Introduction to Bayesian Statistics. The basic concept of Bayesian Statistics and its main difference from classical Statistics. Advantages of Bayesian Statistics. The Bayes Theorem.
Prior distributions. Relative likelihood method, histogram method, fit distribution with a given functional form, conjugate prior distributions, non-informative prior distributions (vague, Jeffreys distributions), Bayes empirical analysis, hierarchical prior distributions.
Posterior distribution: Compute the posterior distribution using various prior distributions. Compute the posterior distribution on data sets extensively used in the bibliography
Bayesian Inference: Elements of Statistical Decision Theory and Bayesian Decision Theory: loss function, risk function, decision rules, Bayes risk, Bayes rule and Bayes decision. Bayes estimators (posterior mean and median), Credible sets, Hypothesis tests (Bayes Factor, Fit of prior distributions for simple hypotheses). Predictive distributions.
Simulation: Pseudo random number simulation, Inverse method, accept - reject method, Importance Sampling. Introduction to Markov Chain Theory, Introduction to Markov Chain Monte Carlo (MCMC) methods, Metropolis - Hastings algorithm, Gibbs Sampler, Hybrid Gibbs Sampler.