what is Bayes theorem? posterior = likelihood * prior / C

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1 who was Bayes? Reverend Thomas Bayes (70-76) part-time mathematician buried in Bunhill Cemetary, Moongate, London famous paper in 763 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 θ Y=0 Y= prior pr(θ) = likelihood pr(y θ)= θ Y ( θ) Y posterior pr(θ Y)=? BS Yandell 005 Plant Microarray Course what is Bayes theorem? posterior = likelihood * prior / C θ pr( parameter data ) = pr( data parameter ) * pr( parameter ) / pr( data) pr( θ, Y) pr( Y θ) pr( θ) pr( θ Y) = = pr( Y) pr( Y) Y=0 Y= prior: probability of parameter before observing data pr( θ ) = pr( parameter ) eual chance of first ball being anywhere on the table posterior: probability of parameter after observing data pr( θ 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 θ ) = pr( data parameter ) basis for classical statistical inference about θ given Y BS Yandell 005 Plant Microarray Course

2 Bayes posterior for normal data n large n small prior prior mean actual mean y = phenotype values n large n small prior prior mean actual mean y = phenotype values small prior variance large prior variance BS Yandell 005 Plant Microarray Course 3 Bayes posterior for normal data model Y i = µ + E i environment E ~ N( 0, σ ), σ known likelihood Y~N( µ, σ ) prior µ ~N( µ 0, κσ ), κ known posterior: mean tends to sample mean single individual µ ~N( µ 0 + B (Y µ 0 ), B σ ) sample of n individuals µ ~ N with Y ( BnY + ( Bn ) µ 0, Bnσ / n) = sum Y / n i { i=,..., n} fudge factor (shrinks to ) B n κn = κn + BS Yandell 005 Plant Microarray Course 4

3 posterior genotypic means G n large n small prior Q QQ y = phenotype values BS Yandell 005 Plant Microarray Course 5 posterior genotypic means G posterior centered on sample genotypic mean but shrunken slightly toward overall mean prior: G ~ N( Y, κσ ) posterior: G ~ N ( BY + ( B ) Y, Bσ / n ) fudge factor: n B = count{ Q κn = κn + i = }, Y = sum Y / n i { Qi = } BS Yandell 005 Plant Microarray Course 6

4 Are strain differences real? strain differences? similar pattern parallel lines no interaction noise negligible? few d.f. per gene Can we trust SD g? SREBP PPARgamma FAS PPARalpha GPAT SCD fat liver muscle islet ACO PEPCK G6Pase BS Yandell 005 Plant Microarray Course 7 Bayesian shrinkage of gene-specific SD gene-specific SD from replication SD g = gene-specific standard deviation (df = ν ) robust abundance-based estimate σ(a g ) = smoothed over mrna depends only on abundance level A g (or constant) combine ideas into gene-specific hybrid prior σ g ~ inv-χ (ν 0, σ(a g ) ) posterior shrinkage estimate ν SD g + ν 0 σ(a g ) ν + ν 0 combines two statistically independent estimates BS Yandell 005 Plant Microarray Course 8

5 .00 SD for strain differences gene-specific σg 0.50 smooth of σg main effects liver σ(ag) interaction fat-liver σ(ag) 0.05 SD = spread fat σ(ag) fat liver muscle islet - BS Yandell average intensity 3 Plant Microarray Course % χ8 limits new (shrunk) σg size of shrinkage νσg + ν0σ(ag) ν + ν0 SD = spread gene-specific σg abundance σ(ag) Shrinkage Estimates of SD fat liver muscle islet BS Yandell Plant Microarray Course - 0 average intensity 3 0

6 How good is shrinkage model? prior for gene-specific σ g ~ inv-χ (ν 0, βσ(a g ) ) fudge β to adjust mean ν σ g + ν 0 βσ(a g ) ν + ν 0 histogram of ratio σ g / σ(a g ) empirical Bayes estimates Density χ approximation ν 0 = 5.45, β = χ approximation with β ν 0 = 3.56, β = BS Yandell 005 Plant Microarray Course Effect of SD Shrinkage on Detection fat-liver interaction shrinkage-based abundance-based 9 genes identified fat liver muscle islet S = (D-center)/spread A = average intensity BS Yandell 005 Plant Microarray Course

7 QTL Mapping (Gary Churchill) Key Idea: Crossing two inbred lines creates linkage diseuilibrium which in turn creates associations and linked segregating QTL QTL Marker Trait BS Yandell 005 Plant Microarray Course 3 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 (ideally > 0) BF pr( model data) / pr( model = pr( model ) / pr( model log( BF ) = log( LR) ( p data) p )log( n) pr(data model) = pr(data model ) BS Yandell 005 Plant Microarray Course 4 )