User:Hussein Alasadi/Notebook/stephens/2013/10/03
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<math> L(f_{i,k,t}^{true}) = P(f_{obs}  f_{i,k,t}^{true}, M) = \frac{P( f_{i,k,t}^{true}  M, f_{obs}) P(f^{obs}M)}{P(f_{i,k,t}^{true}  M)}</math>  <math> L(f_{i,k,t}^{true}) = P(f_{obs}  f_{i,k,t}^{true}, M) = \frac{P( f_{i,k,t}^{true}  M, f_{obs}) P(f^{obs}M)}{P(f_{i,k,t}^{true}  M)}</math>  
  where  +  where <math> f_{i,k,t}^{true}  M </math> ~ <math> N(\mu, \sigma^2 \Sigma) </math> The parameter <math> \sigma^2 </math> allows for overdispersion 
+  
+  where <math> f^{obs} M </math> ~ <math> N_{p1} (\mu_2, \sigma^2 \Sigma_{22} + \epsilon^2I </math>  
Revision as of 22:45, 16 October 2013
analyzing pooled sequenced data with selection  Main project page Next entry 
Notes from MeetingConsider a single lineage for now. X_{j} = frequency of "1" allele at SNP j in the pool (i.e. the true frequency of the 1 allele in the pool)
= number of "0", "1" alleles at SNP j ()
~ Normal approximation to binomial The variance of this distribution results from error due to binomial sampling. To simplify, we just plug in for X_{j}
f_{i,k,j} = frequency of reference allele in group i, replicate and SNP j. vector of frequencies Without loss of generality, we assume that the putative selected site is site j = 1
We assume a prior on our vector of frequencies based on our panel of SNPs (M) of dimension 2mxp ~ MVN(μ,Σ)
where if i = j or if i not equal to j
(f_{i,k,2},....,f_{i,k,p})  f_{i,k,1},M ~ The conditional distribution is easily obtained when we use a result derived here. let X_{2} = (f_{i,k,2},....,f_{i,k,p}) and X_{1} = f_{i,k,1} X_{2}  X_{1},M ~ Thus
let
where ~ N(μ,σ^{2}Σ) The parameter σ^{2} allows for overdispersion where f^{obs}  M ~ N_{p − 1}(μ_{2},σ^{2}Σ_{22} + ε^{2}I
