
1104:
Elementare
Grundbegriffe einer allgemeineren Wahrscheinlichkeitsrechnung I - Intervallwahrscheinlichkeit als umfassendes Konzept
41:
As an interval is compatible with a range of opinions, the analysis ought to be more convincing to a range of different people.
176:
86:'s work—, or when we wish to model probabilities that a group agrees with, rather than those of a single individual.
718:
90:
1884:
Caselton, W. F.; Luo, W. (1992). "Decision making with imprecise probabilities: DempsterâShafer theory and application".
1139:
89:
Perhaps the most common generalization is to replace a single probability specification with an interval specification.
165:, or more generally, lower and upper expectations (previsions), aim to fill this gap. A lower probability function is
2002:
1140:"Imprecise Probabilities > Historical appendix: Theories of imprecise belief (Stanford Encyclopedia of Philosophy)"
850:
764:
593:
713:
665:
1383:
de Cooman, G.; Hermans, F. (2008). "Imprecise probability trees: Bridging two theories of imprecise probability".
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An investigation of the laws of thought on which are founded the mathematical theories of logic and probabilities
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It does raise concerns about inappropriate claims of precision at endpoints, as well as for point values.
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59:
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GĂ€rdenfors, P.; Sahlin, N. E. (1982). "Unreliable probabilities, risk taking, and decision making".
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also give other reasons for intervals, or sets of distributions, representing states of belief.
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624:. At the start of the 1990s, the field started to gather some momentum, with the publication of
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Breese, J. S.; Fertig, K. W. (2013). "Decision making with interval influence diagrams".
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de Cooman, G.; Troffaes, M. C. M.; Miranda, E. (2008). "n-Monotone exact functionals".
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for all non-trivial events represents no constraint at all on the specification of
79:
1692:
Berger, James O. (1984). "The robust
Bayesian viewpoint". In Kadane, J. B. (ed.).
1408:
927:
Smith, Cedric A. B. (1961). "Consistency in statistical inference and decision".
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We then have a flexible continuum of more or less precise models in between.
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1978:
Open source implementation of a classifier based on
Imprecise Probabilities
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Guo, P.; Tanaka, H. (2010). "Decision making with interval probabilities".
1760:
Yager, R. R. (1978). "Fuzzy decision making including unequal objectives".
1977:
1537:
1503:
Possibility Theory - An
Approach to Computerized Processing of Uncertainty
1445:
Zadeh, L. A. (1978). "Fuzzy sets as a basis for a theory of possibility".
1068:"The theory of interval probability as a unifying concept for uncertainty"
1934:
911:
788:
1725:
Seidenfeld, Teddy (1983). "Decisions with indeterminate probabilities".
1947:
1825:
1337:
1988:
Stanford
Encyclopedia of Philosophy article on Imprecise Probabilities
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1917:
1812:
Loui, R. P. (1986). "Decisions with indeterminate probabilities".
1399:
1233:
767:, Walley's formulation of imprecise probabilities is based on the
1973:
The
Society for Imprecise Probability: Theories and Applications
1312:"Upper and lower probabilities induced by a multivalued mapping"
498:. Other related concepts understand the corresponding intervals
1664:"Constructing Probability Boxes and Dempster-Shafer Structures"
1185:"Nonparametric predictive inference and interval probability"
444:{\displaystyle {\underline {P}}(A^{c})=1-{\overline {P}}(A)}
380:, assuming the other one to be naturally defined such that
1274:"Minimax tests and the Neyman-Pearson lemma for capacities"
831:
decision rule. Other approaches appear in the literature.
799:
repeatedly for his interval probabilities, though he and
1972:
719:
lower and upper probabilities, or interval probabilities
1789:
Hard choices: Decision making under unresolved conflict
1982:
224:{\displaystyle {\underline {P}}(A)={\overline {P}}(A)}
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1662:; Lev Ginzburg; David S. Myers; Kari Sentz (2003).
1597:
1528:Troffaes, Matthias C. M.; de Cooman, Gert (2014).
1500:
1355:
1021:Statistical Reasoning with Imprecise Probabilities
1018:
957:. School of Math. and Phys. Sci., Univ. of Sussex.
656:, and so-called two-monotone and totally monotone
630:Statistical Reasoning with Imprecise Probabilities
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490:
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1220:Journal of Mathematical Analysis and Applications
769:subjective variant of the Bayesian interpretation
1305:
1303:
885:
883:
1670:. Sandia National Laboratories. Archived from
1563:"Sul significato soggettivo della probabilitĂ "
1072:International Journal of Approximate Reasoning
980:International Journal of Approximate Reasoning
968:
966:
964:
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944:
640:'s axioms without imposing an interpretation.
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1192:Journal of Statistical Planning and Inference
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8:
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1004:
668:. Moreover, there is a strong connection to
905:
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372:Some approaches, summarized under the name
50:Uncertainty is traditionally modelled by a
1432:Probability and Finance: It's Only a Game!
1916:
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759:Interpretation of imprecise probabilities
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1983:The imprecise probability group at IDSIA
1850:European Journal of Operational Research
930:Journal of the Royal Statistical Society
873:Foundations of the Theory of Probability
1183:Augustin, T.; Coolen, F. P. A. (2004).
876:. New York: Chelsea Publishing Company.
