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Shifted Dirichlet Distributions as Second-Order Probability Distributions that Factors into Marginals
University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för matematik och statistik. (Matematik)
2009 (English)In: Proceedings of the Sixth International Symposium on Imprecise Probability: Theories and Applications, 2009, 405-410 p.Conference 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.

Place, publisher, year, edition, pages
2009. 405-410 p.
National Category
Mathematics
Identifiers
URN: urn:nbn:se:hig:diva-5356ISI: 000280248700042OAI: oai:DiVA.org:hig-5356DiVA: diva2:234366
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: 2016-10-26Bibliographically approved
In thesis
1. The Apparent Arbitrariness of Second-Order Probability Distributions
Open this publication in new window or tab >>The Apparent Arbitrariness of Second-Order Probability Distributions
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. 49 p.
Series
Report Series / Department of Computer & Systems Sciences, ISSN 1101-8526 ; 11-002
National Category
Information Science
Identifiers
urn:nbn:se:hig:diva-8535 (URN)978-91-7447-184-7 (ISBN)
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: 2011-04-06Bibliographically approved

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