C.4 The covariance matrixThe terms of the covariance matrix are defined by:
Its leading diagonal terms may be recognised as equalling the variances of each of our variables; its cross terms measure the correlation between the variables. If a component , it implies that higher estimates of the coefficient make higher estimates of more favourable also; if , the converse is true. It is a standard statistical result that . In the remainder of this section we prove this; readers who are willing to accept this may skip onto Section C.5. Using to denote , we may proceed by rewriting Equation () as:
The normalisation factor in the denominator of this expression, which we denote as , the partition function, may be evaluated by dimensional Gaussian integration, and is a standard result:
Differentiating with respect of any given component of the Hessian matrix yields:
which we may identify as equalling :
This expression may be simplified by recalling that the determinant of a matrix is equal to the scalar product of any of its rows with its cofactors, yielding the result:
where is the cofactor of . Substituting this into Equation () yields:
Recalling that the adjoint of the Hessian matrix is the matrix of cofactors of its transpose, and that is symmetric, we may write:
which proves the result stated earlier.
