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

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 Revision as of 12:38, 21 November 2012 (view source) (→Entry title: first version)← Previous diff Revision as of 13:30, 21 November 2012 (view source) (→Bayesian model of univariate linear regression for QTL detection: add conditional posterior of B)Next diff → 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: $B | \tau \sim \mathcal{N}(\vec{0}, \, \tau^{-1} \Sigma_B) \text{ with } \Sigma_B = diag(\sigma_{\mu}^2, \sigma_a^2, \sigma_d^2)$ $B | \tau \sim \mathcal{N}(\vec{0}, \, \tau^{-1} \Sigma_B) \text{ with } \Sigma_B = diag(\sigma_{\mu}^2, \sigma_a^2, \sigma_d^2)$ + + + * '''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, $Y^TY$, 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 $\Omega^T=\Omega$, which means that $\Omega$ is a [http://en.wikipedia.org/wiki/Symmetric_matrix symmetric matrix]. + This is particularly useful here because we can use the following equality: $\Omega^{-1}\Omega^T=I$. + + $\mathsf{P}(B | Y, X, \tau) \propto exp(B^T \Omega^{-1} B - (X^TY)^T\Omega^{-1}\Omega^TB -B^T\Omega^{-1}\Omega^TX^TY)$ + + 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 [http://en.wikipedia.org/wiki/Kernel_%28statistics%29 kernel] of a Normal distribution, allowing us to write the conditional posterior as: + + $B | Y, X, \tau \sim \mathcal{N}(\Omega X^TY, \tau^{-1} \Omega)$

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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 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:

Y = XB + 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, YTY, 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 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.

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

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:

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