1. Plant Microarray Course
who was Bayes?
Reverend Thomas Bayes ( )
part-time mathematician
buried in Bunhill Cemetary, Moongate, London
famous paper in 1763 Phil Trans Roy Soc London
was Bayes the first with this idea? (Laplace?)
basic idea (from Bayes’ original example)
two billiard balls tossed at random (uniform) on table
where is first ball if the second is to its left (right)? 
first
second
prior pr() = 1
likelihood pr(Y | ) = 1–Y(1–)Y
posterior pr( |Y) = ?
Y=0
Y=1
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Plant Microarray Course

2. Plant Microarray Course
what is Bayes theorem?
posterior = likelihood * prior / C
pr( parameter | data ) =
pr( data | parameter ) * pr( parameter ) / pr( data)
prior: probability of parameter before observing data
pr(  ) = pr( parameter )
equal chance of first ball being anywhere on the table
posterior: probability of parameter after observing data
pr(  | Y ) = pr( parameter | data )
more likely second to left if first is near right end of table
likelihood: probability of data given parameters
pr( Y |  ) = pr( data | parameter )
basis for classical statistical inference about  given Y 
Y=0
Y=1
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Plant Microarray Course

3. Bayes posterior for normal data
actual mean
actual mean
prior mean
prior mean
small prior variance
large prior variance
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Plant Microarray Course

4. Bayes posterior for normal data
model Yi =  + Ei
environment E ~ N( 0, 2 ), 2 known
likelihood Y ~ N( , 2 )
prior  ~ N( 0, 2 ),  known
posterior: mean tends to sample mean
single individual  ~ N( 0 + B1(Y1 – 0), B12)
sample of n individuals
fudge factor
(shrinks to 1)
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Plant Microarray Course

5. posterior genotypic means Gq
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Plant Microarray Course

6. posterior genotypic means Gq
posterior centered on sample genotypic mean
but shrunken slightly toward overall mean
prior:
posterior:
fudge factor:
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Plant Microarray Course

7. Are strain differences real?
similar pattern
parallel lines
no interaction
noise negligible?
few d.f. per gene
Can we trust SDg ?
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Plant Microarray Course

8. Bayesian shrinkage of gene-specific SD
gene-specific SD from replication
SDg = gene-specific standard deviation (df = 1)
robust abundance-based estimate
(Ag) = smoothed over mRNA
depends only on abundance level Ag (or constant)
combine ideas into gene-specific hybrid
“prior” g2 ~ inv-2(0, (Ag)2)
“posterior” shrinkage estimate
1SDg2 + 0(Ag)2
1 + 0
combines two “statistically independent” estimates
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Plant Microarray Course

9. SD for strain differences
gene-specific g
smooth of g
main effects
fat (Ag)
liver (Ag)
interaction
fat-liver (Ag)
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Plant Microarray Course

10. Shrinkage Estimates of SD
gene-specific g
abundance (Ag)
95% 82 limits
new (shrunk) g
size of shrinkage
1g2 + 0(Ag)2
1 + 0
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Plant Microarray Course

11. How good is shrinkage model?
prior for gene-specific
g2 ~ inv-2(0, (Ag)2)
fudge  to adjust mean
1g2 + 0(Ag)2
1 + 0
histogram of ratio
g2 / (Ag)2
empirical Bayes estimates
2 approximation
0 = 5.45,  = 1
2 approximation with 
0 = 3.56,  = .809
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Plant Microarray Course

12. Effect of SD Shrinkage on Detection
fat-liver interaction
shrinkage-based
abundance-based
9 genes identified
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Plant Microarray Course

13. QTL Mapping (Gary Churchill)
Key Idea: Crossing two inbred lines creates linkage disequilibrium which in turn creates associations and linked segregating QTL
QTL
Marker
Trait
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Plant Microarray Course

14. Bayes factors for comparing models
goal of BF: balance model fit with model "complexity“
want “best model” that captures key features (model bias)
want to avoid “overfitting” the data in hand (poor prediction)
what is a Bayes factor (BF)?
ratio of posterior odds to prior odds
ratio of model likelihoods
BF is same as Bayes Information Criteria (BIC)
penalty on likelihood ratio (LR)
want Bayes factor to be much larger than 1 (ideally > 10)
BS Yandell © 2005
Plant Microarray Course