6.6 Finding $\sigma _i$

Throughout the preceding sections, the uncertainties in the supplied target values $f_i$ have been denoted $\sigma _i$ (see section 6.1). The user has the option of supplying these in the source datafile, in which case the provisions of the previous sections are now complete; both best-estimate parameter values and their uncertainties can be calculated. The user may also, however, leave the uncertainties in $f_i$ unstated, in which case, as described in section 6.1, we assume all of the data values to have a common uncertainty $\sigma_\mathrm{data}$, which is an unknown.

In this case, where $\sigma_i = \sigma_\mathrm{data} \forall i$, the best fitting parameter values are independent of $\sigma_\mathrm{data}$, but the same is not true of the uncertainties in these values, as the terms of the Hessian matrix do depend upon $\sigma_\mathrm{data}$. We must therefore undertake a further calculation to find the most probable value of $\sigma_\mathrm{data}$, given the data. This is achieved by maximising $\mathrm{P}\left( \sigma_\mathrm{data} \vert \left\{ \mathbf{x}_i, f_i \right\} \right)$. Returning once again to Bayes' Theorem, we can write:

$$\mathrm{P}\left( \sigma_\mathrm{data} \vert \left\{ \mathbf{x}_i,... ...hrm{P}\left( \left\{ f_i \right\} \vert \left\{ \mathbf{x}_i \right\} \right) }$$ (6.17)

As before, we neglect the denominator, which has no effect upon the maximisation problem, and assume a uniform prior $\mathrm{P}\left( \sigma_\mathrm{data} \vert \left\{ \mathbf{x}_i \right\} \right)$. This reduces the problem to the maximisation of $\mathrm{P}\left( \left\{ f_i \right\} \vert \sigma_\mathrm{data}, \left\{ \mathbf{x}_i \right\} \right)$, which we may write as a marginalised probability distribution over $\mathbf{u}$:

$$\mathrm{P}\left( \left\{ f_i \right\} \vert \sigma_\mathrm{data}, \left\{ \mathbf{x}_i \right\} \right) = \idotsint_{-\infty}^{\infty} \mathrm{P}\left( \left\{ f_i \right\} \vert \sigma_\mathrm{data}, \left\{ \mathbf{x}_i \right\}, \mathbf{u} \right) \times$$ (6.18)

$$\mathrm{P}\left( \mathbf{u} \vert \sigma_\mathrm{data}, \left\{ \mathbf{x}_i \right\} \right) \mathrm{d}^{n_\mathrm{u}}\mathbf{u}$$

Assuming a uniform prior for $\mathbf{u}$, we may neglect the latter term in the integral, but even with this assumption, the integral is not generally tractable, as $\mathrm{P}\left( \left\{ f_i \right\} \vert \sigma_\mathrm{data}, \left\{ \mathbf{x}_i \right\}, \left\{ \mathbf{u}_i \right\} \right)$ may well be multimodal in form. However, if we neglect such possibilities, and assume this probability distribution to be approximate a Gaussian globally, we can make use of the standard result for an $n_\mathrm{u}$-dimensional Gaussian integral:

$$\idotsint_{-\infty}^{\infty} \exp \left( \frac{1}{2}\mathbf{u}^\m... ...\frac{ (2\pi)^{n_\mathrm{u}/2} }{ \sqrt{\mathrm{Det}\left(-\mathbf{A}\right)} }$$ (6.19)

We may thus approximate equation (6.18) as:

$$\mathrm{P}\left( \left\{ f_i \right\} \vert \sigma_\mathrm{data}, \left\{ \mathbf{x}_i \right\} \right) \approx \mathrm{P}\left( \left\{ f_i \right\} \vert \sigma_\mathrm{data}, \left\{ \mathbf{x}_i \right\}, \mathbf{u}^0 \right) \times$$ (6.20)

$$\mathrm{P}\left( \mathbf{u}^0 \vert \sigma_\mathrm{data}, \left\{... ...\frac{ (2\pi)^{n_\mathrm{u}/2} }{ \sqrt{\mathrm{Det}\left(-\mathbf{A}\right)} }$$

As in section 6.2, it is numerically easier to maximise this quantity via its logarithm, which we denote $L_2$, and can write as:

$$L_2 = \sum_{i=0}^{n_\mathrm{d}-1} \left( \frac{ -\left[f_i - f_{\mathbf... ...thrm{data}^2 } - \log_e \left(2\pi\sqrt{\sigma_\mathrm{data}} \right) \right) +$$ (6.21)

$$\log_e \left( \frac{ (2\pi)^{n_\mathrm{u}/2} }{ \sqrt{\mathrm{Det}\left(-\mathbf{A}\right)} } \right)$$

This quantity is maximised numerically, a process simplified by the fact that $\mathbf{u}^0$ is independent of $\sigma_\mathrm{data}$.