a diagonal matrix of the variance in the data.

Hyper-parameters and combining the data sets
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Radio interferometric SZ and X-ray surface brightness data typically have sensitivity to very different angular scales and depend differently on the physical properties that are being constrained. Therefore constructing an analytical physically-based model that provides a satisfactory simultaneous fit to both data sets is difficult, especially if we are unsure about the accuracy of the error estimates. It is therefore good practise to weight the likelihoods for these data sets using two hyper-parameters, which can then be treated as nuisance parameters and marginalised over. The normalised modified Gaussian likelihood for the data now has the following expression $$\begin{aligned}
  L & \equiv & \rmn{Pr}(\bmath{d}|\bmath{\theta},\epsilon,\mathcal{M}_\rmn{j}) \nonumber \\
  & = & {\sqrt{\epsilon^{N}\over(2\pi)^{N}|\mathbfss{C}|}}\exp{\left(-\epsilon{(\bmath{d} - \bmath{m})^\rmn{t}\mathbfss{C}^{-1}(\bmath{d} - \bmath{m})\over2}\right)},\end{aligned}$$ where $\epsilon$ is the hyper-parameter. Its assumed that very little is known about the true value of $\epsilon$, except that it has an expectation value of unity. In this case a suitable prior for each hyper-parameter is given by [@Hobson:2002], where $$\rmn{Pr}(\epsilon) = \exp{(-\epsilon)},$$ and $\epsilon$ has a value between 0 and $\infty$. Note that if the Jeffreys prior were used instead, where $\rmn{Pr}(\epsilon) =
1/\epsilon$, then it would not be 