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

Administration 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).

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

All SNPs have same distribution rrBLUP b~N(0, I σg2) y=x1g1 + x2g2 + … + xpgp + e gBLUP U ~N(0, K σa2)

Selection of priors Distributions of gi Flat Identical normal LSE solve LL solely RR solve REML by EMMA Distributions of gi

Out of control and overfitting? More realistic Out of control and overfitting? … N(0, I σg2) N(0, I σg2) N(0, I σg2) y=x1g1 + x2g2 + … + xpgp + e

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%

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

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

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

Bayesian transformation Posterior distribution of q given y q(parameters) y(data) P(q | y) P(Boy | Pants) ∝ P(Pants | Boy) P(Boy) Likelihood of data given parameters P(y|q) Distribution of parameters (prior) P(q)

Bayesian for hard problem A public school containing 60% males and 40% females. What is the probability to draw four males? -- Probability (12.96%) Four males were draw from a public school. What are the gender proportions? -- Inverse probability (?)

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

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

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 Prior distribution … N(0, I σg12) N(0, I σg22) N(0, I σgp2) y=x1g1 + x2g2 + … + xpgp + e

Prior distributions of gi Selection of priors Flat Others RR Bayes Prior distributions of gi

One choice is inverse Chi-Square σgi2~X-1(v, S) Hyper parameters … N(0, I σg12) N(0, I σg22) N(0, I σgp2) y=x1g1 + x2g2 + … + xpgp + e

P(y | gi, σgi2, σe2 v, s) P(gi, σgi2, σe2 v, s) Bayesian likelihood P(gi, σgi2, σe2 v, s | y) = P(y | gi, σgi2, σe2 v, s) P(gi, σgi2, σe2 v, s)

Variation of assumption σgi2>0 for all i Bayes A } σgi2=0 with probability π Bayes B σgi2~X-1(v, S) with probability 1-π

Genomic Effect Variance Bayes alphabet Method marker effect Genomic Effect Variance Residual variance Unknown parameter Bayes A All SNPs X-2(v,S) X-2(0,-2) Bayes B P(1-π) Bayes Cπ X-2(v,S’) π Bayes Dπ S π BayesianLASSO Double exponential effects λ t BayesMulti, BayesR Multiple normal distributions γ

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