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

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<math>\mathcal{L}(\tau, B) = \mathsf{P}(Y  X, \tau, B)</math>  <math>\mathcal{L}(\tau, B) = \mathsf{P}(Y  X, \tau, B)</math>  
  <math>\mathcal{L}(\tau, B) = \left(\frac{\tau}{2 \pi}\right)^{  +  <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> 
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<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>  <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>  
+  
+  Again, we use the priors and likelihoods specified above (but everything inside the integral is kept inside it, even if it doesn't depend on <math>B</math>!):  
+  
+  <math>\mathsf{P}(\tau  Y, X) \propto \tau^{\frac{\kappa}{2}  1} e^{\frac{\lambda}{2} \tau} \int \tau^{1/2} \tau^{N/2} exp(\frac{\tau}{2} B^T \Sigma_B^{1} B) exp(\frac{\tau}{2} (Y  XB)^T (Y  XB)) \mathsf{d}B</math>  
+  
+  As we used a conjugate prior for <math>\tau</math>, we know that we expect a Gamma distribution for the posterior.  
+  Therefore, we can take <math>\tau^{N/2}</math> out of the integral and start guessing what looks like a Gamma distribution.  
+  We also factorize inside the exponential:  
+  
+  <math>\mathsf{P}(\tau  Y, X) \propto \tau^{\frac{N+\kappa}{2}  1} e^{\frac{\lambda}{2} \tau} \int \tau^{1/2} exp \left[\frac{\tau}{2} \left( (B  \Omega X^T Y)^T \Omega^{1} (B  \Omega X^T Y)  Y^T X \Omega X^T Y + Y^T Y \right) \right] \mathsf{d}B</math>  
+  
+  We recognize the conditional posterior of <math>B</math>.  
+  This allows us to use the fact that the pdf of the Normal distribution integrates to one:  
+  
+  <math>\mathsf{P}(\tau  Y, X) \propto \tau^{\frac{N+\kappa}{2}  1} e^{\frac{\lambda}{2} \tau} exp\left[\frac{\tau}{2} (Y^T X \Omega X^T Y + Y^T Y) \right]</math>  
+  
+  We finally recognize the following Gamma distribution:  
+  
+  <math>\tau  Y, X \sim \Gamma \left( \frac{N+\kappa}{2}, \; \frac{1}{2} (Y^T X \Omega X^T Y + Y^T Y + \lambda) \right)</math>  
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Revision as of 18:22, 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:
Again, we use the priors and likelihoods specified above (but everything inside the integral is kept inside it, even if it doesn't depend on B!):
As we used a conjugate prior for τ, we know that we expect a Gamma distribution for the posterior. Therefore, we can take τ^{N / 2} out of the integral and start guessing what looks like a Gamma distribution. We also factorize inside the exponential:
We recognize the conditional posterior of B. This allows us to use the fact that the pdf of the Normal distribution integrates to one:
We finally recognize the following Gamma distribution:
