Bayesova linearna regresija : delo diplomskega seminarja

Peček, Urh (author), Smrekar, Jaka (mentor)

Bayesova linearna regresija je dobila ime po angleškem statistiku Thomasu Bayesu, ki je živel v prvi polovici 18. stoletja. Namen naloge je predstaviti temeljne ideje Bayesovega linearnega modeliranja, začenši s teorijo, ki stoji za Bayesovo statistiko, pa tudi nekatere praktične primere Bayesove linearne regresije. Na začetku je opisana regresijska analiza in njena uporaba. Podrobneje je obravnavana Bayesova statistika in izpeljava obrazca, ki temelji na Bayesovem izreku in je osnova, na kateri temelji Bayesovo sklepanje. Vključena je tudi praktična uporaba Bayesovega izreka ter posodabljanja na primeru testiranja prisotnosti bolezenskega stanja pri pacientu. Opisana in na poučnem primeru meta kovanca je predstavljena ocena parametrov z Bayesovim pristopom in njene razlike ter prednosti v primerjavi s frekventističnim pristopom ocenjevanja parametrov. Podan je normalni model linearne regresije. Na njem je predstavljena klasična linearna regresija, kjer je za oceno parametrov uporabljena metoda največjega verjetja. Kot glavni del naloge, je na normalnem modelu linearne regresije opisana Bayesova linearna regresija, kjer za oceno parametrov uporabimo Bayesov pristop. Izpeljana je skupna aposteriorna porazdelitev parametrov normalnega modela prek konjugiranih družin. Za konec sta Bayesovo posodabljanje in aposteriorna porazdelitev parametrov predstavljena tudi na primeru vpliva statističnih podatkov košarkašev lige NBA na njihovo plačo.

• Language: Slovenian
• Keywords: linearna regresija, Bayesova statistika, Bayesov izrek, apriorna in aposteriorna porazdelitev, Bayesovo sklepanje, frekventistično sklepanje, Bayesovo posodabljanje
• Work type: Final seminar paper
• Typology: 2.11 - Undergraduate Thesis
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2020
• PID: 20.500.12556/RUL-119740
• UDC: 519.2
• COBISS.SI-ID: 58664963
• Publication date in RUL: 11.09.2020

PEČEK, Urh, 2020, Bayesova linearna regresija : delo diplomskega seminarja [online]. Bachelor’s thesis. [Accessed 17 May 2025]. Retrieved from: https://repozitorij.uni-lj.si/IzpisGradiva.php?lang=eng&id=119740

Secondary language

• Language: English
• Title: Bayesian linear regression

Abstract

The Bayesian linear regression was named after the English statistician Thomas Bayes, who lived in the first half of the 18th century. The purpose of this bachelors' thesis is to present the basic ideas of Bayesian linear modeling, starting with the theory behind Bayesian statistics, as well as some practical examples of Bayesian linear regression. Regression analysis and its application are described at the beginning. In more detail is discussed the Bayesian statistics and the derivation of the form, which is based on the Bayesian theorem and is the basis on which Bayes' reasoning is based. Also included is the practical application of Bayesian theorem and updating in the case of testing for the presence of a disease state in a patient. Described and on an instructive example of a coin flip is presented the estimation of parameters with the Bayesian approach and its differences and advantages in comparison with the frequency approach of parameter estimation. A normal linear regression model is given. A classical linear regression is presented on it, where the maximum probability method is used to estimate the parameters. As the main part of the thesis, the Bayesian linear regression is described on a normal linear regression model, where we use the Bayesian approach to estimate the parameters. A joint a posteriori distribution of the parameters of the normal model over conjugated families is derived. Finally, Bayes 'update and a posteriori distribution of parameters are also presented in the case of the impact of NBA basketball players' statistics on their salary.

• Keywords: linear regression, Bayesian statistics, Bayesian theorem, a priori and a posteriori distribution, Bayesian inference, frequency inference, Bayesian updating