| Abstract | U ovom diplomskom radu smo se bavili bayesovskom statistikom odnosno bayesovskim linearnim regresijskim modelima. Ukratko, u linearnoj regresiji pokusavamo objasniti varijabilnost zavisne varijable \(Y\) pomoću jedne ili više nezavisnih varijabli \(X\). Koristimo pretpostavku da je regresijski šum nezavisno normalno distribuiran, zavisna varijabla \(Y\) također ima normalnu distribuciju, a \(\textbf{X}\) je matrica dizajna, prvi stupac sadrži jedinice. Sad kad je model postavljen, bayesovska analiza traži aposteriornu distribuciju parametara i prediktivnu distribuciju za model. Analiza započinje apriornom distribucijom. Parametre (\(\beta, \sigma^2\)) označimo sa (\(B, \Sigma^2\)) kao slučajne veličine. U radu smo koristili pretpostavku da parametar (\(B, \Sigma^2\)) ima nepravilnu apriornu distribuciju \(\pi (\beta, \sigma^2) \propto \frac{1}{\sigma^2}\), te dobili da je aposteriorna distribucija za \(\beta\) uvjetno na \(\Sigma^2=\sigma^2\) normalna, a aposteriorna distribucija za \(\Sigma^2 Inv\chi^2\)-distribucija. Drugi slučaj koji smo pokazali je slučaj apriorne distribucije. Pretpostavili smo da \(B\) ima normalnu apriornu distribuciju (uvjetnu na \(\sigma^2\)), a \(\sigma^2\) inverznu \(\chi^2\) -apriornu distribuciju. Dobili smo da je i aposteriorna distribucija za \(B\) dana sa \(\pi ( \beta|\) y, X, \(\sigma^2\)) također normalna, a aposteriorna distribucija od \(\Sigma^2\) ponovno inverzna \(\chi^2\) distribucija. Pokazali smo i slučaj nejednakih varijanci, kad imamo dva modela, \(Y_1=X_1 \beta + \epsilon_1\) i \(Y_2=X_2 \beta + \epsilon_2\). Neinformativna apriorna distribucija se može utvrditi, pod pretpostavkom da su parametri nezavisni. Tada je apriorna distribucija zapisana kao: \(\pi(\beta, \sigma_1, \sigma_2 ) \propto \frac{1}{\sigma_1 \sigma_2}\). Pokazuje se da je marginalna aposteriorna distribucija od \(B\) produkt dvije multivarijatne Studentove t-gustoće. Također je pokazano da se marginalna aposteriorna distribucija od \(B\) može aproksimirati sa normalnom distribucijom. Slično smo pokazali i za multivarijatni slučaj. |
| Abstract (english) | This thesis is concerned with a Bayesian statistics and Bayesian linear regression models. In short, in linear regression we are trying to explain the variability in dependent variable \(Y\) with the help of one or more independent variables \(X\). We assume that regression disturbances are independently and identically disturbed with normal distribution, the dependent variable \(Y\) has also normal distribution and \(\textbf{X}\) is design matrix, first column of \(\textbf{X}\) is a column of ones. Once the model is specified, the Bayesian analysis seeks the posterior distribution of the parameters and a predictive distribution for the model's prediction. The analysis begins with a prior distribution. Parameters ((\(\beta, \sigma^2\)) are denoted with (\(B, \Sigma^2\)) like random variables. We used assumption that parameter (\(B, \Sigma^2\)), has improper prior \(\pi (\beta, \sigma^2) \propto \frac{1}{\sigma^2}\), it follows that posterior distribution of \(\beta\) conditional on \(\Sigma^2=\sigma^2\) is normal, and posterior distribution of \(\Sigma^2\) is \(Inv\chi^2\) distribution. Another case, which we have shown is case of informative prior distribution. We assume that \(B\) has a normal prior distribution (conditional on \(\sigma^2\)), and \(\Sigma^2\) inverted \(\chi^2\) − prior distribution. It follows that posterior distribution for \(B\) is given with: \(\pi ( \beta|\) y, X, \(\sigma^2\)) is also normal and posterior distribution of \(\Sigma^2\) is also inverted \(\chi^2\) distribution. We also have shown the case of unequal variance, when we have two models, \(Y_1=X_1 \beta + \epsilon_1\) and \(Y_2=X_2 \beta + \epsilon_2\). A non-informative prior distribution can be asserted, by assuming that the parameters are independent. The prior is written, then, as: \(\pi(\beta, \sigma_1, \sigma_2 ) \propto \frac{1}{\sigma_1 \sigma_2}\). It is shown that the marginal posterior distribution of \(B\) is the product of two multivariate Student's t-densities. Also, it is shown that the marginal posterior of \(B\) can be approximated with a normal distribution. Similarly, we have shown for the multivariate case. |
