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Likelihood

Note that from equations 1 and 2 with $ l_{it}=0$, our data, $ y_{it}$, is related to our spatio-temporal noise process, $ q_{it}$, by:

$\displaystyle y_{it}=r_{it}+q_{it}$ (11)

where $ r_{it}$ is the signal component. This equation, along with equations 78 and 9, describes our likelihood, $ p(\vec{y}\vert\vec{r,\alpha,\beta,\phi_\epsilon })$. We now need to specify the priors on the noise parameters $ \{\vec{
\alpha,\beta,\phi_\epsilon}\}$. For this we assume independence between the priors for these parameters, i.e. $ p(\vec{
\alpha,\beta,\phi_\epsilon}) = p(\vec{
\alpha})p(\vec{\beta})p(\vec{\phi_\epsilon})$. It is worth noting that assuming independence in the prior does not impose independence between the same parameters in the posterior distribution.