C.3 Estimating the error in
To estimate the error in the best-fitting parameter values that we find, we assume to be approximated by an -dimensional Gaussian distribution around . Taking a Taylor expansion of about , we can write:
Since the logarithm of a Gaussian distribution is a parabola, the quadratic terms in the above expansion encode the Gaussian component of the probability distribution about .1 We may write the sum of these terms, which we denote , in matrix form:
where the superscript represents the transpose of the vector displacement from , and is the Hessian matrix of , given by:
This is the Hessian matrix which is output by the fit command. In general, an -dimensional Gaussian distribution such as that given by Equation () yields elliptical contours of equi-probability in parameter space, whose principal axes need not be aligned with our chosen coordinate axes – the variables . The eigenvectors of are the principal axes of these ellipses, and the corresponding eigenvalues equal , where is the standard deviation of the probability density function along the direction of these axes.
This can be visualised by imagining that we diagonalise , and expand Equation () in our diagonal basis. The resulting expression for is a sum of square terms; the cross terms vanish in this basis by definition. The equations of the equi-probability contours become the equations of ellipses:
where is some constant. By comparison with the equation for the logarithm of a Gaussian distribution, we can associate with in our eigenvector basis.
The problem of evaluating the standard deviations of our variables is more complicated, however, as we are attempting to evaluate the width of these elliptical equi-probability contours in directions which are, in general, not aligned with their principal axes. To achieve this, we first convert our Hessian matrix into a covariance matrix.