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Sundgren, David

Open this publication in new window or tab >>Discrete Second-order Probability Distributions that Factor into Marginals### Sundgren, David

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##### Abstract [en]

##### Place, publisher, year, edition, pages

SIPTA, 2011
##### Keywords

Discrete probability, second-order probability, imprecise probability, multivariate Pólya distribution, conjugate prior, compound hypergeometric likelihood.
##### National Category

Probability Theory and Statistics
##### Identifiers

urn:nbn:se:hig:diva-9807 (URN)000323983600036 ()2-s2.0-84883215401 (Scopus ID)978-3-902652-40-9 (ISBN)
##### Conference

7th International Symposium on Imprecise Probabilities: Theories and Applications (ISIPTA), July 25-28, 2011, Innsbruck, Austria
#####

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Available from: 2011-09-01 Created: 2011-07-30 Last updated: 2018-03-13Bibliographically approved

University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Mathematics.

In realistic decision problems there is more often than not uncertainty in the background information. As for representation of uncertain or imprecise probability values, second-order probability, i.e. probability distributions over probabilities, offers an option. With a subjective view of probability second-order probability would seem to be impractical since it is hard for a person to construct a second-order distributions that reflects his or her beliefs. From the perspective of probability as relative frequency the task of constructing or updating a second-order probability distribution from data is somewhat easier. Here a very simple model for updating lower bounds of probabilities is employed. But the difficulties in choosing second-order distributions may be further alleviated if structural properties are considered. Either some of the probability values are dependent in some way, e.g. that they are known to be almost equal, or they are not dependent in any other way than what follows from that the values sum to one. In this work we present the unique family of discrete second-order probability distributions that correspond to the case where dependence is limited. These distributions are shown to have the property that the joint distributions are equal to normalised products of marginal distributions. The distribution family introduced here is a generalisation of a special case of the multivariate Pólya distribution and is shown to be conjugate prior to a compound hypergeometric distribution.

Open this publication in new window or tab >>The Apparent Arbitrariness of Second-Order Probability Distributions### Sundgren, David

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_1_j_idt208_some",{id:"formSmash:j_idt204:1:j_idt208:some",widgetVar:"widget_formSmash_j_idt204_1_j_idt208_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_1_j_idt208_otherAuthors",{id:"formSmash:j_idt204:1:j_idt208:otherAuthors",widgetVar:"widget_formSmash_j_idt204_1_j_idt208_otherAuthors",multiple:true}); 2011 (English)Doctoral thesis, comprehensive summary (Other academic)
##### Abstract [en]

##### Abstract [la]

##### Abstract [sv]

##### 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: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

### Utkin, Lev

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##### Supervisors

### Ekenberg, Love

### Danielson, Mats

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Available from: 2011-04-06 Created: 2011-03-04 Last updated: 2018-03-13Bibliographically approved

University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Mathematics.

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.

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.

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.

St Petersburg Forest Technical Academy, Dept. of Computer Science.

Institutionen för data- och systemvetenskap, Stockholms universitet.

Institutionen för data- och systemvetenskap, Stockholms universitet.

Open this publication in new window or tab >>Expected Utility from Multinomial Second-order Probability Distributions### Sundgren, David

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_2_j_idt208_some",{id:"formSmash:j_idt204:2:j_idt208:some",widgetVar:"widget_formSmash_j_idt204_2_j_idt208_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_2_j_idt208_otherAuthors",{id:"formSmash:j_idt204:2:j_idt208:otherAuthors",widgetVar:"widget_formSmash_j_idt204_2_j_idt208_otherAuthors",multiple:true}); 2010 (English)In: Polibits, ISSN 1870-9044, no 42, p. 71-75Article in journal (Refereed) Published
##### Abstract [en]

##### Place, publisher, year, edition, pages

Mexico City: Centro de Innovacíon y Desarrollo Tecnológico en Cómputo, Instituto Politécnico Nacional, 2010
##### Keywords

Imprecise probability. second-order probability, discrete probability distributions, multinomial distributions, expected utilty.
##### National Category

Computer Sciences
##### Identifiers

urn:nbn:se:hig:diva-7970 (URN)
#####

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Available from: 2010-11-12 Created: 2010-11-12 Last updated: 2018-03-13Bibliographically approved

University of Gävle, Faculty of Engineering and Sustainable Development, Department of Electronics, Mathematics and Natural Sciences, Mathematics.

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.

Open this publication in new window or tab >>Shifted Dirichlet Distributions as Second-Order Probability Distributions that Factors into Marginals### Sundgren, David

### Ekenberg, Love

### Danielson, Mats

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_3_j_idt208_some",{id:"formSmash:j_idt204:3:j_idt208:some",widgetVar:"widget_formSmash_j_idt204_3_j_idt208_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_3_j_idt208_otherAuthors",{id:"formSmash:j_idt204:3:j_idt208:otherAuthors",widgetVar:"widget_formSmash_j_idt204_3_j_idt208_otherAuthors",multiple:true}); 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]

##### 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
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Available from: 2009-09-08 Created: 2009-09-08 Last updated: 2018-03-13Bibliographically approved

University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för matematik och statistik.

