User:Timothee Flutre/Notebook/Postdoc/2011/11/10
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Bayesian model of univariate linear regression for QTL detectionSee Servin & Stephens (PLoS Genetics, 2007).
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\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})} where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \beta_1} is in fact the additive effect of the SNP, noted Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle a} from now on, and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \beta_2} is the dominance effect of the SNP, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle d = a k} . Let's now write the model in matrix notation: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle Y = X B + E \text{ where } B = [ \mu \; a \; d ]^T} This gives the following multivariate Normal distribution for the phenotypes: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle Y  X, \tau, B \sim \mathcal{N}(XB, \tau^{1} I_N)} Even though we can write the likelihood as a multivariate Normal, I still keep the term "univariate" in the title because the covariance matrix of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle Y  X, \tau, B} is in fact parametrized by a single real number, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \tau} . The likelihood of the parameters given the data is therefore: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \mathcal{L}(\tau, B) = \mathsf{P}(Y  X, \tau, B)} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\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)}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \mathsf{P}(\tau, B) = \mathsf{P}(\tau) \mathsf{P}(B  \tau)} A Gamma distribution for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \tau} : Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \tau \sim \Gamma(\kappa/2, \, \lambda/2)} which means: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \mathsf{P}(\tau) = \frac{\frac{\lambda}{2}^{\kappa/2}}{\Gamma(\frac{\kappa}{2})} \tau^{\frac{\kappa}{2}1} e^{\frac{\lambda}{2} \tau}} And a multivariate Normal distribution for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle B} : Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\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)} which means: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\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)}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \mathsf{P}(\tau, B  Y, X) = \mathsf{P}(\tau  Y, X) \mathsf{P}(B  Y, X, \tau)}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \mathsf{P}(B  Y, X, \tau) = \frac{\mathsf{P}(B, Y  X, \tau)}{\mathsf{P}(Y  X, \tau)}} Let's neglect the normalization constant for now: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \mathsf{P}(B  Y, X, \tau) \propto \mathsf{P}(B  \tau) \mathsf{P}(Y  X, \tau, B)} Similarly, let's keep only the terms in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle B} for the moment: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \mathsf{P}(B  Y, X, \tau) \propto exp(B^T \Sigma_B^{1} B) exp((YXB)^T(YXB))} We expand: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \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: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \mathsf{P}(B  Y, X, \tau) \propto exp(B^T (\Sigma_B^{1} + X^TX) B  Y^TXB B^TX^TY)} Importantly, let's define: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \Omega = (\Sigma_B^{1} + X^TX)^{1}} We can see that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \Omega^T=\Omega} , which means that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \Omega} is a symmetric matrix. This is particularly useful here because we can use the following equality: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \Omega^{1}\Omega^T=I} . Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\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)} This now becomes easy to factorizes totally: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \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: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle B  Y, X, \tau \sim \mathcal{N}(\Omega X^TY, \tau^{1} \Omega)}
Similarly to the equations above: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \mathsf{P}(\tau  Y, X) \propto \mathsf{P}(\tau) \mathsf{P}(Y  X, \tau)} But now, to handle the second term, we need to integrate over Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle B} , thus effectively taking into account the uncertainty in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle B} : Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \mathsf{P}(\tau  Y, X) \propto \mathsf{P}(\tau) \int \mathsf{P}(B  \tau) \mathsf{P}(Y  X, \tau, B) \mathsf{d}B} 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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle B} !): Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\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} As we used a conjugate prior for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \tau} , we know that we expect a Gamma distribution for the posterior. Therefore, we can take Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \tau^{N/2}} out of the integral and start guessing what looks like a Gamma distribution. We also factorize inside the exponential: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\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} We recognize the conditional posterior of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle B} . This allows us to use the fact that the pdf of the Normal distribution integrates to one: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\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 Y  Y^T X \Omega X^T Y) \right]} We finally recognize a Gamma distribution, allowing us to write the posterior as: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \tau  Y, X \sim \Gamma \left( \frac{N+\kappa}{2}, \; \frac{1}{2} (Y^T Y  Y^T X \Omega X^T Y + \lambda) \right)}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle B, \tau  Y, X \sim \mathcal{N}IG(\Omega X^TY, \; \tau^{1}\Omega, \; \frac{N+\kappa}{2}, \; \frac{\lambda^\ast}{2})} where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \lambda^\ast = Y^T Y  Y^T X \Omega X^T Y + \lambda}
Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \mathsf{P}(B  Y, X) = \int \mathsf{P}(\tau) \mathsf{P}(B  Y, X, \tau) \mathsf{d}\tau} Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \mathsf{P}(B  Y, X) = \frac{\frac{\lambda^\ast}{2}^{\frac{N+\kappa}{2}}}{(2\pi)^\frac{3}{2} \Omega^{\frac{1}{2}} \Gamma(\frac{N+\kappa}{2})} \int \tau^{\frac{N+\kappa+3}{2}1} exp \left[\tau \left( \frac{\lambda^\ast}{2} + (B  \Omega X^TY)^T \Omega^{1} (B  \Omega X^TY) \right) \right] \mathsf{d}\tau} Here we recognize the formula to integrate the Gamma function: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \mathsf{P}(B  Y, X) = \frac{\frac{\lambda^\ast}{2}^{\frac{N+\kappa}{2}} \Gamma(\frac{N+\kappa+3}{2})}{(2\pi)^\frac{3}{2} \Omega^{\frac{1}{2}} \Gamma(\frac{N+\kappa}{2})} \left( \frac{\lambda^\ast}{2} + (B  \Omega X^TY)^T \Omega^{1} (B  \Omega X^TY) \right)^{\frac{N+\kappa+3}{2}}} And we now recognize a multivariate Student's tdistribution: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle \mathsf{P}(B  Y, X) = \frac{\Gamma(\frac{N+\kappa+3}{2})}{\Gamma(\frac{N+\kappa}{2}) \pi^\frac{3}{2} \lambda^\ast \Omega^{\frac{1}{2}} } \left( 1 + \frac{(B  \Omega X^TY)^T \Omega^{1} (B  \Omega X^TY)}{\lambda^\ast} \right)^{\frac{N+\kappa+3}{2}}} We hence can write: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://api.formulasearchengine.com/v1/":): {\displaystyle B  Y, X \sim \mathcal{S}_{N+\kappa}(\Omega X^TY, \; (Y^T Y  Y^T X \Omega X^T Y + \lambda) \Omega)}
invariance properties motivate the use of limits for some "unimportant" hyperparameters average BF over grid
