Difference between revisions of "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>

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Bayesian model of univariate linear regression for QTL detection

See Servin & Stephens (PLoS Genetics, 2007).


  • Data: let's assume that we obtained data from N individuals. We note the (quantitative) phenotypes (e.g. expression level at a given gene), and the genotypes at a given SNP (as allele dose, 0, 1 or 2).


  • Goal: we want to assess the evidence in the data for an effect of the genotype on the phenotype.


  • Assumptions: the relationship between genotype and phenotype is linear; the individuals are not genetically related; there is no hidden confounding factors in the phenotypes.


  • Likelihood:

where is in fact the additive effect of the SNP, noted from now on, and is the dominance effect of the SNP, .

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:


  • Priors: we use the usual conjugate prior


  • Joint posterior:


  • Conditional posterior of B:

Here and in the following, we neglect all constants (e.g. normalization constant, , etc):

We use the prior and likelihood and keep only the terms in :

We expand:

We factorize some terms:

Let's define . We can see that , which means that is a symmetric matrix. This is particularly useful here because we can use the following equality: .

This now becomes easy to factorizes totally:

We recognize the kernel of a Normal distribution, allowing us to write the conditional posterior as:


  • Posterior of :

Similarly to the equations above:

But now, to handle the second term, we need to integrate over , thus effectively taking into account the uncertainty in :