User:Timothee Flutre/Notebook/Postdoc/2011/11/10: Difference between revisions
(→Bayesian model of univariate linear regression for QTL detection: finish posterior tau) |
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* '''Data''': let's assume that we obtained data from N individuals. We note <math>y_1,\ldots,y_N</math> the (quantitative) phenotypes (e.g. expression | * '''Data''': let's assume that we obtained data from N individuals. We note <math>y_1,\ldots,y_N</math> the (quantitative) phenotypes (e.g. expression levels at a given gene), and <math>g_1,\ldots,g_N</math> the genotypes at a given SNP (encoded as allele dose: 0, 1 or 2). | ||
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* '''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> | * '''Likelihood''': we start by writing the usual linear regression for one individual | ||
<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> | |||
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>. | ||
Let's now write in matrix notation: | Let's now write the model in matrix notation: | ||
<math>Y = X B + E \text{ where } B = [ \mu \; a \; d ]^T</math> | <math>Y = X B + E \text{ where } B = [ \mu \; a \; d ]^T</math> | ||
This gives the following [http://en.wikipedia.org/wiki/Multivariate_normal_distribution multivariate Normal distribution] for the phenotypes: | |||
<math>Y | X | <math>Y | X, \tau, B \sim \mathcal{N}(XB, \tau^{-1} I_N)</math> | ||
The likelihood of the parameters given the data is therefore: | The likelihood of the parameters given the data is therefore: | ||
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<math>\mathcal{L}(\tau, B) = \mathsf{P}(Y | X, \tau, B)</math> | <math>\mathcal{L}(\tau, B) = \mathsf{P}(Y | X, \tau, B)</math> | ||
<math>\mathcal{L}(\tau, B) = \left(\frac{\tau}{2 \pi}\right)^{N | <math>\mathcal{L}(\tau, B) = \left(\frac{\tau}{2 \pi}\right)^{\frac{N}{2}} exp \left( -\frac{\tau}{2} (Y - XB)^T (Y - XB) \right)</math> | ||
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<math>\mathsf{P}(\tau, B) = \mathsf{P}(\tau) \mathsf{P}(B | \tau)</math> | <math>\mathsf{P}(\tau, B) = \mathsf{P}(\tau) \mathsf{P}(B | \tau)</math> | ||
A [http://en.wikipedia.org/wiki/Gamma_distribution Gamma distribution] for <math>\tau</math>: | |||
<math>\tau \sim \Gamma(\kappa/2, \, \lambda/2)</math> | <math>\tau \sim \Gamma(\kappa/2, \, \lambda/2)</math> | ||
which means: | |||
<math>\mathsf{P}(\tau) = \frac{\frac{\lambda}{2}^{\kappa/2}}{\Gamma(\frac{\kappa}{2})} \tau^{\frac{\kappa}{2}-1} e^{-\frac{\lambda}{2} \tau}</math> | |||
And a multivariate Normal distribution for <math>B</math>: | |||
<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> | <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> | ||
which means: | |||
<math>\mathsf{P}(B | \tau) = \left(\frac{\tau}{2 \pi}\right)^{\frac{3}{2}} |\Sigma_B|^{-\frac{1}{2}} exp \left(-\frac{\tau}{2} B^T \Sigma_B^{-1} B \right)</math> | |||
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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 <math>B</math>!): | 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 <math>B</math>!): | ||
<math>\mathsf{P}(\tau | Y, X) \propto \tau^{\frac{\kappa}{2} - 1} e^{-\frac{\lambda}{2} \tau} \int \tau^{ | <math>\mathsf{P}(\tau | Y, X) \propto \tau^{\frac{\kappa}{2} - 1} e^{-\frac{\lambda}{2} \tau} \int \tau^{3/2} \tau^{N/2} exp(-\frac{\tau}{2} B^T \Sigma_B^{-1} B) exp(-\frac{\tau}{2} (Y - XB)^T (Y - XB)) \mathsf{d}B</math> | ||
As we used a conjugate prior for <math>\tau</math>, we know that we expect a Gamma distribution for the posterior. | As we used a conjugate prior for <math>\tau</math>, we know that we expect a Gamma distribution for the posterior. | ||
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We also factorize inside the exponential: | We also factorize inside the exponential: | ||
<math>\mathsf{P}(\tau | Y, X) \propto \tau^{\frac{N+\kappa}{2} - 1} e^{-\frac{\lambda}{2} \tau} \int \tau^{ | <math>\mathsf{P}(\tau | Y, X) \propto \tau^{\frac{N+\kappa}{2} - 1} e^{-\frac{\lambda}{2} \tau} \int \tau^{3/2} exp \left[-\frac{\tau}{2} \left( (B - \Omega X^T Y)^T \Omega^{-1} (B - \Omega X^T Y) - Y^T X \Omega X^T Y + Y^T Y \right) \right] \mathsf{d}B</math> | ||
We recognize the conditional posterior of <math>B</math>. | We recognize the conditional posterior of <math>B</math>. | ||
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<math>\mathsf{P}(\tau | Y, X) \propto \tau^{\frac{N+\kappa}{2} - 1} e^{-\frac{\lambda}{2} \tau} exp\left[-\frac{\tau}{2} (Y^T X \Omega X^T Y + Y^T Y) \right]</math> | <math>\mathsf{P}(\tau | Y, X) \propto \tau^{\frac{N+\kappa}{2} - 1} e^{-\frac{\lambda}{2} \tau} exp\left[-\frac{\tau}{2} (Y^T X \Omega X^T Y + Y^T Y) \right]</math> | ||
We finally recognize | We finally recognize a Gamma distribution, allowing us to write the posterior as: | ||
<math>\tau | Y, X \sim \Gamma \left( \frac{N+\kappa}{2}, \; \frac{1}{2} (Y^T X \Omega X^T Y + Y^T Y + \lambda) \right)</math> | <math>\tau | Y, X \sim \Gamma \left( \frac{N+\kappa}{2}, \; \frac{1}{2} (Y^T X \Omega X^T Y + Y^T Y + \lambda) \right)</math> | ||
Revision as of 19:24, 21 November 2012
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Bayesian model of univariate linear regression for QTL detectionSee Servin & Stephens (PLoS Genetics, 2007).
