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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 $\displaystyle y_1,\ldots,y_N$ the (quantitative) phenotypes (e.g. expression levels at a given gene), and $\displaystyle g_1,\ldots,g_N$ the genotypes at a given SNP (encoded 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: we start by writing the usual linear regression for one individual

$\displaystyle \forall i \in \{1,\ldots,N\}, \; y_i = \mu + \beta_1 g_i + \beta_2 \mathbf{1}_{g_i=1} + \epsilon_i \text{ with } \epsilon_i \overset{i.i.d}{\sim} \mathcal{N}(0,\tau^{-1})$

where $\displaystyle \beta_1$ is in fact the additive effect of the SNP, noted $\displaystyle a$ from now on, and $\displaystyle \beta_2$ is the dominance effect of the SNP, $\displaystyle d = a k$ .

Let's now write the model in matrix notation:

$\displaystyle Y = X B + E \text{ where } B = [ \mu \; a \; d ]^T$

This gives the following multivariate Normal distribution for the phenotypes:

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

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 $\displaystyle Y | X, \tau, B$ remains a single real number, $\displaystyle \tau$ .

The likelihood of the parameters given the data is therefore:

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

$\displaystyle \mathcal{L}(\tau, B) = \left(\frac{\tau}{2 \pi}\right)^{\frac{N}{2}} exp \left( -\frac{\tau}{2} (Y - XB)^T (Y - XB) \right)$

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

A Gamma distribution for $\displaystyle \tau$ :

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

which means:

$\displaystyle \mathsf{P}(\tau) = \frac{\frac{\lambda}{2}^{\kappa/2}}{\Gamma(\frac{\kappa}{2})} \tau^{\frac{\kappa}{2}-1} e^{-\frac{\lambda}{2} \tau}$

And a multivariate Normal distribution for $\displaystyle B$ :

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

which means:

$\displaystyle \mathsf{P}(B | \tau) = \left(\frac{\tau}{2 \pi}\right)^{\frac{3}{2}} |\Sigma_B|^{-\frac{1}{2}} exp \left(-\frac{\tau}{2} B^T \Sigma_B^{-1} B \right)$

• Joint posterior (1):

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

• Conditional posterior of B:

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

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

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

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

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

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

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

We expand:

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

We factorize some terms:

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

Importantly, let's define:

$\displaystyle \Omega = (\Sigma_B^{-1} + X^TX)^{-1}$

We can see that $\displaystyle \Omega^T=\Omega$ , which means that $\displaystyle \Omega$ is a symmetric matrix. This is particularly useful here because we can use the following equality: $\displaystyle \Omega^{-1}\Omega^T=I$ .

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

This now becomes easy to factorizes totally:

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

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

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

• Posterior of $\displaystyle \tau$ :

Similarly to the equations above:

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

But now, to handle the second term, we need to integrate over $\displaystyle B$ , thus effectively taking into account the uncertainty in $\displaystyle B$ :

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

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 $\displaystyle B$ !):

$\displaystyle \mathsf{P}(\tau | Y, X) \propto \tau^{\frac{\kappa}{2} - 1} e^{-\frac{\lambda}{2} \tau} \int \tau^{3/2} \tau^{N/2} exp(-\frac{\tau}{2} B^T \Sigma_B^{-1} B) exp(-\frac{\tau}{2} (Y - XB)^T (Y - XB)) \mathsf{d}B$

As we used a conjugate prior for $\displaystyle \tau$ , we know that we expect a Gamma distribution for the posterior. Therefore, we can take $\displaystyle \tau^{N/2}$ out of the integral and start guessing what looks like a Gamma distribution. We also factorize inside the exponential:

$\displaystyle \mathsf{P}(\tau | Y, X) \propto \tau^{\frac{N+\kappa}{2} - 1} e^{-\frac{\lambda}{2} \tau} \int \tau^{3/2} exp \left[-\frac{\tau}{2} \left( (B - \Omega X^T Y)^T \Omega^{-1} (B - \Omega X^T Y) - Y^T X \Omega X^T Y + Y^T Y \right) \right] \mathsf{d}B$

We recognize the conditional posterior of $\displaystyle B$ . This allows us to use the fact that the pdf of the Normal distribution integrates to one:

$\displaystyle \mathsf{P}(\tau | Y, X) \propto \tau^{\frac{N+\kappa}{2} - 1} e^{-\frac{\lambda}{2} \tau} exp\left[-\frac{\tau}{2} (Y^T X \Omega X^T Y + Y^T Y) \right]$

We finally recognize a Gamma distribution, allowing us to write the posterior as:

$\displaystyle \tau | Y, X \sim \Gamma \left( \frac{N+\kappa}{2}, \; \frac{1}{2} (Y^T X \Omega X^T Y + Y^T Y + \lambda) \right)$

• Joint posterior (2): sometimes it is said that the joint posterior follows a Normal Inverse Gamma distribution:

$\displaystyle B, \tau | Y, X \sim \mathcal{N}IG(\Omega X^TY, \; \tau^{-1}\Omega, \; \frac{N+\kappa}{2}, \; \frac{\lambda^\ast}{2})$

where $\displaystyle \lambda^\ast = (Y^T X \Omega X^T Y + Y^T Y + \lambda)$

• Marginal posterior of B: we can now integrate out $\displaystyle \tau$ :

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

$\displaystyle \mathsf{P}(B | Y, X) = \frac{\frac{\lambda^\ast}{2}^{\frac{N+\kappa}{2}}}{(2\pi)^\frac{3}{2} |\Omega|^{\frac{1}{2}} \Gamma(\frac{N+\kappa}{2})} \int \tau^{\frac{N+\kappa+3}{2}-1} exp \left[-\tau \left( \frac{\lambda^\ast}{2} + (B - \Omega X^TY)^T \Omega^{-1} (B - \Omega X^TY) \right) \right] \mathsf{d}\tau$

Here we recognize the formula to integrate the Gamma function:

$\displaystyle \mathsf{P}(B | Y, X) = \frac{\frac{\lambda^\ast}{2}^{\frac{N+\kappa}{2}} \Gamma(\frac{N+\kappa+3}{2})}{(2\pi)^\frac{3}{2} |\Omega|^{\frac{1}{2}} \Gamma(\frac{N+\kappa}{2})} \left( \frac{\lambda^\ast}{2} + (B - \Omega X^TY)^T \Omega^{-1} (B - \Omega X^TY) \right)^{-\frac{N+\kappa+3}{2}}$

And we now recognize a multivariate Student's t-distribution:

$\displaystyle \mathsf{P}(B | Y, X) = \frac{\Gamma(\frac{N+\kappa+3}{2})}{\Gamma(\frac{N+\kappa}{2}) \pi^\frac{3}{2} |\lambda^\ast \Omega|^{\frac{1}{2}} } \left( 1 + \frac{(B - \Omega X^TY)^T \Omega^{-1} (B - \Omega X^TY)}{\lambda^\ast} \right)^{-\frac{N+\kappa+3}{2}}$

We hence can write:

$\displaystyle B | Y, X \sim \mathcal{S}_{N+\kappa}(\Omega X^TY, \; (Y^T X \Omega X^T Y + Y^T Y + \lambda) \Omega)$

• Bayes Factor: to do

• In practice: to do

invariance properties motivate the use of limits for some "unimportant" hyperparameters

average BF over grid

• R code: to do