1. An Iterative Monte Carlo Method for Nonconjugate Bayesian Analysis B. P. Carlin and A. E. Gelfand Statistics and Computing 1991 A Generic Approach to Posterior Integration and Gibbs Sampling P. Muller Alternatives to the Gibbs Sampling Schemes P. Muller Metropolized Gibbs Sampler: An Improvement Jun S. Liu Presented by: Mingyuan Zhou Duke University, ECE July 25, 2011

2. Outline Introduction Gibbs sampler Tailored rejection method An Iterative Monte Carlo Method for Nonconjugate Bayesian Analysis B. P. Carlin and A. E. Gelfand Statistics and Computing 1991

3. Introduction Gibbs sampler (Geman and Geman, 1984; Gelfand and Smith, 1990) requires conjugacy Sampling under nonconjugacy –Rejection algorithm –Tailored general rejection method

4. Random variables: Conditional densities: Gibbs sampler provides an iterative Markovian updating scheme which enables us to make sample-based estimates of the marginal densities. Gelfand and Smith (1990) show that is better than a kernel density estimate for Gibbs sampler

5. Rejection algorithm

6. Split-normal and split-t envelope function Tailored rejection method

7. Split-normal

8. Split-t

9. Split-normal and Split-t

10. Split-normal

11. Split-t

12. Summary of tailored rejection method

14. Outline Introduction Algorithm Applications to Gibbs Sampler Examples A Generic Approach to Posterior Integration and Gibbs Sampling P. Muller

15. Algorithm

16. Implementation

17. Initialization Candidate generating =, Updating mean and covariance Accessing convergence Posterior inference

18. Convergence

19. Application to Gibbs Sampler

22. Alternatives to the Gibbs Sampling Schemes P. Muller

24. Metropolized Gibbs Sampler: An Improvement Jun S. Liu