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

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  * '''Likelihood''':  +  * '''Likelihood''': <math>\forall i \in \{1,\ldots,N\}, \; y_i = \mu + \beta_1 g_i + \beta_2 \mathbf{1}_{g_i=1} + \epsilon_i \text{ with } \epsilon_i \overset{i.i.d}{\sim} \mathcal{N}(0,\tau^{1})</math> 
  +  
  <math>\forall i \in \{1,\ldots,N\}, \; y_i = \mu + \beta_1 g_i + \beta_2 \mathbf{1}_{g_i=1} + \epsilon_i  +  
  +  
  with  +  
where <math>\beta_1</math> is in fact the additive effect of the SNP, noted <math>a</math> from now on, and <math>\beta_2</math> is the dominance effect of the SNP, <math>d = a k</math>.  where <math>\beta_1</math> is in fact the additive effect of the SNP, noted <math>a</math> from now on, and <math>\beta_2</math> is the dominance effect of the SNP, <math>d = a k</math>.  
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Let's now write in matrix notation:  Let's now write in matrix notation:  
  <math>Y = X B + E  +  <math>Y = X B + E \text{ where } B = [ \mu \; a \; d ]^T</math> 
  +  
  where  +  
which gives the following conditional distribution for the phenotypes:  which gives the following conditional distribution for the phenotypes:  
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<math>B  Y, X, \tau \sim \mathcal{N}(\Omega X^TY, \tau^{1} \Omega)</math>  <math>B  Y, X, \tau \sim \mathcal{N}(\Omega X^TY, \tau^{1} \Omega)</math>  
+  
+  
+  * '''Posterior of <math>\tau</math>''':  
+  
+  Similarly to the equations above:  
+  
+  <math>\mathsf{P}(\tau  Y, X) \propto \mathsf{P}(\tau) \mathsf{P}(Y  X, \tau)</math>  
+  
+  But now, to handle the second term, we need to integrate over <math>B</math>, thus effectively taking into account the uncertainty in <math>B</math>:  
+  
+  <math>\mathsf{P}(\tau  Y, X) \propto \mathsf{P}(\tau) \int \mathsf{P}(B  \tau) \mathsf{P}(Y  X, \tau, B) \mathsf{d}B</math>  
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Revision as of 12:46, 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:
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:
