Asimptotska svojstva Bayesovog procjenitelja

master's thesis

• Author: Josip Kiralj
• Mentor: Miljenko Huzak (mentor)
• Committee member: Miljenko Huzak (predsjednik povjerenstva)
• Committee member: Bojan Basrak (član povjerenstva)
• Committee member: Nikola Sandrić (član povjerenstva)
• Committee member: Igor Pažanin (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2018-09-26, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

U ovom radu smo se bavili asimptotskom teorijom u bayesovskoj statistici. Uveli smo osnovne pojmove bayesovske statistike te dali metodu konstruiranja aposteriorne distribucije. Opisali smo Bayesove procjenitelje obzirom na kvadratnu i apsolutnu funkciju gubitka. Zatim smo uveli pojam konzistentnosti aposteriorne distribucije koji nam na neki način daje opravdanje bayesovskog pristupa sa gledišta klasične statistike. U sklopu konzistentnosti smo promatrali kako se asimptoski ponašaju dvije konzistentne aposteriorne distribucije koje su proizašle iz različitih apriornih distribucija. Pokazali smo da su, uz neke pretpostavke, asimptoski ekvivalentne. Zatim smo iskazali i dokazali Doobov teorem koji nam govori da je aposteriorna distribucija konzistentna na cijelom nosaču apriorne distribucije. Također smo opisali jednu metodu procjene proizašle iz klasične statistike: metodu maksimalne vjerodostojnosti. Pokazali smo da uz određene uvjete imamo istovremeno jaku konzistentnost procjenitelja maksimalne vjerodostojnosti i konzistentnost aposteriorne distribucije, ali smo i naveli primjer kada to ne vrijedi. Nakon toga, bavili smo se asimptotskim svojstvima aposteriorne distribucije i Bayesovih procjenitelja. Teoremom Bernstein-von Mises opisali smo asimptotsko ponašanje aposteriorne distribucije u regularnim statističkim modelima, rezultatom koji je zapravo analogon teorema o asimptotskoj normalnosti procjenitelja maksimalne vjerodostojnosti. Posljedično tome, dokazali smo asimptotsku ekvivalenciju Bayesovih procjenitelja sa procjeniteljem maksimalne vjerodostojnosti, odnosno dokazali smo njihovu asimptotsku normalnost. Rezultati i razmatranja su potkrijepljeni primjerima. Na kraju, primjerom smo ilustrirali asimptotsku normalnost aposteriorne distribucije i u jednom neregularnom modelu.

Abstract (english)

In this thesis we dealt with large sample theory in Bayesian statistics. We introduced some basic concepts in Bayesian statistics and described how the posterior distribution is constructed. Then we described the Bayes estimator under the square error and absolute loss functions. Furthermore, the posterior consistency was introduced, a sort of frequentist validation of the Bayesian approach. We observed the asymptotic behaviour of two consistent posteriors that arose from different priors and showed that, under some assumptions, they are asymptotically equivalent. Next, we proved Doobs theorem that states that the posterior is consistent on the support of the prior. We also described one frequentist estimation method: maximum likelihood estimation. Then we showed that under certain conditions the strong consistency of the maximum likelihood estimator and the consistency of the posterior distribution occurs. However, we showed in an example that the previous statement is not valid in general. Afterwards, we dealt with the asymptotic properties of the posterior distribution and Bayes estimators. We stated the Bernstein-von Mises theorem about the asymptotic normality of the posterior when the regularity conditions hold. It is an analogous result to the asymptotic normality of the maximum likelihood estimator. Moreover, the asymptotic equivalence of the Bayes estimators and the maximum likelihood estimator was proven. Consequently, the asymptotic normality of the Bayes estimator was obtained. We provided examples that illustrate the main results described in the thesis. Lastly, we gave an example of a result in asymptotic normality of the posterior when the distribution is nonregular.

• Keywords: bayesovska statistika; bayesov procjenitelj; Doobov teorem; teorem Bernstein-von Mises; aposteriorna distribucija
• Keywords (english): Bayesian statistics; Bayes estimator; Doobs theorem; Bernstein-von Mises theorem; posterior distribution
• Language: croatian
• URN:NBN: urn:nbn:hr:217:643629
• 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: 2019-05-17 08:47:29