862:
16:Probability theory for low quality data
1627:"The axioms of subjective probability"
700:, or sets of probability distributions
652:. Included are also concepts based on
251:is equivalent to a precise probability
1430:Shafer, Glenn; Vladimir Vovk (2001).
1316:The Annals of Mathematical Statistics
288:{\displaystyle {\underline {P}}(A)=0}
7:
1499:Dubois, Didier; Henri Prade (1988).
1484:Dubois, Didier; Henri Prade (1985).
1172:. D. RĂos Insua. New York: Springer.
660:, which have become very popular in
562:for all events as the basic entity.
330:{\displaystyle {\overline {P}}(A)=1}
807:Issues with imprecise probabilities
122:{\displaystyle {\underline {P}}(A)}
1272:Huber, P. J.; V. Strassen (1973).
729:possibility and necessity measures
666:(DempsterâShafer) belief functions
578:and probability. In the 1920s, in
158:{\displaystyle {\overline {P}}(A)}
14:
1358:A Mathematical Theory of Evidence
1048:Non-additive Measure and Integral
975:"Notes on conditional previsions"
738:comparative probability orderings
1696:Robustness of Bayesian Analyses
1153:Kuznetsov, Vladimir P. (1991).
955:Notes on conditional previsions
714:DempsterâShafer evidence theory
689:of extensions of the theory of
1791:. Cambridge University Press.
1362:. Princeton University Press.
787:that happen to be financially
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54:distribution, as developed by
1:
1727:Behavioral and Brain Sciences
1700:. Elsevier Science. pp.
1157:. Moscow: Radio i Svyaz Publ.
1122:Keynes, John Maynard (1921).
1085:10.1016/S0888-613X(00)00032-3
1066:Weichselberger, Kurt (2000).
916:. London: Walton and Maberly.
795:. This gap is also given by
91:Lower and upper probabilities
1774:10.1016/0165-0114(78)90010-6
1604:. New York: Academic Press.
1461:10.1016/0165-0114(78)90029-5
1409:10.1016/j.artint.2008.03.001
1025:. London: Chapman and Hall.
741:partial preference orderings
535:
427:
376:, directly use one of these
307:
207:
141:
1155:Interval Statistical Models
1128:. London: Macmillan And Co.
1102:Weichselberger, K. (2001).
973:Williams, Peter M. (2007).
953:Williams, Peter M. (1975).
851:Imprecise Dirichlet process
765:probability interpretations
2024:
2008:Statistical approximations
1863:10.1016/j.ejor.2009.07.020
1596:Fine, Terrence L. (1973).
1561:de Finetti, Bruno (1931).
1507:. New York: Plenum Press.
1251:10.1016/j.jmaa.2008.05.071
1204:10.1016/j.jspi.2003.07.003
1168:Ruggeri, Fabrizio (2000).
1046:Denneberg, Dieter (1994).
994:10.1016/j.ijar.2006.07.019
890:de Finetti, Bruno (1974).
870:Kolmogorov, A. N. (1950).
734:lower and upper previsions
678:game-theoretic probability
1739:10.1017/S0140525X0001582X
1125:A Treatise on Probability
744:sets of desirable gambles
650:non-parametric statistics
581:A Treatise on Probability
374:nonadditive probabilities
1886:Water Resources Research
1625:Fishburn, P. C. (1986).
1310:Dempster, A. P. (1967).
1279:The Annals of Statistics
1170:Robust Bayesian Analysis
28:probability distribution
1600:Theories of Probability
1582:10.4064/fm-17-1-298-329
1568:Fundamenta Mathematicae
1486:Théorie des possibilité
1386:Artificial Intelligence
1329:10.1214/aoms/1177698950
662:artificial intelligence
1762:Fuzzy Sets and Systems
1448:Fuzzy Sets and Systems
1354:Shafer, Glenn (1976).
1293:10.1214/aos/1176342363
1106:. Heidelberg: Physica.
1017:Walley, Peter (1991).
910:Boole, George (1854).
846:Robust decision making
793:asymmetric information
556:
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173:the special case with
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1644:10.1214/ss/1177013611
1538:10.1002/9781118762622
892:Theory of Probability
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471:{\displaystyle A^{c}}
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20:Imprecise probability
1050:. Dordrecht: Kluwer.
753:robust Bayes methods
634:interval probability
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384:
359:{\displaystyle P(A)}
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1814:Theory and Decision
1631:Statistical Science
1243:2008JMAA..347..143D
684:Mathematical models
654:Choquet integration
2003:Probability theory
1948:10.1007/BF00486156
1826:10.1007/BF00134099
1470:10338.dmlcz/135193
894:. New York: Wiley.
841:Ambiguity aversion
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478:the complement of
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32:expert elicitation
24:probability theory
1898:10.1029/92WR01818
1892:(12): 3071â3083.
1787:Levi, I. (1990).
1711:978-0-444-86209-9
1660:Vladik Kreinovich
1611:978-0-12-256450-5
1547:978-0-470-72377-7
1514:978-0-306-42520-2
1393:(11): 1400â1427.
1369:978-0-691-08175-5
1032:978-0-412-28660-5
709:Random set theory
646:robust statistics
610:Peter M. Williams
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638:Kolmogorov
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