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Lecture 27: Bayesian theorem

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

2 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).

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

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

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

6 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

7 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

8 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%

9 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)

10 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%

11 P(Pants | Boy) P(Boy) + P(Pants | Girl) P(Girl)
P(Boy|Pants) 60%*100 = % 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)

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

13 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)

14 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 (?)

15 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 (?)

16 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)

17 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=""))

18 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=""))

19 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=""))

20 Depend what you believe

21 Ten are all males

22 Control of unknown parameters
Prior distribution … N(0, I σg12) N(0, I σg22) N(0, I σgp2) y=x1g1 + x2g2 + … + xpgp + e

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

24 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

25 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)

26 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

27 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

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

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