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.

Open this publication in new window or tab >>Warp effects on calculating interval probabilities### Sundgren, David

### Danielson, Mats

### Ekenberg, Love

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_4_j_idt208_some",{id:"formSmash:j_idt204:4:j_idt208:some",widgetVar:"widget_formSmash_j_idt204_4_j_idt208_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_4_j_idt208_otherAuthors",{id:"formSmash:j_idt204:4:j_idt208:otherAuthors",widgetVar:"widget_formSmash_j_idt204_4_j_idt208_otherAuthors",multiple:true}); 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]

##### Place, publisher, year, edition, pages

Elsevier B.V., 2009
##### Keywords

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)
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Available from: 2009-09-08 Created: 2009-09-08 Last updated: 2018-03-13Bibliographically approved

University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för matematik och statistik.

Deptartment of Computer and Systems Sciences, Stockholm University and TH, Kista, Sweden.

Deptartment of Computer and Systems Sciences, Stockholm University and TH, Kista, Sweden.

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.

Open this publication in new window or tab >>Recognition of video sequences using low frequencies and color### Åhlén, Julia

### Sundgren, David

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_5_j_idt208_some",{id:"formSmash:j_idt204:5:j_idt208:some",widgetVar:"widget_formSmash_j_idt204_5_j_idt208_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_5_j_idt208_otherAuthors",{id:"formSmash:j_idt204:5:j_idt208:otherAuthors",widgetVar:"widget_formSmash_j_idt204_5_j_idt208_otherAuthors",multiple:true}); 2008 (English)In: Advanced topics on signal processing, robotics and automation: proceedings of the 7th WSEAS International Conference on Signal Processing, Robotics and Automation (ISPRA '08), Cambridge, UK, February 20-22 2008, WSEAS , 2008, p. 204-207Conference paper, Published paper (Refereed)
##### Abstract [en]

##### Place, publisher, year, edition, pages

WSEAS, 2008
##### Identifiers

urn:nbn:se:hig:diva-1627 (URN)978-960-6766-44-2 (ISBN)
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Available from: 2008-03-26 Created: 2008-03-26 Last updated: 2023-02-17Bibliographically approved

University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för datavetenskap.

University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för matematik och statistik.

This paper presents a method for descriptive feature matching between two video streams of the same scene. The second video is a corrupted copy of the first video. Examples of reliable recognition of an original scene that is subjected for compression and noise are presented. The algorithm is invariant to image scale and rotation, providing robust matching across a substantial range of video signals. This paper shows an approach to using descriptive features for randomly chosen frames from video stream. By involving color information as a feature we increase the robustness of the identification process. The median of low frequencies is measured, followed by an average of spectral information from the same frame to identify clusters belonging to a scene in a film. Finally, a comparison analysis is done to verify the originality of the film.

Open this publication in new window or tab >>Recognition of video signal by matching with the original video sequence### Åhlén, Julia

### Sundgren, David

University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för matematik och statistik.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_6_j_idt208_some",{id:"formSmash:j_idt204:6:j_idt208:some",widgetVar:"widget_formSmash_j_idt204_6_j_idt208_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_6_j_idt208_otherAuthors",{id:"formSmash:j_idt204:6:j_idt208:otherAuthors",widgetVar:"widget_formSmash_j_idt204_6_j_idt208_otherAuthors",multiple:true}); 2008 (English)In: Proceedings on CD-ROM, IADIS International Conference, 2008Conference paper, Published paper (Refereed)
##### Abstract [en]

##### Identifiers

urn:nbn:se:hig:diva-1667 (URN)
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Available from: 2008-05-08 Created: 2008-05-08 Last updated: 2023-02-17Bibliographically approved

University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för datavetenskap.

An adequate analysis of the originality of the video is of a great importance in video processing field. This paper presents a method for feature matching between two video streams. One of the video streams is subjected for severe compression, noise and rotation. These two copies represent the same movie but only one of the versions is original. Examples of reliable recognition of an original scene that is subjected for compression and noise are presented. The algorithm is invariant to image scale and rotation, providing robust matching across a substantial range of video signals. This paper shows an approach to using descriptive features for randomly chosen unstructured frames from video stream. By involving color information as a feature together with intensity profile for frame blocks we increase the robustness of the identification process. The median of low frequencies is measured, followed by an average of spectral information from the same frame to identify clusters belonging to a scene in a film. Finally, a comparison of mean squares of the difference is done to verify the originality of the film.

