Bayesovsko zaključivanje za linearnu regresiju

master's thesis

• Title (english): Bayesian inference for linear regression
• Author: Agata Kolar
• Mentor: Snježana Lubura Strunjak (mentor)
• Committee member: Snježana Lubura Strunjak (predsjednik povjerenstva)
• Committee member: Zvonko Iljazović (član povjerenstva)
• Committee member: Matko Ljulj (član povjerenstva)
• Committee member: Hrvoje Planinić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2023-05-24, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom radu opisuju se metode Bayesovskog zaključivanja za linearnu regresiju. U prvom poglavlju uvodimo osnovne definicije i rezultate iz teorije vjerojatnosti i matematičke statistike koji su bitni za razumijevanje Bayesovog teorema i temeljnih koncepata iz Bayesovske statistike. U drugom poglavlju donosimo iskaze diskretne i neprekidne verzije Bayesovog teorema, zajedno s dokazima i primjerima. Navedeni teoremi čine temeljnu okosnicu za Bayesovsko statističko zaključivanje. Uvode se ključni pojmovi u Bayesovskoj statistici: apriorna i aposteriorna distribucija. U trećem poglavlju uvodimo model jednostavne i višestruke linearne regresije te korište-njem metode najmanjih kvadrata određujemo procjene parametara za modele. Također, navodimo i osnovne pretpostavke o vjerojatnosnoj distribuciji slučajnih grešaka u modelu bez kojih ne bismo mogli donositi validne zaključke o nepoznatim parametrima modela linearne regresije. U posljednjem, petom poglavlju, predstavljamo primjenu obrađenih metoda Bayesovskog zaključivanja kroz nekoliko primjera. Zadaci su riješeni u programskom jeziku R.

Abstract (english)

In this paper, we describe Bayesian inference methods for linear regression. In the first chapter, we introduce basic definitions and results from probability theory and mathematical statistics that are essential for understanding Bayes’ theorem and fundamental concepts of Bayesian statistics. In the second chapter, the discrete and continuous versions of Bayes’ theorem are presented, along with proofs and examples. These theorems form the basic framework for Bayesian statistical inference. Key concepts in Bayesian statistics are introduced: prior and posterior distribution. In the third chapter, we introduce the model of simple and multiple linear regression, and, using the least squares method, we determine estimates of the parameters for the models. We also state basic assumptions about the probability distribution of random errors in the regression model, without which we would not be able to make valid conclusions about the unknown parameters of the linear regression model. The examples are solved in the programming language R.

• Keywords: teorija vjerojatnosti; matematička statistika; Bayesov teorem; Bayesovska statistika; programskom jeziku R
• Keywords (english): probability theory; mathematical statistics; Bayes theorem; Bayesian statistics; programming language R
• Language: croatian
• URN:NBN: urn:nbn:hr:217:383055
• Study programme: Title: Financial and Business Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: sveučilišni magistar matematike (sveučilišni magistar matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2024-02-09 11:07:08