6.1 Notation

I shall assume that we have some function $f()$, which takes $n_\mathrm{x}$ parameters, $x_0$... $x_{n_\mathrm{x}-1}$, the set of which may collectively be written as the vector $\mathbf{x}$. We are supplied a datafile, containing a number $n_\mathrm{d}$ of datapoints, each consisting of a set of values for each of the $n_\mathrm{x}$ parameters, and one for the value which we are seeking to make $f(\mathbf{x})$ match. I shall call of parameter values for the $i$th datapoint $\mathbf{x}_i$, and the corresponding value which we are trying to match $f_i$. The datafile may contain error estimates for the values $f_i$, which I shall denote $\sigma _i$. If these are not supplied, then I shall consider these quantities to be unknown, and equal to some constant $\sigma_\mathrm{data}$.

Finally, I assume that there are $n_\mathrm{u}$ coefficients within the function $f()$ that we are able to vary, corresponding to those variable names listed after the via statement in the fit command. I shall call these coefficients $u_0$... $u_{n_\mathrm{u}-1}$, and refer to them collectively as $\mathbf{u}$.

I model the values $f_i$ in the supplied datafile as being noisy Gaussian-distributed observations of the true function $f()$, and within this framework, seek to find that vector of values $\mathbf{u}$ which is most probable, given these observations. The probability of any given $\mathbf{u}$ is written $\mathrm{P}\left( \mathbf{u} \vert \left\{ \mathbf{x}_i, f_i, \sigma_i \right\} \right)$.