who was Bayes?
• Reverend Thomas Bayes (1702-1761)
–
–
–
–
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
Y=0
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Y=1
prior
pr() = 1
likelihood pr(Y | ) = 1–Y(1–)Y
posterior pr( |Y) = ?
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what is Bayes theorem?
• posterior = likelihood * prior / C

pr( parameter | data ) =
pr( data | parameter ) * pr( parameter ) / pr( data)
pr ( , Y ) pr (Y |  )  pr ( )
pr ( | Y ) 

pr (Y )
pr (Y )
Y=0
Y=1
• 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
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8
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prior mean
actual mean
n small prior
n large
n large
prior mean
n small
prior
actual mean
Bayes posterior for normal data
12
14
16
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y = phenotype values
10
12
14
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y = phenotype values
small prior variance
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large prior variance
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Bayes posterior for normal data
model
environment
likelihood
prior
Yi =  + Ei
E ~ N( 0, 2 ), 2 known
Y ~ N( , 2 )
 ~ N( 0, 2 ),  known
posterior:
single individual
mean tends to sample mean
 ~ N( 0 + B1(Y1 – 0), B12)
sample of n individuals
 ~ N BnY  (1  Bn )  0 , Bn 2 / n 
with Y  sum Yi / n
{i 1,..., n}
fudge factor
(shrinks to 1)
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Bn 
n
n  1
1
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n large
n small prior
posterior genotypic means Gq
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qq
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10
Qq
12
y = phenotype values
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QQ
5
posterior genotypic means Gq
posterior centered on sample genotypic mean
but shrunken slightly toward overall mean


prior:
Gq ~ N Y ,  2
posterior:
Gq ~ N BqYq  (1  Bq )Y , Bq 2 / nq


nq  count {Qi  q}, Yq  sum Yi / nq
{Qi  q}
fudge factor:
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Bq 
nq
nq  1
1
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Are strain differences real?
SREBP1
BTBR
SCD1
2.6
2.3
B6
BTBR
BTBR
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BTBR
G6Pase
0.0
1.0
2.0
1.0 1.4 1.8 2.2
2.2
1.8
1.4
B6
B6
3.0
PEPCK
BTBR
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BTBR
2.9
2
-2 -1 0
BTBR
islet
BTBR
B6
PPARalpha
ACO
B6
muscle
BTBR
1
0.5
B6
B6
B6
PPARgamma
-0.5 0.0
few d.f. per gene
Can we trust SDg ?
1.2
1.9
1.7
-2.5
B6
noise negligible?
liver
GPAT
1.6
-1.5
similar pattern
parallel lines
no interaction
fat
FAS
2.1
strain differences?
B6
BTBR
B6
BTBR
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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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1.00
SD for strain differences
gene-specific g
0.50
smooth of g
main effects
liver (Ag)
interaction
fat-liver (Ag)
B6
0.05
SD = spread
0.10
0.20
fat (Ag)
B6
fat
liver
muscle
islet
BTBR
BTBR
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-2
-1
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0
1
average intensity
2
3
9
B6
0.05
95% 82 limits
new (shrunk) g
size of shrinkage
1g2 + 0(Ag)2
1 + 0
SD = spread
0.10
0.20
gene-specific g
abundance (Ag)
0.50
1.00
Shrinkage Estimates of SD
B6
fat
liver
muscle
islet
BTBR
BTBR
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-2
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-1
0
1
average intensity
2
3
10
How good is shrinkage model?
0.8
prior for gene-specific
0 = 5.45,  = 1
2 approximation with 
0 = 3.56,  = .809
0.4
0.2
0.0
2 approximation
Density
histogram of ratio
g2 / (Ag)2
empirical Bayes estimates
0.6
g2 ~ inv-2(0, (Ag)2)
fudge  to adjust mean
1g2 + 0(Ag)2
1 + 0
0
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6
8
10
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B6
B6
liver
muscle
10
5
0
-5
islet
BTBR
-15
fat
-10
fat-liver interaction
shrinkage-based
abundance-based
9 genes identified
S = (D-center)/spread
15
Effect of SD Shrinkage on Detection
0.02
BTBR
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0.10
0.50
2.00
10.00
A = average intensity
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QTL Mapping (Gary Churchill)
Key Idea: Crossing two inbred lines creates
linkage disequilibrium which in turn creates
associations and linked segregating QTL
QTL
Marker
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Trait
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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)
pr ( model 1 | data ) / pr ( model 2 | data ) pr (data | model 1 )
BF12 

pr ( model 1 ) / pr ( model 2 )
pr (data | model 2 )
 2 log( BF12 )  2 log( LR)  ( p2  p1 ) log( n)
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