Bayesovsko zaključivanje za očekivanje normalne distribucije

master's thesis

• Title (english): Bayesian inference for the expectation of normal distribution
• Author: Dora Dunković
• Mentor: Snježana Lubura Strunjak (mentor)
• Committee member: Snježana Lubura Strunjak (predsjednik povjerenstva)
• Committee member: Tina Bosner (član povjerenstva)
• Committee member: Marcela Hanzer (član povjerenstva)
• Committee member: Rudi Mrazović (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2021-12-02, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom radu opisuju se metode za zaključivanje o vrijednosti parametra očekivanja normalne distribucije. Na početku se navode osnovni pojmovi teorije vjerojatnosti i statistike kako bi se izgradila teorija potrebna za uvodenje Bayesovog teorema i osnova Bayesovskog zaključivanja temeljenih na teoremu. Uvode se diskretna i neprekidna verzija Bayesovog teorema, zajedno s dokazima, te pojmovi usko vezani uz primjenu teorema poput apriorne i aposteriorne distribucije. Za kraj uvodnog dijela objašnjava se ideja Bayesovskog zaključivanja. U nastavku se obrađuju metode zaključivanja, temeljene na diskretnoj verziji Bayesovog teorema, u slučaju kada je skup mogućih vrijednosti nepoznatog parametra diskretan. Zatim se obrađuju i metode temeljene na neprekidnoj verziji Bayesovog teorema za slučajeve kada pretpostavljamo da nepoznati parametar može biti jednak bilo kojoj vrijednosti unutar nekog intervala. Uvodi se i pojam Bayesovog intervala vjerodostojnosti za parametar očekivanja, koji je analogon standardnog frekvencionističkog intervala pouzdanosti, te njegova interpretacija u Bayesovskoj statistici te metode računanja intervala. Nadalje, opisuju se razlike Bayesovskog i frekvencionističkog pristupa. Razmatraju se standardne metode za točkovnu procjenu, intervalnu procjenu i testiranje hipoteza te se uspoređuju s odgovarajućim Bayesovskim metodama. Konačno, navedene metode se primjenuju na odabranim zadacima koji se rješavaju u programskom jeziku R.

Abstract (english)

In this paper, we describe methods of Bayesian inference for mean of normal distribution. First, we introduce basic concepts and definitions of probability theory and statistics for the purpose of building theory needed to introduce and prove Bayes theorem and the basics of Bayesian inference. We introduce discrete and continuous version of Bayes theorem, along with their proofs, as well as concepts regarding theorem’s use such as prior and posterior distribution. At the end of the introductory part, we explain the idea of Bayesian inference that is based on the theorem. We continue with methods of inference, based on the discrete version of Bayes theorem, that is used in cases when the set of possible values for the mean is discrete. Next, we introduce the methods of inference based on the continuous version of Bayes theorem, used under the assumption that the mean can be equal to any value in an interval. We also introduce the concept of Bayesian credible interval for normal mean as well as its interpretation in Bayesian statistics and methods for its computation. Furthermore, we explain the differences between Bayesian and frequentist inference. We compare frequentist and Bayesian point estimators, intervals and testing hypothesis. Finally, we use the explained methods on selected problems and solve the problems in programming language R.

• Keywords: normalna distribucija; teorija vjerojatnosti; statistika; apriorna distribucija; aposteriorna distribucija; Bayesov interval vjerodostojnosti; programski jezik R
• Keywords (english): normal distribution; probability theory; statistics; a priori distribution; posterior distribution; Bayesian credibility interval; programming language R
• Language: croatian
• URN:NBN: urn:nbn:hr:217:716963
• Study programme: Title: Finance and Business Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2021-12-14 13:15:40