1. Bayesian Density Regression Author: David B. Dunson and Natesh Pillai Presenter: Ya Xue April 28, 2006

2. Outline Key idea Proof Application to HME

3. Bayesian Density Regression with Standard DP The regression model: (i=1,...,n) Two cases: 1. 2. Parametric model Standard Dirichlet process mixture model

4. Bayesian Density Regression with Standard DP Model The algorithm automatically finds the shrinkage of parameters

5. Polya Urn Model Standard Polya urn model This paper proposed a generalized Polya urn model. where is a kernel function. monotonically as increases. (1)

6. Idea – Spatial DP Equation (1) implies The prior probability of setting decreases as increases. The prior probability of increases as more neighbors are added that have predictor values x j close to x i. The expected prior probability of increases in proportion to the hyperparameter.

7. Outline Key idea Proof Application to HME

8. Spatial Varying Regression Model At a given location in the feature space, A mixture of an innovation random measure and neighboring random measures j~i indexes samples

9. Theorem 1

10. Hierarchical Model The hierarchical form

11. Let denote an index set for the subjects drawn from the jth mixture component, for j=1,...,n. Then we have for Conditioning on Z, we can use the Polya urn result to obtain the conditional prior Only the subvector of elements of belonging to are informative. Conditional Distribution (2)

12. Marginalize over Z We obtain the following generalization of the Polya urn scheme (a) (b) if sample i and j belong to the same mixture component.

13. Example (a)(b) For example, n=4, p(m i )

14. Rewrite Equation (2) Let Then Eqn.(2) can be expressed as (3)

15. Theorem 4 Hence, Eqn. (3) is equivalent to

16. Predictive distribution

17. Outline Key idea Proof Application to HME

18. Mixture Model We simulate data from a mixture of two normal linear regression models Poor results obtained by using the standard DP mixture model.