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The Apparent Arbitrariness of Second-Order Probability Distributions
University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Mathematics. (matematik)
2011 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Adequate representation of imprecise probabilities is a crucial and non-trivial problem in decision analysis. Second-order probability distributions is the model for imprecise probabilitoes whose merits are discussed in this thesis.

That imprecise probabilities may be represented by second-order probability distributions is well known but there has been little attention to specific distributions. Since different probability distributions have different properties, the study of the desired properties of models of imprecise probabilities with respect to second-order models require analysis of particular second-order distributions.

An often held objection to second-order probabilities is the apparent arbitrarines in the choice of distribution. We find some evidence that the structure of second-order distributions is an important factor that prohibits arbitrary choice of distributions. In particular, the properties of two second-order distributions are investigated; the uniform joint distribution and a variant of the Dirichlet distribution that has the property of being the normalised product of its own marginal distributions.

The joint uniform distribution is in this thesis shown to have marginal distributions that belie the supposed non-informativeness of a uniform distribution. On the other hand, the modified Dirichlet distribution  discovered here has its information content evenly divided among the joint and marginal distributions in that the total correlation of the variables is minimal.

It is also argued in the thesis that discrete distributions, as opposed to the continuous distributions mentioned above, would have the advantage of providing a natural setting for updating of lower bounds, and computation of expected utility is made more efficient.

Abstract [la]

In placitorum scrutatione maxima et mehercle minime levis difficultas eo spectat, quomodo probabilitates dubiae bene ostendantur. In hac thesi de utilitate distributionum probabilitatum secundi ordinis disseremus, in quantum ad probabilitates dubias ostendendas valeant.

Omnibus fere notum est probabilitates dubias ostendi posse per distributiones probabilitatum secundi ordinis, sed pauci operam distributionibus singulis contulerunt. Cum tamen distributiones probabilitatum valde inter se diversae sint, si quis proprietatibus desideratis probabilitatum dubiarum secundi ordinis studium conferre vult, primum debet quasdam praescriptas distributiones secundi ordinis investigare.

Sed fortasse, quod saeponumero fieri solet, quispiam dixerit probabilitates secundi ordinis nulla, ut videtur, ratione habita quasi vagari quoad delectum distributionis. Nos tamen nonnulla indicia comperimus quibus freto confirmare audemus ipsam formam distributionum secundi ordinis multum valere ad praedictum distributionum secundi ordinis delectum rationabiliter peragendum. Imprimus proprietates duarum distributionum secundi ordinis investigabimus, nimirum distributionis uniformis coniunctae et alterius cuisdam speciei distributionis quae `Dirichleti'vocatur, quae ex ipsius distributionibusnmarginalibus ad normam correcta oritur.

In hac thesi probamus illam coniunctam uniformem distributionem continere distributiones marginales eius modi quae illos refellant qui negant distributionem uniformem quicquam alicuius moment afferre. Attamen in illa distributione Dirichleti paulo mutata, quam hoc loco patefacimus, omnia aequaliter inter coniunctas et marginales distributiones divisa sunt, in quantum tota ratio quae inter variantia intercessit ad minimum reducitur.

Insuper in hac thesi confirmamus distributiones discretas potius quam antedictas distributiones continuas in hoc utiliores esse, quod per eas limiets inferiores in melius mutare licet, et beneficia exsepectata accuratius computari possunt.

Abstract [sv]

Adekvat representation av osäkra eller imprecisa sannolikheter är ett avgörande och icke-trivialt problem i beslutsanalys. I denna avhandilng diskuteras förtjänsterna hos andra ordningens sannolikheter som en modell för imprecisa sannolikheter.

Att imprecisa sannolikheter kan representeras med andra ordningens sannolikheter ä välkänt, men hittills har särskilda andra ordningens fördelningarinte ägnats någon större uppmärksamhet. Då olika sannolikhetsfördelningar har olika egenskaper kräver studiet av önskvärda egenskaper hos modeller för imprecisa sannolikheter en granskning av specifika andra ordningens fördelningar.

Den godtycklighet som tycks vidhäfta valet av andra ordningens sannolikhetsfördelningar är en ofta förekommande invändning mot andra ordningens sannolikhetsfördelningar. Vi finner vissa belägg för att strukturen hos andra ordningens fördelningar är en omständighet som hindrar godtyckligt val av fördelningar. I synnerhet undersöks egenskaper hos två andra ordningens fördelningar; den likformiga simultana fördelningen och en variant av Dirichletfördelningen med egenskapen att vara lika med den normliserade produkten av sina egna marginalfördelningar.

Den likformiga simultana fördelningen visas i avhandlinegn ha marginalfördelningar som motsäger den förmodat icke-informativa strukturen hos en likformig fördelning. Å andra sidan gäller för den modifierade Dirichletfördelningen som upptäckts här att informationsinnehållet är jämnt fördelat mellan den simultana fördelningen och marginalfördelningarna; den totala korrelationen mellan variablerna är minimal.

