User:Timothee Flutre/Notebook/Postdoc/2011/11/10
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(→Bayesian model of univariate linear regression for QTL detection: start posterior of tau) 
(→Bayesian model of univariate linear regression for QTL detection: add lik and joint prior) 

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<math>Y  X, B, \tau \sim \mathcal{N}(XB, \tau^{1} I_N)</math>  <math>Y  X, B, \tau \sim \mathcal{N}(XB, \tau^{1} I_N)</math>  
+  The likelihood of the parameters given the data is therefore:  
  * '''Priors''': conjugate  +  <math>\mathcal{L}(\tau, B) = \mathsf{P}(Y  X, \tau, B)</math> 
+  
+  <math>\mathcal{L}(\tau, B) = \left(\frac{\tau}{2 \pi}\right)^{n/2} exp \left( \frac{\tau}{2} (Y  XB)^T (Y  XB) \right)</math>  
+  
+  
+  * '''Priors''': we use the usual conjugate prior  
+  
+  <math>\mathsf{P}(\tau, B) = \mathsf{P}(\tau) \mathsf{P}(B  \tau)</math>  
<math>\tau \sim \Gamma(\kappa/2, \, \lambda/2)</math>  <math>\tau \sim \Gamma(\kappa/2, \, \lambda/2)</math> 
Revision as of 12:57, 21 November 2012
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Bayesian model of univariate linear regression for QTL detectionSee Servin & Stephens (PLoS Genetics, 2007).
where β_{1} is in fact the additive effect of the SNP, noted a from now on, and β_{2} is the dominance effect of the SNP, d = ak. Let's now write in matrix notation:
which gives the following conditional distribution for the phenotypes:
The likelihood of the parameters given the data is therefore:
Here and in the following, we neglect all constants (e.g. normalization constant, Y^{T}Y, etc):
We use the prior and likelihood and keep only the terms in B:
We expand:
We factorize some terms:
Let's define . We can see that Ω^{T} = Ω, which means that Ω is a symmetric matrix. This is particularly useful here because we can use the following equality: Ω^{ − 1}Ω^{T} = I.
This now becomes easy to factorizes totally:
We recognize the kernel of a Normal distribution, allowing us to write the conditional posterior as:
Similarly to the equations above:
But now, to handle the second term, we need to integrate over B, thus effectively taking into account the uncertainty in B:
