Bayesian statistics and the Dirichlet process

U ovom radu uvodimo pojam Dirichletovog procesa u kontekstu bayesovske statistike. Navedeni su osnovni pojmovi i rezultati teorije vjerojatnosti koji će se koristiti u nastavku rada, a neke pomoćne tvrdnje su i dokazane. Posebno, definiran je pojam slučajnog procesa te je opisan način na koji se on zadaje pomoću suglasne familije konačno-dimenzionalnih distribucija. Zatim je definirana uvjetna funkcija gustoće, pokazana je Bayesova formula te su objašnjeni pojmovi apriorne i aposteriorne gustoće. Kroz nekoliko je primjera opisano zaključivanje u tzv. parametarskoj bayesovskoj statistici. Konačno, u kontekstu tzv. neparametarske bayesovske statistike opisan je pojam slučajne vjerojatnosne mjere, specijalnog slučaja slučajnoga procesa. Uvedena je Dirichletova distribucija, a Dirichletov proces definiran je kao ona slučajna vjerojatnost čije su konačno-dimenzionalne distribucije upravo Dirichletove. Definicija je opravdana provjerom Kolmogorovljevih uvjeta suglasnosti. Teorija je zaokružena primjerima i malom simulacijskom studijom.In this thesis, we introduce the term Dirichlet process in the context of Bayesian statistics. The basic terms and results of the probability theory that will be used in the rest of the paper are listed, and some auxiliary propositions are also proven. In particular, the concept of a stochastic process is defined and the way in which it is assigned using a consistent collection of finite-dimensional distributions is described. Then the conditional probability density function is defined, the Bayes’ formula is shown and the terms prior and posterior density functions are explained. Several examples describe the conclusion in the so-called parametric Bayesian statistics. Finally, in the context of the so-called non-parametric Bayesian statistics the concept of a random probability measure, a special case of a stochastic process, is described. The Dirichlet distribution is introduced, and the Dirichlet process is defined as that random probability measure whose finite-dimensional distributions are exactly Dirichlet. The definition is justified by checking the conditions required by the Kolmogorov extension theorem. The theory is rounded off with examples and a small simulation study