Det hävdas också i avhandlingen att diskreta sannolikhetsfördelningar i motsats till de kontinuerliga fördelningar som nämnts ovan har fördelen att utgöra en naturlig miljö för uppdatering av undre gränser och dessutom tillåta en mer effektiv beräkning av förväntad nytta.

Place, publisher, year, edition, pages
Stockholm: Department of Computer and Systems Sciences at Stockholm University , 2011. , p. 49
Series
Report Series / Department of Computer & Systems Sciences, ISSN 1101-8526 ; 11-002
National Category
Information Systems
Identifiers
URN: urn:nbn:se:hig:diva-8535ISBN: 978-91-7447-184-7 (print)OAI: oai:DiVA.org:hig-8535DiVA, id: diva2:405744
Public defence
2011-03-18, Sal C, Institutionen för Data- och Systemvetenskap, Forum, Isafjordsgatan 39, Kista, 13:00 (English)
Opponent
Supervisors
Available from: 2011-04-06 Created: 2011-03-04 Last updated: 2018-03-13Bibliographically approved
List of papers
1. Cross-disciplinary research in analytic decision support systems
Open this publication in new window or tab >>Cross-disciplinary research in analytic decision support systems
Show others...
2006 (English)In: ITI 2006: Proceedings of the 28th International Conference on Information Technology Interfaces, Zagreb: University Computing Centre SRCE, University of Zagreb , 2006, p. 123-128Conference paper, Published paper (Refereed)
Abstract [en]

A main problem in nearly all contexts is that unguided decision making is tremendously difficult and can lead to inefficient decision processes and undesired consequences. Therefore, decision support systems (DSSs) are of prime concern to any organization and there have been numerous approaches to such from, e.g., computational, mathematical, financial, philosophical, psychological, and sociological angles. However, a key observation is that efficient decision making is not easily performed by using methods from one discipline only. The case is rather that if real world decision making is taken seriously, several aspects must be included. This article describes some efforts of the DECIDE research group for approaching decision making and developing DSSs in a cross-disciplinary environment.

Place, publisher, year, edition, pages
Zagreb: University Computing Centre SRCE, University of Zagreb, 2006
National Category
Computer and Information Sciences
Identifiers
urn:nbn:se:hig:diva-2454 (URN)000239576000023 ()953-7138-05-4 (ISBN)
Conference
28th International Conference on Information Technology Interfaces, Cavtat, Croatia, Jun 19-22, 2006
Available from: 2007-04-30 Created: 2007-04-30 Last updated: 2018-03-13Bibliographically approved
2. Structure information in decision trees and similar formalisms
Open this publication in new window or tab >>Structure information in decision trees and similar formalisms
2007 (English)In: Proceedings of the Twentieth International Florida Artificial Intelligence Research Society Conference, Menlo Park, CA: AAAI Press , 2007, p. 62-67Conference paper, Published paper (Refereed)
Abstract [en]

In attempting to address real-life decision problems, where uncertainty about input data prevails, some kind of representation of imprecise information is important and several have been proposed over the years. In particular, first-order representations of imprecision, such as sets of probability measures, upper and lower probabilities, and interval probabilities and utilities of various kinds, have been suggested for enabling a better representation of the input sentences. A common problem is, however, that pure interval analyses in many cases cannot discriminate sufficiently between the various strategies under consideration, which, needless to say, is a substantial problem in real-life decision making in agents as well as decision support tools. This is one reason prohibiting a more wide-spread use. In this article we demonstrate that in many situations, the discrimination can be made much clearer by using information inherent in the decision structure. It is discussed using

second-order probabilities which, even when they are implicit, add information when handling aggregations of imprecise representations, as is the case in decision trees and probabilistic networks. The important conclusion is that since structure carries information, the structure of the decision problem influences evaluations of all interval representations and is quantifiable.

Place, publisher, year, edition, pages
Menlo Park, CA: AAAI Press, 2007
Identifiers
urn:nbn:se:hig:diva-2476 (URN)978-1-57735-319-5 (ISBN)
Available from: 2007-05-15 Created: 2007-05-15 Last updated: 2018-03-13Bibliographically approved
3. Warp effects on calculating interval probabilities
Open this publication in new window or tab >>Warp effects on calculating interval probabilities
2009 (English)In: International Journal of Approximate Reasoning, ISSN 0888-613X, E-ISSN 1873-4731, Vol. 50, no 9, p. 1360-1368Article in journal (Refereed) Published
Abstract [en]

In real-life decision analysis, the probabilities and utilities of consequences are in general vague and imprecise. One way to model imprecise probabilities is to represent a probability with the interval between the lowest possible and the highest possible probability, respectively. However, there are disadvantages with this approach; one being that when an event has several possible outcomes, the distributions of belief in the different probabilities are heavily concentrated toward their centres of mass, meaning that much of the information of the original intervals are lost. Representing an imprecise probability with the distribution’s centre of mass therefore in practice gives much the same result as using an interval, but a single number instead of an interval is computationally easier and avoids problems such as overlapping intervals. We demonstrate why second-order calculations add information when handling imprecise representations, as is the case of decision trees or probabilistic networks. We suggest a measure of belief density for such intervals. We also discuss properties applicable to general distributions. The results herein apply also to approaches which do not explicitly deal with second-order distributions, instead using only first-order concepts such as upper and lower bounds.

