Gibbs - Mixture Models and Gibbs Sampling Readings: Hoff...

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Unformatted text preview: Mixture Models and Gibbs Sampling Readings: Hoff CHapter 6 October 15, 2010 Eyes Example Bowmaker et al. (1985) analyze data on the peak sensitivity wavelengths for individual microspectophotometric records on a small set of monkeys eyes. WinBUGs Examples Volume II gives the data for one monkey. Histogram of Eyes Data Y Frequency 530 535 540 545 550 555 1 2 3 4 5 6 7 Mixture Model Model the data using a Mixture of 2 Normals: Y i | 1 , 2 , 2 1 , 2 2 , 1 , 2 ind 1 N( 1 , 2 1 ) + 2 N( 2 , 2 2 ) Which is equivalent to Y i | T i , 1 , 2 , 2 1 , 2 2 ind N( T i , 2 T i ) T i iid Cat( T , ) where T i is a latent variable indicating which group observation i belongs to i.e. T i { 1 , 2 } and P( T i = j ) = j , and j j = 1 Prior Distributions Based on WinBUGS example, adopt noninformative prior distributions j iid N(0 , 1 . 10 6 ) 1 / 2 j iid G(0 . 001 , . 001) ( 1 , 2 ) Dirichlet(1 , 1) 1 Beta(1 , 1)) Proper prior distributions are necessary for Mixture Models; if prior on or 2 is improper, then the posterior will also be improper if all observations are in one group! False sense of security with vague but proper priors... Single Component Gibbs Sampler Find full conditional distributions for I 1 | 2 , 2 1 , 2 2 , 1 , 2 , T 1 , . . . , T N , Y (normal) I 2 | 1 , 2 1 , 2 2...
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Gibbs - Mixture Models and Gibbs Sampling Readings: Hoff...

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