Bayesovsko hijerarhijsko modeliranje

master's thesis

• Title (english): Bayesian hierarchical modeling
• Author: Nina Drobac
• Mentor: Azra Tafro (mentor)
• Committee member: Azra Tafro (predsjednik povjerenstva)
• Committee member: Dijana Ilišević (član povjerenstva)
• Committee member: Marko Vrdoljak (član povjerenstva)
• Committee member: Zvonko Iljazović (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2023-03-02, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U statistici često koristimo pretpostavku da su podaci međusobno nezavisni, no ona nije sasvim opravdana kada radimo s opažanjima koja su, zbog načina na koji su prikupljeni ili prirode problema kojeg opisuju, strukturirani u nekoliko grupa. Zbog toga uvodimo hijerarhijske modele, kod kojih je distribucija podataka za svaku grupu parametrizirana s θj, j = 1, 2, . . . , J, a odnos među parametrima modeliran je zajedničkom vjerojatnosnom distribucijom nad θ1, . . . , θJ. U okviru hijerarhijskih modela, posebno se ističe prednost bayesovskog pristupa, koji omogućava da pri donošenju zaključaka za jednu grupu iskoristimo i podatke o preostalim grupama. Bayesovsko hijerarhijsko modeliranje podrazumijeva da je i zajednička distribucija od θ1, . . . , θJ parametrizirana preko hiperparametara ϕ, za koje također promatramo apriornu i aposeriornu distribuciju i za koje donosimo statističke zaključke slijedeći bayesovske principe. U ovom smo radu iznijeli osnove bayesovske statistike, motivirali uvođenje hijerarhijskih modela te obradili bayesovsko hijerarhijsko modeliranje, s naglaskom na postavljanju modela i analizi dobivenih rezultata. Na primjerima su ilustrirane prednosti i glavne značajke bayesovskog hijerarhijskog pristupa modeliranju klasteriranih podataka, kao i metode računarske statistike korisne za prikaz rezultata.

Abstract (english)

When doing statistical analysis, we often assume that data points are mutually independent. However, due to the way it was collected or the nature of the problem it came from, data is often structured into groups or clusters. In these situations, the assumption of independence comes into question. To tackle that, we introduce hierarchical models, where the probability distribution of data for each of the J groups is defined by the parameter θj and the relationship between these parameters is modeled by the joint distribution of θ1, . . . , θJ. Bayesian inference further broadens the use of hierarchical models because it enables us to make statistical conclusions about a certain group while incorporating information contained in the data belonging to other groups. In Bayesian hierarchical models, the joint distribution of θ1, . . . , θJ is defined by the hyperparameters ϕ, for which we also set a prior and calculate the posterior distribution using Bayesian inference. In this master’s thesis, we present the main theoretical results of Bayesian statistics, we discuss the motivation behind hierarchical modeling and give a detailed description of Bayesian hierarchical models, focusing on the procedure of setting up and analysing these models. We highlight the key advantages and principles of Bayesian hierarchical modeling using examples in which we also showcase computational methods for displaying results.

• Keywords: statistika; hijerarhijski modeli; bayesovski pristup; apriorna distribucija; aposeriorna distribucija; bayesovska statistika
• Keywords (english): statistics; hierarchical models; Bayesian approach; a priori distribution; aposterior distribution; Bayesian statistics
• Language: croatian
• URN:NBN: urn:nbn:hr:217:426134
• Study programme: Title: Mathematical Statistics; 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: 2023-03-24 11:06:12