Part I Probability --------------------- Outcomes, Events & Probability. Conditional Probability & Independence. Bayes' Theorem: Interpretation & Use. Random Variables and Random Vectors. Directed Graphical Models. Expected Value, Variance and Covariance. Univariate and Multivariate Distributions. Generating Functions and Convergence Theorems. Part II Statistics --------------------- The Empirical Distribution and the Bootstrap Statistical Modelling: the Likelihood Function. Parameter Estimation: Point and Interval Estimation. Hypothesis Testing. Alternative Inferential Frameworks: Frequentist vs Bayesian Inference.

Riferimenti principali: F.M. Dekking, C. Kraaikamp, H.P. Lopuhaa, L.E. Meester (2007). A Modern Introduction to Probability and Statistics. Springer. L. Wasserman (2004). All of Statistics: A Concise Course in Statistical Inference. Springer.

• introduction to the basics of Bayesian inference • conjugate Bayesian models • examples of non-conjugate models • introduction to Monte Carlo methods as approximation strategy • Monte Carlo methods for Bayesian inference • pseudo-random number generation • uniform distributions and common classes of parametric distributions • general classes of algorithms for simulating from a known density: inverse transform sampling, acceptancerejection algorithm, fundamental theorem of simulation • classical asymptotic theorems and Monte Carlo methods: convergence and error control • importance sampling techniques • alternative Monte Carlo strategies for approximating marginal likelihood and Bayes Factor • introduction to Markov chains on a finite state space • introduction to Markov chains on general state spaces • transition kernels and transition densities • Markov chains, stationarity, invariant measures • limiting distributions and rate of convergence • general algorithms for Markov chain simulation with a prescribed invariant distribution • Gibbs sampling • Metropolis Hastings • MH, alternative proposal distributions, tuning • basic examples of GS • basic examples of MH • reversibility • hybrid methods: kernel composition, kernel mixtures • GS and MH implementation on real data examples • Bayesian hierarchical linear models, generalized linear model (examples) • linear mixed-effect models • multimodel inference: model choice via marginal likelihood and DIC only and model averaging

Teaching material will be also delivered through the Sapienza elearning platform Moodle at the following address: https://elearning.uniroma1.it/course/view.php?id=13089 Course Material and Main Reference Books • Course lecture notes (slide available at the https://elearning2.uniroma1.it/course/view.php?id=7253) • R or R+Jags commented codes • Peter Hoff, A First Course in Bayesian Statistical Methods. Springer-Verlag Inc, 2009. • Jean-Michel Marin and Christian P. Robert, Bayesian Core: A Practical Approach to Computational Bayesian Statistics, Springer, 2007 • Christian P. Robert and George Casella. Monte Carlo statistical methods (2nd ed.. Springer-Verlag Inc, 2004. • Ioannis Ntzoufras, Bayesian Modeling Using WinBUGS. Wiley, 2009. • Peter Congdon. Bayesian Statistical Modelling (2nd ed.). Wiley, 2006