Precision Parameter Hyperpriors

As we are employing a fully Bayesian approach we do not assume predetermined or known values for precisions in the model. Thus far in the noise modelling, these include the noise precision $\phi _{\epsilon _i}$, the MRF parameter $\phi _{\alpha _p}$, the ARD precisions $\phi _{\beta _{ij}}$ and $\phi _{\alpha _{pi}}$. We use a standard conjugate Gamma hyperprior. For the noise precision we have:

$$\phi_{\epsilon_i}\vert \tilde{a}_{\epsilon}, \tilde{b}_{\epsilon} \sim Ga(\tilde{a}_{\epsilon}, \tilde{b}_{\epsilon})$$ (16)

for the MRF precisions:

$$\phi_{\beta}\vert \tilde{a}_{\beta}, \tilde{b}_{\beta} \sim Ga(\tilde{a}_{\beta}, \tilde{b}_{\beta})$$

$$\phi_{\alpha_p}\vert \tilde{a}_{\alpha}, \tilde{b}_{\alpha} \sim Ga(\tilde{a}_{\alpha}, \tilde{b}_{\alpha})$$ (17)

and for the ARD precisions:

$$\phi_{\beta_{ij}}\vert \tilde{a}_{\beta}, \tilde{b}_{\beta} \sim Ga(\tilde{a}_{\beta}, \tilde{b}_{\beta})$$

$$\phi_{\alpha_{pi}}\vert \tilde{a}_{\alpha}, \tilde{b}_{\alpha} \sim Ga(\tilde{a}_{\alpha}, \tilde{b}_{\alpha})$$ (18)

where $Ga(a_{\phi},b_{\phi})$ is the Gamma distribution, and the $a_{\phi}$ and $b_{\phi}$ are known hyperparameters of the Gamma distribution. Setting $a_{\phi}=0$ and $b_{\phi}=0$ would be equivalent to a uniform prior $log(\sigma^2 = 1/\phi) \sim U(-\infty,+\infty)$, which would be problematic since it would result in an improper posterior with an infinite spike at $\sigma=0$. As long as $a_{\phi}$ and $b_{\phi}$ are set positive then a proper posterior will result. With no prior information about the variance, $a_{\phi}$ and $b_{\phi}$ the approach taken is to choose a very disperse prior, i.e. with mean based on an empirical initial estimate and a very large variance so that the choice of mean hardly affects the posterior distribution.