Prof. L. Tardella

semester II

6 credits

List of topics:

- introduction to the basics of Bayesian inference
- 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

- classical asymptotic theorems and Monte Carlo methods: convergence and error control
- importance sampling techniques
- other techniques for improving error control: antithetic variates, control variates
- 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 (logistic and probit) 
- finite mixture models
- model choice and model averaging; transdimensional methods 

- ergodic theorems for MCMC
- error control and variance estimation of ergodic Markov Chains, effective sample size
- convergence monitoring
- software tools for MCMC implementation: R packages and BUGS

- sequential methods and particle filters (*)
- ABC (Approximate Bayesian Computation) (*)
- dynamic linear models (*)