1. Bayesian Models for Gene expression With DNA Microarray Data Joseph G. Ibrahim, Ming-Hui Chen, and Robert J. Gray Presented by Yong Zhang

2. Goals: 1)To build a model to compare between normal and tumor tissues and to find the genes that best distinguish between tissue types. 2)to develop model assessment techniques so as to assess the fit of a class of competing models.

3. Outline General Model Gene Selection Algo. Prior Distributions L measures(assessment) example

4. Data structure x: the expression level for a given gene C 0 : threshold value for which a gene is considered as not expressed Let p = P(x=c 0 ), then where y is the continuous part for x.

5. j=1, 2 index the tissue type(normal vs. tumor) i=1,2,…n j, ith individual g=1,…G, gth gene x jig : the gene expression mixture random variable for the jth tissue type for the ith individual and the gth gene.

6. The General Model Assume δ jig = 1(x jig =c 0 ) p jg =P(x jig =c 0 )=P(δ jig = 1)

7.  =( ,  2,p) Data D=(x 111,…x 2, n2,G,  ) Likelihood function for  : L(  |D)= In order to find which genes best discriminate between the normal and tumor tissues, let

8. Then we set such that we can use  g to judge them.

9. Prior Distributions  jg 2 ~ Inverse Gamma(a j0,b j0 )  j0 ~ N(m j0,v j0 2 ), j=1,2

10. b j0 ~ gamma(q j0,t j0 ) e jg ~ N(u j0,k j0 w j0 2 )

11. Gene Selection Algo. 1)For each gene, compute  g and 2)Select a “threshold” value, say r 0, to decide which genes are different. If 3)Once the gth genes are declared different, set  1g   2g, otherwise set  1g =  2g   g, where  g is treated as unknown.

12. Gene Selection Algo. 4) Create several submodels using several values of r 0. 5) Use L measure to decide which submodel is the best one(smallest L measure).

13. The properties of this approach 1)Model the gene expression level as a mixture random variable. 2)Use a lognormal model for the continuous part of the mixture. 3)Use L measure statistic for evaluating models.

14. L measure for model assessment It relies on the notion of an imaginary replicate experiment. Let z= (z 111, …, z 2,n2,G ) denote future values of a replicate experiment.

15. L measure is the expected squared Euclidean distance between x and z, A more general is The r.s. of the last formula can be got by MCMC.

18. Computational Algo.(MCMC) 1.

23. For 1–4 and 6, the generation is straightforward. For 5, we can use an adaptive rejection algorithm(Gilks and Wild, 1992) because the corresponding conditional posterior densities are log-concave.

24. Discussion That model development and prior distributions in this paper can be easily extended to handle three or more tissue types. More general classes of priors The gene selection criterions