Washington State University Lecture 26 Bayesian theory Homework 6 last due April 28 Friday 310PM Final exam May 4 Thursday 120 minutes 310510PM 50 questions Party April 28 Friday 430730 food at 500 130 Johnson Hall ID: 779473
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Slide1
Statistical Genomics
Zhiwu Zhang
Washington State University
Lecture 26: Bayesian theory
Slide2Homework 6 (last) due April 28, Friday, 3:10PMFinal exam: May 4 (Thursday), 120 minutes (3:10-5:10PM),
50 questionsParty: April 28, Friday, 4:30-7:30 (food at 5:00), 130 Johnson HallCourse evaluation starts on next Wednesday, April 17Administration
Slide3Outline
Concept development for genomic selectionBayesian theorem
Bayesian transformationBayesian likelihoodBayesian alphabet for genomic selection
Slide4All SNPs have same distribution
y=x1
g1 + x2g2 +
… + xpgp + e ~N(0, gi~N(0, I σg
2
)
U
K
σ
a
2
)
rrBLUP
g
BLUP
Slide5Selection of priors
D
istributions of
gi LSEsolve LL solelyFlatIdentical normalRRsolve REML by EMMA
σ
g
2
Slide6More realistic
y=x
1g1 + x2g
2 + … + xpgp + eN(0, I σg12)N(0, I σ
gp
2
)
N(0, I σ
g2
2
)
…
Out of control and
overfitting
?
Slide7Need 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
Slide8An 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%
Slide9Probability
P(pants)=60%*100+40%50%=80%
P(Boy)*P(Pants | Boy) + P(Girl)*P(Pants | Girl)
Slide10Inverse 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)
Slide11P(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)
Slide12Bayesian theorem
P(Boy|Pants)P(Pants)=P(Pants|Boy)P(Boy)
y(data)
q(parameters)X
Slide13Bayesian transformation
P(Boy | Pants)
P(Pants | Boy) P(Boy)
y(data)q(parameters) Likelihood of data given parametersP(y|q)Distribution of parameters (prior)P(
q
)
P(
q
|
y
)
Posterior distribution of
q
given
y
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 (?)
Slide15Prior knowledge
U
nsure
Reject100% male
G
ender distribution
100% female
unlikely
L
ikely
Safe
Four males were draw from a
public
schoo
l
. What are
the
gender proportions? --
Inverse
p
robability (?)
Slide16P(
G|y)
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(G)
Transform hard problem to easy one
Slide17P(
y|G)
p=seq(0, 1
, .01)n=4k=npyp=dbinom(k,n,p)theMax=pyp==max(pyp)pMax
=
p
[
theMax
]
plot
(
p
,
pyp
,
type
=
"
b
",main=paste(
"Data=", pMax,sep
=""))
Probability of having 4 males given male proportion
Slide18P(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
robability of male proportion
Slide19P(
y|G
) P(G)
ppy=pd*pyptheMax=ppy==max(ppy)pMax=p[theMax]plot(p,ppy,
type
=
"b"
,
main
=paste(
"Optimum="
,
pMax
,
sep
=
""
))
Probability of male proportion
given 4 males drawn
Slide20Depend what you believe
Male=Female
More Male
Slide21Ten are all males
Male=Female
More Male
Much more malevs. 57%
Slide22Bayesian likelihood
P(Boy | Pants)
P(Pants | Boy) P(Boy)
y(data)q(parameters) Likelihood of data given parametersP(y|q)Distribution of parameters (prior)P(q
)
P(
q
|
y
)
Posterior distribution of
q
given
y
Highlight
Concept development for genomic selectionBayesian theorem
Bayesian transformationBayesian likelihoodBayesian alphabet for genomic selection