User:Timothee Flutre/Notebook/Postdoc/2011/11/10: Difference between revisions
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* '''Likelihood''': | * '''Likelihood''': <math>\forall i \in \{1,\ldots,N\}, \; y_i = \mu + \beta_1 g_i + \beta_2 \mathbf{1}_{g_i=1} + \epsilon_i \text{ with } \epsilon_i \overset{i.i.d}{\sim} \mathcal{N}(0,\tau^{-1})</math> | ||
<math>\forall i \in \{1,\ldots,N\}, \; y_i = \mu + \beta_1 g_i + \beta_2 \mathbf{1}_{g_i=1} + \epsilon_i | |||
with | |||
where <math>\beta_1</math> is in fact the additive effect of the SNP, noted <math>a</math> from now on, and <math>\beta_2</math> is the dominance effect of the SNP, <math>d = a k</math>. | where <math>\beta_1</math> is in fact the additive effect of the SNP, noted <math>a</math> from now on, and <math>\beta_2</math> is the dominance effect of the SNP, <math>d = a k</math>. | ||
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Let's now write in matrix notation: | Let's now write in matrix notation: | ||
<math>Y = X B + E | <math>Y = X B + E \text{ where } B = [ \mu \; a \; d ]^T</math> | ||
where | |||
which gives the following conditional distribution for the phenotypes: | which gives the following conditional distribution for the phenotypes: | ||
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<math>B | Y, X, \tau \sim \mathcal{N}(\Omega X^TY, \tau^{-1} \Omega)</math> | <math>B | Y, X, \tau \sim \mathcal{N}(\Omega X^TY, \tau^{-1} \Omega)</math> | ||
* '''Posterior of <math>\tau</math>''': | |||
Similarly to the equations above: | |||
<math>\mathsf{P}(\tau | Y, X) \propto \mathsf{P}(\tau) \mathsf{P}(Y | X, \tau)</math> | |||
But now, to handle the second term, we need to integrate over <math>B</math>, thus effectively taking into account the uncertainty in <math>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> | |||
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Revision as of 10:46, 21 November 2012
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Bayesian model of univariate linear regression for QTL detectionSee Servin & Stephens (PLoS Genetics, 2007).
where [math]\displaystyle{ \beta_1 }[/math] is in fact the additive effect of the SNP, noted [math]\displaystyle{ a }[/math] from now on, and [math]\displaystyle{ \beta_2 }[/math] is the dominance effect of the SNP, [math]\displaystyle{ d = a k }[/math]. Let's now write in matrix notation: [math]\displaystyle{ Y = X B + E \text{ where } B = [ \mu \; a \; d ]^T }[/math] which gives the following conditional distribution for the phenotypes: [math]\displaystyle{ Y | X, B, \tau \sim \mathcal{N}(XB, \tau^{-1} I_N) }[/math]
[math]\displaystyle{ \tau \sim \Gamma(\kappa/2, \, \lambda/2) }[/math] [math]\displaystyle{ 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]
[math]\displaystyle{ \mathsf{P}(\tau, B | Y, X) = \mathsf{P}(\tau | Y, X) \mathsf{P}(B | Y, X, \tau) }[/math]
[math]\displaystyle{ \mathsf{P}(B | Y, X, \tau) = \mathsf{P}(B, Y | X, \tau) }[/math] [math]\displaystyle{ \mathsf{P}(B | Y, X, \tau) = \frac{\mathsf{P}(B, Y | X, \tau)}{\mathsf{P}(Y | X, \tau)} }[/math] [math]\displaystyle{ \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]\displaystyle{ Y^TY }[/math], etc): [math]\displaystyle{ \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]\displaystyle{ B }[/math]: [math]\displaystyle{ \mathsf{P}(B | Y, X, \tau) \propto exp(B^T \Sigma_B^{-1} B) exp((Y-XB)^T(Y-XB)) }[/math] We expand: [math]\displaystyle{ \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]\displaystyle{ \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]\displaystyle{ \Omega = (\Sigma_B^{-1} + X^TX)^{-1} }[/math]. We can see that [math]\displaystyle{ \Omega^T=\Omega }[/math], which means that [math]\displaystyle{ \Omega }[/math] is a symmetric matrix. This is particularly useful here because we can use the following equality: [math]\displaystyle{ \Omega^{-1}\Omega^T=I }[/math]. [math]\displaystyle{ \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]\displaystyle{ \mathsf{P}(B | Y, X, \tau) \propto exp((B^T - \Omega X^TY)^T\Omega^{-1}(B - \Omega X^TY)) }[/math] We recognize the kernel of a Normal distribution, allowing us to write the conditional posterior as: [math]\displaystyle{ B | Y, X, \tau \sim \mathcal{N}(\Omega X^TY, \tau^{-1} \Omega) }[/math]
Similarly to the equations above: [math]\displaystyle{ \mathsf{P}(\tau | Y, X) \propto \mathsf{P}(\tau) \mathsf{P}(Y | X, \tau) }[/math] But now, to handle the second term, we need to integrate over [math]\displaystyle{ B }[/math], thus effectively taking into account the uncertainty in [math]\displaystyle{ B }[/math]: [math]\displaystyle{ \mathsf{P}(\tau | Y, X) \propto \mathsf{P}(\tau) \int \mathsf{P}(B | \tau) \mathsf{P}(Y | X, \tau, B) \mathsf{d}B }[/math] |