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# Bayesian Density Regression - PowerPoint PPT Presentation

Bayesian Density Regression. Author: David B. Dunson and Natesh Pillai Presenter: Ya Xue April 28, 2006. Outline. Key idea Proof Application to HME. Bayesian Density Regression with Standard DP. The regression model: (i=1,...,n) Two cases:. Parametric model.

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### Bayesian Density Regression

Author: David B. Dunson and Natesh Pillai

Presenter: Ya Xue

April 28, 2006

• Key idea

• Proof

• Application to HME

• The regression model: (i=1,...,n)

• Two cases:

Parametric model

Standard Dirichlet process mixture model

• Model

• The algorithm automatically finds the shrinkage of parameters

• Standard Polya urn model

• This paper proposed a generalized Polya urn model.

(1)

where is a kernel function.

monotonically as increases.

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 xj close to xi.

• The expected prior probability of increases in proportion to the hyperparameter .

• Key idea

• Proof

• Application to HME

• At a given location in the feature space,

A mixture of an innovation random measure

and neighboring random measures

j~i indexes samples

• The hierarchical form

• 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.

(2)

• We obtain the following generalization of the Polya urn scheme

(a)

(b)

if sample i and j belong to the same mixture component.

For example, n=4,

(a)

(b)

p(mi)

• Let

• Then Eqn.(2) can be expressed as

(3)

Hence, Eqn. (3) is equivalent to

• Key idea

• Proof

• Application to HME

• We simulate data from a mixture of two normal linear regression models

• Poor results obtained by using the standard DP mixture model.