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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 :

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 !):

As we used a conjugate prior for , we know that we expect a Gamma distribution for the posterior. Therefore, we can take 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 . This allows us to use the fact that the pdf of the Normal distribution integrates to one:

We finally recognize the following Gamma distribution: