X-ray data and obtain probability distributions for the derived physical quantities. The joint probability distribution of a set of model parameters ($\bmath{\theta}$), given the data ($\bmath{d}$) and the model ($\mathcal{M}_\rmn{j}$), can be calculated from Bayes’ Theorem, $$\label{equation:bayes_theorem}
  \rmn{Pr}(\bmath{\theta}|\bmath{d},\mathcal{M}_\rmn{j}) =
      {\rmn{Pr}(\bmath{d}|\bmath{\theta},\mathcal{M}_\rmn{j})\rmn{Pr}(\bmath{\theta}|\mathcal{M}_\rmn{j})\over
        \rmn{Pr}(\bmath{d}|\mathcal{M}_\rmn{j})}.$$ The probability of the data given the set of model parameters $\rmn{Pr}(\bmath{d}|\bmath{\theta},\mathcal{M}_\rmn{j})$ (the likelihood, $L$) can be calculated based on assumptions of the distribution of the error in the data. When the data set is large and therefore quasi-continuous (such as the noise generated in radio instrumentation), one can approximate the likelihood by the form given for Gaussian multivariate data [see e.g. @Sivia:2006] $$\begin{aligned}
\label{equation:gaussian_likelihood}
  L & \equiv & \rmn{Pr}(\bmath{d}|\bmath{\theta},\mathcal{M}_\rmn{j}) \nonumber \\
  & = &  {1\over\sqrt{(2\pi)^{N}|\mathbfss{C}|}}\exp{\left(-{(\bmath{d} - \bmath{m})^\rmn{t}\mathbfss{C}^{-1}(\bmath{d} - \bmath{m})\over2}\right)},\end{aligned}$$ where $N$ is equal to the size of $\bmath{d}$, $\textbfss{C}$ is the covariance matrix of the data, $|\mathbfss{C}|$ is the determinant of the covariance matrix and $\bmath{m}$ is the vector of model data. Since we wish to perform joint fits to $k$ independent data sets, such that $\bmath{d}$ = $\sum{\bmath{d}_\rmn{k}}$, then the total likelihood is just given by the product of the individual likelihoods.

The probability of the parameter values given the model, $\rmn{Pr}(\bmath{\theta}|\mathcal{M}_\rmn{j})$, is known as the prior 