1. Introduction: Metropolis-Hasting Sampler Purpose--To draw samples from a probability distribution There are three steps 1Propose a move from x to y 2Accept the move from x to y 3. Move from x to y

2. Detailed Balance Equations Rate from x to y Rate from y to x

3. Detailed Balance Equation

6. When is Metropolis-Hasting Sampler useful? No explicit formula for Yet the likelihood ratio has an explicit formula

7. Example 1 Bayesian Posterior

8. Frequency Statistics Suppose that the likelihood of Where are latent variables Use MCMC to simulate Then

9. 1 2 3 4 5 6 7 8 9 10 11

10. 1 2 3 4 5 6 7 8 9 10 11

11. 1 2 3 4 5 6 7 8 9 10 11

12. 1 2 3 4 5 6 7 8 9 10 11 4 switched with 2

17. Importance Sampling

18. Sample from the proposal distribution Importance Sampling