[math]\displaystyle{ \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] 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 the model in matrix notation: [math]\displaystyle{ Y = X B + E \text{ where } B = [ \mu \; a \; d ]^T }[/math] This gives the following multivariate Normal distribution for the phenotypes: [math]\displaystyle{ Y | X, \tau, B \sim \mathcal{N}(XB, \tau^{-1} I_N) }[/math] The likelihood of the parameters given the data is therefore: [math]\displaystyle{ \mathcal{L}(\tau, B) = \mathsf{P}(Y | X, \tau, B) }[/math] [math]\displaystyle{ \mathcal{L}(\tau, B) = \left(\frac{\tau}{2 \pi}\right)^{\frac{N}{2}} exp \left( -\frac{\tau}{2} (Y - XB)^T (Y - XB) \right) }[/math]
[math]\displaystyle{ \mathsf{P}(\tau, B) = \mathsf{P}(\tau) \mathsf{P}(B | \tau) }[/math] A Gamma distribution for [math]\displaystyle{ \tau }[/math]: [math]\displaystyle{ \tau \sim \Gamma(\kappa/2, \, \lambda/2) }[/math] which means: [math]\displaystyle{ \mathsf{P}(\tau) = \frac{\frac{\lambda}{2}^{\kappa/2}}{\Gamma(\frac{\kappa}{2})} \tau^{\frac{\kappa}{2}-1} e^{-\frac{\lambda}{2} \tau} }[/math] And a multivariate Normal distribution for [math]\displaystyle{ B }[/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] which means: [math]\displaystyle{ \mathsf{P}(B | \tau) = \left(\frac{\tau}{2 \pi}\right)^{\frac{3}{2}} |\Sigma_B|^{-\frac{1}{2}} exp \left(-\frac{\tau}{2} B^T \Sigma_B^{-1} B \right) }[/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] 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 [math]\displaystyle{ B }[/math]!): [math]\displaystyle{ \mathsf{P}(\tau | Y, X) \propto \tau^{\frac{\kappa}{2} - 1} e^{-\frac{\lambda}{2} \tau} \int \tau^{3/2} \tau^{N/2} exp(-\frac{\tau}{2} B^T \Sigma_B^{-1} B) exp(-\frac{\tau}{2} (Y - XB)^T (Y - XB)) \mathsf{d}B }[/math] As we used a conjugate prior for [math]\displaystyle{ \tau }[/math], we know that we expect a Gamma distribution for the posterior. Therefore, we can take [math]\displaystyle{ \tau^{N/2} }[/math] out of the integral and start guessing what looks like a Gamma distribution. We also factorize inside the exponential: [math]\displaystyle{ \mathsf{P}(\tau | Y, X) \propto \tau^{\frac{N+\kappa}{2} - 1} e^{-\frac{\lambda}{2} \tau} \int \tau^{3/2} exp \left[-\frac{\tau}{2} \left( (B - \Omega X^T Y)^T \Omega^{-1} (B - \Omega X^T Y) - Y^T X \Omega X^T Y + Y^T Y \right) \right] \mathsf{d}B }[/math] We recognize the conditional posterior of [math]\displaystyle{ B }[/math]. This allows us to use the fact that the pdf of the Normal distribution integrates to one: [math]\displaystyle{ \mathsf{P}(\tau | Y, X) \propto \tau^{\frac{N+\kappa}{2} - 1} e^{-\frac{\lambda}{2} \tau} exp\left[-\frac{\tau}{2} (Y^T X \Omega X^T Y + Y^T Y) \right] }[/math] We finally recognize a Gamma distribution, allowing us to write the posterior as: [math]\displaystyle{ \tau | Y, X \sim \Gamma \left( \frac{N+\kappa}{2}, \; \frac{1}{2} (Y^T X \Omega X^T Y + Y^T Y + \lambda) \right) }[/math] | |
