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An objective Bayesian approach for discrete scenarios

Villa, Cristiano (2013) An objective Bayesian approach for discrete scenarios. Doctor of Philosophy (PhD) thesis, University of Kent. (doi:10.22024/UniKent/01.02.94708) (KAR id:94708)

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Official URL:
https://doi.org/10.22024/UniKent/01.02.94708

Abstract

Objective prior distributions represent a fundamental part of Bayesian infer­ence. Although several approaches for continuous parameter spaces have been developed, Bayesian theory lacks of a general method that allows to obtain priors for the discrete case. In the present work we propose a novel idea, based on losses, to derive objec­tive priors for discrete parameter spaces. We objectively measure the worth of each parameter values, and link it to the prior probability by means of the self­information loss function. The worth is measured by taking into consideration the surroundings of each element of the parameter space. Bayes theorem is then re-interpreted, where prior and posterior beliefs are not expressed as probabilities, but as losses. The approach allows to retain meaning from the beginning to the end of the Bayesian updating process. The prior distribution obtained with the above approach is identified as the Villa-Walker prior. We illustrate the approach by applying it to various scenarios. We derive ob­jective priors for five specific models: a population size model, the Hypergeometric and multivariate Hypergeometric models, the Binomial-Beta model, and the Bi­nomial model. We also derive the Villa-Walker prior for the number of degrees of freedom of a t distribution. An important result in this last case, is that the objective prior has to be truncated. We finally apply the idea to discrete scenarios other that parameter spaces: model selection, and variable selection for linear regression models. We show how an objective model prior can be obtained, by applying our approach, on the basis of the importance that each model has with respect to the other ones. We illustrate various cases: nested and non-nested models, models with discrete and continuous supports, uniparameter and multiparameter models. For the variable selection scenario, the prior includes a loss component due to the complexity of each regression model.

Item Type: Thesis (Doctor of Philosophy (PhD))
Thesis advisor: Walker, Stephen G.
DOI/Identification number: 10.22024/UniKent/01.02.94708
Additional information: This thesis has been digitised by EThOS, the British Library digitisation service, for purposes of preservation and dissemination. It was uploaded to KAR on 25 April 2022 in order to hold its content and record within University of Kent systems. It is available Open Access using a Creative Commons Attribution, Non-commercial, No Derivatives (https://creativecommons.org/licenses/by-nc-nd/4.0/) licence so that the thesis and its author, can benefit from opportunities for increased readership and citation. This was done in line with University of Kent policies (https://www.kent.ac.uk/is/strategy/docs/Kent%20Open%20Access%20policy.pdf). If you feel that your rights are compromised by open access to this thesis, or if you would like more information about its availability, please contact us at ResearchSupport@kent.ac.uk and we will seriously consider your claim under the terms of our Take-Down Policy (https://www.kent.ac.uk/is/regulations/library/kar-take-down-policy.html).
Subjects: Q Science > QA Mathematics (inc Computing science)
Divisions: Divisions > Division of Computing, Engineering and Mathematical Sciences > School of Mathematics, Statistics and Actuarial Science
SWORD Depositor: SWORD Copy
Depositing User: SWORD Copy
Date Deposited: 22 Sep 2022 11:51 UTC
Last Modified: 22 Sep 2022 11:51 UTC
Resource URI: https://kar.kent.ac.uk/id/eprint/94708 (The current URI for this page, for reference purposes)

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