Open this publication in new window or tab >>Some properties of aggregated distributions over expected values### Sundgren, David

University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för matematik och statistik.### Ekenberg, Love

### Danielson, Mats

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_7_j_idt208_some",{id:"formSmash:j_idt204:7:j_idt208:some",widgetVar:"widget_formSmash_j_idt204_7_j_idt208_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_7_j_idt208_otherAuthors",{id:"formSmash:j_idt204:7:j_idt208:otherAuthors",widgetVar:"widget_formSmash_j_idt204_7_j_idt208_otherAuthors",multiple:true}); 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]

##### Place, publisher, year, edition, pages

Berlin, Heidelberg: Springer, 2008
##### Series

Lecture Notes in Computer Science, ISSN 1611-3349 ; 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)
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Available from: 2008-11-06 Created: 2008-11-06 Last updated: 2021-09-13Bibliographically approved

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.

Open this publication in new window or tab >>Application of underwater hyperspectral data for color correction purposes### Åhlén, Julia

### Sundgren, David

University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för matematik och statistik.### Bengtsson, Ewert

PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_8_j_idt208_some",{id:"formSmash:j_idt204:8:j_idt208:some",widgetVar:"widget_formSmash_j_idt204_8_j_idt208_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_8_j_idt208_otherAuthors",{id:"formSmash:j_idt204:8:j_idt208:otherAuthors",widgetVar:"widget_formSmash_j_idt204_8_j_idt208_otherAuthors",multiple:true}); 2007 (English)In: Pattern recognition and image analysis, ISSN 1555-6212, Vol. 17, no 1, p. 170-173Article in journal (Refereed) Published
##### Abstract [en]

##### Identifiers

urn:nbn:se:hig:diva-1626 (URN)10.1134/S105466180701021X (DOI)
#####

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Available from: 2008-03-26 Created: 2008-03-26 Last updated: 2023-02-17Bibliographically approved

University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för datavetenskap.

CBA.

Color correction of underwater images has been considered a difficult task for a number of reasons. Those include severe absorption of the water column, the unpredictable behavior of light under the water surface, limited access to reliable data for correction purposes, and the fact that we are only able to process three spectral channels, which is insufficient for most color correction applications. Here, the authors present a method to estimate a hyperspectral image from an RGB image and pointwise hyperspectral data. This is then used to color correct the hyperspectral underwater image and transform it back into RGB color space.

Open this publication in new window or tab >>Distribution of expected utility in second-order decision analysis### Sundgren, David

University of Gävle, Department of Mathematics, Natural and Computer Sciences, Ämnesavdelningen för matematik och statistik.PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_9_j_idt208_some",{id:"formSmash:j_idt204:9:j_idt208:some",widgetVar:"widget_formSmash_j_idt204_9_j_idt208_some",multiple:true}); PrimeFaces.cw("AccordionPanel","widget_formSmash_j_idt204_9_j_idt208_otherAuthors",{id:"formSmash:j_idt204:9:j_idt208:otherAuthors",widgetVar:"widget_formSmash_j_idt204_9_j_idt208_otherAuthors",multiple:true}); 2007 (English)Licentiate thesis, comprehensive summary (Other academic)
##### Abstract [en]

##### Abstract [la]

##### Place, publisher, year, edition, pages

Kista: Data- och systemvetenskap, Kungliga Tekniska högskolan, 2007. p. vi, 20
##### Series

DSV Report Series, ISSN 1101-8526 ; 07:006
##### Keywords

beslutsanalys, andra ordningens fördelningar, förväntad nytta
##### National Category

Information Systems
##### Identifiers

urn:nbn:se:hig:diva-2491 (URN)
##### Presentation

(English)
#####

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Available from: 2007-06-07 Created: 2007-06-07 Last updated: 2018-03-13Bibliographically approved

In decision analysis maximising the expected utility is an often used approach in choosing the optimal alternative. But when probabilities and utilities are vague or imprecise expected utility is fraught with complications. Studying second-order effects on decision analysis casts light on the importance of the structure of decision problems, pointing out some pitfalls in decision making and suggesting an easy to implement and easy to understand method of comparing decision alternatives. The topic of this thesis is such second-order effects of decision analysis, particularly with regards to expected utility and interval-bound probabilities. Explicit expressions for the second-order distributions inherent in interval-bound probabilities in general and likewise for distributions of expected utility for small decision problems are produced. By investigating these distributions the phenomenon of warping, that is concentration of belief, is studied.

In explicatione consiliorum, maxima facere communis utilitas saepe trita ratio deligendi meliorem optionem est. Verum si probabilitates et utilitates incertae vel dubiae sint, communis utilitas perturbationes affert. Studium secundi ordinis effectuum in explicatione consiliorum explanat momentum structurae quaestionium consilii, insidias aliquas ad consilium capiendum indicat et facilem ad efficiendum et intellegendum rationem comparandi varia consilia suadet. Haec thesis tractat de secundi ordinis effectibus explicationis consilii, praesertim de commune utilitate et de probabilitatibus coniunctis intervallo. Voces apertae distributionum ordinis secundi in probabilitatibus intervallo conjunctis insitarum omnino et item distributionum utilitatis expectatae in parvis quaestionibus consiliorum eduntur. His distributionibus cognitis studetur res inflexionis, aliter dictu intentio fidei.