Place, publisher, year, edition, pages
Elsevier B.V., 2009
Keyword
Decision analysis, Probability, Intervals, Second-order distributions
National Category
Mathematics
Identifiers
urn:nbn:se:hig:diva-5370 (URN)10.1016/j.ijar.2009.04.008 (DOI)000272341300004 ()2-s2.0-70350567791 (Scopus ID)
Available from: 2009-09-08 Created: 2009-09-08 Last updated: 2018-03-13Bibliographically approved
4. Some properties of aggregated distributions over expected values
Open this publication in new window or tab >>Some properties of aggregated distributions over expected values
2008 (English)In: MICAI 2008: Advances in Artificial Intelligence: 7th Mexican International Conference on Artificial Intelligence, Atizapán de Zaragoza, Mexico, October 27-31, 2008, Proceedings, Berlin, Heidelberg: Springer , 2008, p. 699-709Conference paper, Published paper (Refereed)
Abstract [en]

Software agents and humans alike face severe difficulties in making decisions in uncertain contexts. One approach is to formalise the decision situation by means of decision theory, i.e. probabilities and utilities leading to the principle of maximising the expected utility. Expected utility is here considered as a stochastic variable; under the assumption that all utility values are equally likely, and that each vector of probability values is equally likely, the probability distribution of expected utility is calculated for two, three, and four possible outcomes. The effect of these probability distributions concentrating around the middle value is explored and its significance for making decisions.

Place, publisher, year, edition, pages
Berlin, Heidelberg: Springer, 2008
Series
Lecture Notes in Computer Science, ISSN 1611-3349 (Online) ; 5317
National Category
Computer Sciences
Identifiers
urn:nbn:se:hig:diva-2250 (URN)10.1007/978-3-540-88636-5_66 (DOI)000261873400066 ()978-3-540-88635-8 (ISBN)
Available from: 2008-11-06 Created: 2008-11-06 Last updated: 2018-03-13Bibliographically approved
5. Shifted Dirichlet Distributions as Second-Order Probability Distributions that Factors into Marginals
Open this publication in new window or tab >>Shifted Dirichlet Distributions as Second-Order Probability Distributions that Factors into Marginals
2009 (English)In: Proceedings of the Sixth International Symposium on Imprecise Probability: Theories and Applications, 2009, p. 405-410Conference paper, Published paper (Refereed)
Abstract [en]

In classic decision theory it is assumed that a decision-maker can assign precise numerical values corresponding to the true value of each consequence, as well as precise numerical probabilities for their occurrences. In attempting to address real-life problems, where uncertainty in the input data prevails, some kind of representation of imprecise information is important. Second-order distributions, probability distributions over probabilities, is one way to achieve such a representation. However, it is hard to intuitively understand statements in a multi-dimensional space and user statements must be provided more locally. But the information-theoretic interplay between joint and marginal distributions may give rise to unwanted effects on the global level. We consider this problem in a setting of second-order probability distributions and find a family of distributions that normalised over the probability simplex equals its own product of marginals. For such distributions, there is no flow of information between the joint distributions and the marginal distributions other than the trivial fact that the variables belong to the probability simplex. marginal distributions may give rise to unwanted effects on the global level.

National Category
Mathematics
Identifiers
urn:nbn:se:hig:diva-5356 (URN)000280248700042 ()
Conference
Sixth International Symposium on Imprecise Probability: Theories and Applications, ISIPTA '09, Durham, 14-18 July 2009
Available from: 2009-09-08 Created: 2009-09-08 Last updated: 2018-03-13Bibliographically approved
6. Expected Utility from Multinomial Second-order Probability Distributions
Open this publication in new window or tab >>Expected Utility from Multinomial Second-order Probability Distributions
2010 (English)In: Polibits, ISSN 1870-9044, no 42, p. 71-75Article in journal (Refereed) Published
Abstract [en]

We consider the problem of maximizing expected utility when utilities and probabilities are given by discrete probability dis- tributions so that expected utility is a discrete stochastic variable. As for discrete second-order distributions, that is probability distributions where the variables are themselves probabilities, the multinomial family is a reasonable choice at least if first-order probabilities are interpreted as relative frequencies. We suggest a decision rule that reflects the uncertainty present in distribution-based probabilities and utilities and we show an example of this rule in action with multinomial second-order distributions.

 

Place, publisher, year, edition, pages
Mexico City: Centro de Innovacíon y Desarrollo Tecnológico en Cómputo, Instituto Politécnico Nacional, 2010
Keyword
Imprecise probability. second-order probability, discrete probability distributions, multinomial distributions, expected utilty.
National Category
Computer Sciences
Identifiers
urn:nbn:se:hig:diva-7970 (URN)
Available from: 2010-11-12 Created: 2010-11-12 Last updated: 2018-03-13Bibliographically approved

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