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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 l.jpg

Bayesian Density Regression

Author: David B. Dunson and Natesh Pillai

Presenter: Ya Xue

April 28, 2006


Outline l.jpg
Outline

  • Key idea

  • Proof

  • Application to HME


Bayesian density regression with standard dp l.jpg
Bayesian Density Regression with Standard DP

  • The regression model: (i=1,...,n)

  • Two cases:

Parametric model

Standard Dirichlet process mixture model


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Bayesian Density Regression with Standard DP

  • Model

  • The algorithm automatically finds the shrinkage of parameters


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Polya Urn Model

  • Standard Polya urn model

  • This paper proposed a generalized Polya urn model.

(1)

where is a kernel function.

monotonically as increases.


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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 xj close to xi.

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


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Outline

  • Key idea

  • Proof

  • Application to HME


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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



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Hierarchical Model

  • The hierarchical form


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Conditional Distribution

  • 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)


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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.


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Example

For example, n=4,

(a)

(b)

p(mi)


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Rewrite Equation (2)

  • Let

  • Then Eqn.(2) can be expressed as

(3)


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Theorem 4

Hence, Eqn. (3) is equivalent to



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Outline

  • Key idea

  • Proof

  • Application to HME


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Mixture Model

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

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