Lecture 26: Bayesian theory

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Presentation transcript:

Lecture 26: Bayesian theory Statistical Genomics Lecture 26: Bayesian theory Zhiwu Zhang Washington State University

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

All SNPs have same distribution rrBLUP gi~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 σg12) N(0, I σg22) N(0, I σgp2) 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 q(parameters) X P(Boy|Pants)P(Pants)=P(Pants|Boy)P(Boy) Constant y(data)

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 (0.6^4=12.96%) Four males were draw from a public school. What are the male proportion? -- Inverse probability (?)

Prior knowledge Gender distribution 100% male 100% female Safe unlikely Likely Unsure Reject Four males were draw from a public school. What is the male proportion? -- 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) Probability of having 4 males given male proportion 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=""))

Probability of male proportion P(G) Probability of male proportion 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(G|y)∝ P(y|G) P(G) Probability of male proportion given 4 males drawn ppy=pd*pyp theMax=ppy==max(ppy) pMax=p[theMax] plot(p,ppy,type="b",main=paste("Optimum=", pMax,sep=""))

Depend what you believe Male=Female More Male

Ten are all males More Male Male=Female Much more male vs. 57%

Bayesian likelihood ∝ 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)

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