User:Timothee Flutre/Notebook/Postdoc/2011/11/10

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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 $y_1,\ldots,y_N$ the (quantitative) phenotypes (e.g. expression level at a given gene), and $g_1,\ldots,g_N$ 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:

$\forall i \in \{1,\ldots,N\}, \; y_i = \mu + \beta_1 g_i + \beta_2 \mathbf{1}_{g_i=1} + \epsilon_i$

with: $\epsilon_i \overset{i.i.d}{\sim} \mathcal{N}(0,\tau^{-1})$

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, $d = a k$ .

Let's now write in matrix notation:

$Y = X B + E$

where $B = [ \mu \; a \; d ]^T$

which gives the following conditional distribution for the phenotypes:

$Y | X, B, \tau \sim \mathcal{N}(XB, \tau^{-1} I_N)$

Priors: conjugate

$\tau \sim \Gamma(\kappa/2, \, \lambda/2)$

$B | \tau \sim \mathcal{N}(\vec{0}, \, \tau^{-1} \Sigma_B) \text{ with } \Sigma_B = diag(\sigma_{\mu}^2, \sigma_a^2, \sigma_d^2)$

Joint posterior:

$\mathsf{P}(\tau, B | Y, X) = \mathsf{P}(\tau | Y, X) \mathsf{P}(B | Y, X, \tau)$

Conditional posterior of B:

$\mathsf{P}(B | Y, X, \tau) = \mathsf{P}(B, Y | X, \tau)$

$\mathsf{P}(B | Y, X, \tau) = \frac{\mathsf{P}(B, Y | X, \tau)}{\mathsf{P}(Y | X, \tau)}$

$\mathsf{P}(B | Y, X, \tau) = \frac{\mathsf{P}(B | \tau) \mathsf{P}(Y | X, B, \tau)}{\int \mathsf{P}(B | \tau) \mathsf{P}(Y | X, \tau, B) \mathsf{d}B}$

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

$\mathsf{P}(B | Y, X, \tau) \propto \mathsf{P}(B | \tau) \mathsf{P}(Y | X, \tau, B)$

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

$\mathsf{P}(B | Y, X, \tau) \propto exp(B^T \Sigma_B^{-1} B) exp((Y-XB)^T(Y-XB))$

We expand:

$\mathsf{P}(B | Y, X, \tau) \propto exp(B^T \Sigma_B^{-1} B - Y^TXB -B^TX^TY + B^TX^TXB)$

We factorize some terms:

$\mathsf{P}(B | Y, X, \tau) \propto exp(B^T (\Sigma_B^{-1} + X^TX) B - Y^TXB -B^TX^TY)$

Let's define $\Omega = (\Sigma_B^{-1} + X^TX)^{-1}$ . 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: $Ω - 1 Ω T = I$ .

$\mathsf{P}(B | Y, X, \tau) \propto exp(B^T \Omega^{-1} B - (X^TY)^T\Omega^{-1}\Omega^TB -B^T\Omega^{-1}\Omega^TX^TY)$

This now becomes easy to factorizes totally:

$\mathsf{P}(B | Y, X, \tau) \propto exp((B^T - \Omega X^TY)^T\Omega^{-1}(B - \Omega X^TY))$

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

$B | Y, X, \tau \sim \mathcal{N}(\Omega X^TY, \tau^{-1} \Omega)$

@@ Line 15: / Line 15: @@
-* '''Goal''': we want (i) to assess the evidence in the data for an effect of the genotype on the phenotype, and (ii) estimate the posterior distribution of this effect.
+* '''Goal''': we want to assess the evidence in the data for an effect of the genotype on the phenotype.
@@ Line 45: / Line 45: @@
 <math>B | \tau \sim \mathcal{N}(\vec{0}, \, \tau^{-1} \Sigma_B) \text{ with } \Sigma_B = diag(\sigma_{\mu}^2, \sigma_a^2, \sigma_d^2)</math>
+* '''Joint posterior''':
+<math>\mathsf{P}(\tau, B | Y, X) = \mathsf{P}(\tau | Y, X) \mathsf{P}(B | Y, X, \tau)</math>
+* '''Conditional posterior of B''':
+<math>\mathsf{P}(B | Y, X, \tau) = \mathsf{P}(B, Y | X, \tau)</math>
+<math>\mathsf{P}(B | Y, X, \tau) = \frac{\mathsf{P}(B, Y | X, \tau)}{\mathsf{P}(Y | X, \tau)}</math>
+<math>\mathsf{P}(B | Y, X, \tau) = \frac{\mathsf{P}(B | \tau) \mathsf{P}(Y | X, B, \tau)}{\int \mathsf{P}(B | \tau) \mathsf{P}(Y | X, \tau, B) \mathsf{d}B}</math>
+Here and in the following, we neglect all constants (e.g. normalization constant, <math>Y^TY</math>, etc):
+<math>\mathsf{P}(B | Y, X, \tau) \propto \mathsf{P}(B | \tau) \mathsf{P}(Y | X, \tau, B)</math>
+We use the prior and likelihood and keep only the terms in <math>B</math>:
+<math>\mathsf{P}(B | Y, X, \tau) \propto exp(B^T \Sigma_B^{-1} B) exp((Y-XB)^T(Y-XB))</math>
+We expand:
+<math>\mathsf{P}(B | Y, X, \tau) \propto exp(B^T \Sigma_B^{-1} B - Y^TXB -B^TX^TY + B^TX^TXB)</math>
+We factorize some terms:
+<math>\mathsf{P}(B | Y, X, \tau) \propto exp(B^T (\Sigma_B^{-1} + X^TX) B - Y^TXB -B^TX^TY)</math>
+Let's define <math>\Omega = (\Sigma_B^{-1} + X^TX)^{-1}</math>. We can see that <math>\Omega^T=\Omega</math>, which means that <math>\Omega</math> is a [http://en.wikipedia.org/wiki/Symmetric_matrix symmetric matrix].
+This is particularly useful here because we can use the following equality: <math>\Omega^{-1}\Omega^T=I</math>.
+<math>\mathsf{P}(B | Y, X, \tau) \propto exp(B^T \Omega^{-1} B - (X^TY)^T\Omega^{-1}\Omega^TB -B^T\Omega^{-1}\Omega^TX^TY)</math>
+This now becomes easy to factorizes totally:
+<math>\mathsf{P}(B | Y, X, \tau) \propto exp((B^T - \Omega X^TY)^T\Omega^{-1}(B - \Omega X^TY))</math>
+We recognize the [http://en.wikipedia.org/wiki/Kernel_%28statistics%29 kernel] of a Normal distribution, allowing us to write the conditional posterior as:
+<math>B | Y, X, \tau \sim \mathcal{N}(\Omega X^TY, \tau^{-1} \Omega)</math>
 <!-- ##### DO NOT edit below this line unless you know what you are doing. ##### -->

User:Timothee Flutre/Notebook/Postdoc/2011/11/10

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Revision as of 13:30, 21 November 2012

Bayesian model of univariate linear regression for QTL detection

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