Statistical Genomics Zhiwu Zhang Washington State University Lecture 27: Bayesian theorem

 Homework 6 (last) posted, due April 29, Friday, 3:10PM  Final exam: May 3, 120 minutes (3:10-5:10PM), 50  Evaluation due May 6 (7 out of 19 received). Administration

Outline  Concept development for genomic selection  Bayesian theorem  Bayesian transformation  Bayesian likelihood  Bayesian alphabet for genomic selection

All SNPs have same distribution y=x 1 g 1 + x 2 g 2 + … + x p g p + e ~N(0, b~N(0, I σ g 2 ) UK σa2)σa2) rrBLUP gBLUP

Selection of priors Distributions of g i LSE solve LL solely Flat Identical normal RR solve REML by EMMA σg2σg2

More realistic y=x 1 g 1 + x 2 g 2 + … + x p g p + e N(0, I σ g 2 ) … Out of control and overfitting?

Need help from Thomas Bayes "An Essay towards solving a Problem in the Doctrine of Chances" which was read to the Royal Society in 1763 after Bayes' death by Richard Price

An example from middle school A school by 60% boys and 40% girls. All boy wear pants. Half girls wear pants and half wear skirt. What is the probability to meet a student with pants? P(Pants)=60%*100+40%50%=80%

Probability P(pants)=60%*100+40%50%=80% P(Boy)*P(Pants | Boy) + P(Girl)*P(Pants | Girl)

Inverse question A school by 60% boys and 40% girls. All boy wear pants. Half girls wear pants and half wear skirt. Meet a student with pants. What is the probability the student is a boy? 60%*100+40%50% 60%*100% = 75% P(Boy | Pants)

P(Pants | Boy) P(Boy) + P(Pants | Girl) P(Girl) 60%*100+40%50% 60%*100 = 75% P(Pants | Boy) P(Boy) P(Pants) P(Pants | Boy) P(Boy)

Bayesian theorem P(Boy|Pants)P(Pants)=P(Pants|Boy)P(Boy)

Bayesian transformation P(Boy | Pants) P(Pants | Boy) P(Boy) y(data)  parameters  Likelihood of data given parameters P(y|  ) Distribution of parameters (prior) P(  ) P(  | y) Posterior distribution of  given y

Bayesian for hard problem A public school containing 60% males and 40% females. What is the probability to draw four males? -- Probability (36%) Four males were draw from a public school. What are the gender proportions? -- Inverse probability (?)

Prior knowledge Unsure Reject 100% male Gender distribution 100% female unlikely Likely Safe Four males were draw from a public school. What are the gender proportions? -- Inverse probability (?)

P(G|y) Probability of unknown given data (hard to solve) Probability of observed given unknown (easy to solve) Prior knowledge of unknown (freedom) P(G) Transform hard problem to easy one

P(y|G) p=seq(0, 1,.01) n=4 k=n pyp=dbinom(k,n,p) theMax=pyp==max(pyp) pMax=p[theMax] plot(p,pyp,type="b",main=paste("Data=", pMax,sep=""))

P(G) ps=p*10-5 pd=dnorm(ps) theMax=pd==max(pd) pMax=p[theMax] plot(p,pd,type="b",main=paste("Prior=", pMax,sep=""))

P(y|G) P(G) ppy=pd*pyp theMax=ppy==max(ppy) pMax=p[theMax] plot(p,ppy,type="b",main=paste("Optimum=", pMax,sep=""))

Depend what you believe

Ten are all males

Control of unknown parameters y=x 1 g 1 + x 2 g 2 + … + x p g p + e N(0, I σ g1 2 )N(0, I σ gp 2 )N(0, I σ g2 2 ) … Prior distribution

Selection of priors Prior distributions of g i RR Flat Others Bayes

One choice is inverse Chi-Square y=x 1 g 1 + x 2 g 2 + … + x p g p + e N(0, I σ g1 2 )N(0, I σ gp 2 )N(0, I σ g2 2 ) … σ gi 2 ~X -1 (v, S)Hyper parameters

Bayesian likelihood P(g i, σ gi 2, σ e 2 v, s | y) = P(y | g i, σ gi 2, σ e 2 v, s) P(g i, σ gi 2, σ e 2 v, s)

Variation of assumption σ gi 2 ~X -1 (v, S) with probability 1-π σ gi 2 =0 with probability π σ gi 2 >0 for all i Bayes A Bayes B }

Methodmarker effect Genomic Effect Variance Residual variance Unknown parameter Bayes AAll SNPsX -2 (v,S) X -2 (0,-2) Bayes BP(1-π)X -2 (v,S)X -2 (0,-2) Bayes Cπ P(1-π)X -2 (v,S’)X -2 (0,-2) π Bayes Dπ P(1-π)X -2 (v,S)X -2 (0,-2) S π BayesianLASSO P(1-π) Double exponential effects λ t BayesMulti, BayesR P(1-π) Multiple normal distributions γ Bayes alphabet

Highlight  Concept development for genomic selection  Bayesian theorem  Bayesian transformation  Bayesian likelihood  Bayesian alphabet for genomic selection