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Bayesian model of univariate linear regression for QTL detection
See Servin & Stephens (PLoS Genetics, 2007).
where β1 is in fact the additive effect of the SNP, noted a from now on, and β2 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, YTY, etc):
We use the prior and likelihood and keep only the terms in B:
We factorize some terms:
Let's define . We can see that ΩT = Ω, which means that Ω is a symmetric matrix. This is particularly useful here because we can use the following equality: Ω − 1ΩT = 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: