Bayesian inference for a proportion of the binomial distribution

U ovom radu opisuje se Bayesovsko zaključivanje za proporciju binomne razdiobe. U uvodnom poglavlju uvedeni su osnovni pojmovi iz vjerojatnosti i statistike koji su potrebni za izgradnju teorije na kojoj se temelji Bayesov teorem. Također, u sklopu prvog poglavlja iskazana je diskretna i neprekidna verzija Bayesova teorema te su navedene distribucije koje smo koristili. Uvode se i ključni pojmovi vezani za teorem: apriorna i aposteriorna distribucija. U drugom poglavlju opisuje se Bayesovsko zaključivanje za diskretne slučajne varijable - binomnu i Poissonovu. Prikazani primjeri predstavljaju primjenu teorema i vode do zaključka da podatke možemo analizirati istovremeno ili sekvencijalno te u oba slučaja doći do istih aposteriornih distribucija. Treće poglavlje predstavlja okosnicu ovog rada, Bayesovsko zaključivanje za proporciju binomne razdiobe. Opisani su načini odabira apriornih distribucija, a obrađene su uniformna i Beta apriorna distribucija. Uvodi se bitan pojam konjugirane familije distribucija te se pokazuje da je Beta familija konjugirana za podatke koji dolaze iz binomne raspodjele. Objašnjeno je kako donijeti zaključke o aposteriornim distribucijama. Također, uvode se Bayesovski intervali vjerodostojnosti. U četvrtom poglavlju uspoređujemo Bayesovske i frekvencionističke zaključke o parametru do kojih dolazimo točkovnom procjenom, procjenom intervala i testiranjem hipoteza. Razmatra se razlika u interpretaciji parametra, definiraju se procjenitelji, uspoređuju se intervali vjerodostojnosti i pouzdanosti te se promatraju jednostrane i dvostrane hipoteze s oba stajališta. U posljednjem poglavlju nalaze se zadaci na kojima su primjenjene proučene metode. Zadaci su riješeni uz pomoć programskog jezika R, a kodovi su priloženi na kraju rada.This paper describes Bayesian inference for a proportion of the binomial distribution. In the introductory chapter we introduce the basic concepts from probability and statistics that are needed to build the theory on which Bayes theorem is based. Also, in the first chapter, discrete and continuous versions of Bayes theorem are presented, along with the distributions we used. Key concepts related to the theorem are also introduced: a prior and a posterior distributions. The second chapter describes Bayesian inference for discrete random variables, binomial and Poisson. The presented examples represent the application of the theorem and lead to the conclusion that we can analyze the data simultaneously or sequentially and in both cases arrive at the same posterior distributions. The third chapter presents the backbone of this paper, Bayesian inference for a proportion of the binomial distribution. This chapter describes ways of selecting a prior distributions, and the uniform and Beta prior distributions are processed. The important notion of the conjugate family of distributions is introduced and it is shown that the Beta family is conjugate for data coming from the binomial distribution. It is explained how to draw conclusions about posterior distributions. Also, Bayesian credibility intervals are introduced. In the fourth chapter, we compare Bayesian and frequentist conclusions about the parameter that we reach by point estimation, interval estimation and hypothesis testing. The difference in parameter interpretation is considered, estimators are defined, credibility and confidence intervals are compared, and one-side and and two-side hypotheses are observed from both points of view. In the last chapter, there are problems to which the studied methods were applied. Problems were solved using the programming language R, and the codes are attached at the end of the paper