C.4 The covariance matrix

The terms of the covariance matrix $V_{ij}$ are defined by:

  \begin{equation}  V_{ij} = \left< \left(u_ i - u^0_ i\right) \left(u_ j - u^0_ j\right) \right> \label{eqn:def_ covar} \end{equation}   (C.8)

Its leading diagonal terms may be recognised as equalling the variances of each of our $n_\mathrm {u}$ variables; its cross terms measure the correlation between the variables. If a component $V_{ij} > 0$, it implies that higher estimates of the coefficient $u_ i$ make higher estimates of $u_ j$ more favourable also; if $V_{ij} < 0$, the converse is true.

It is a standard statistical result that $\mathbf{V} = (-\mathbf{A})^{-1}$. In the remainder of this section we prove this; readers who are willing to accept this may skip onto Section C.5.

Using $\Delta u_ i$ to denote $\left(u_ i - u^0_ i\right)$, we may proceed by rewriting Equation () as:

  $\displaystyle  V_{ij}  $ $\displaystyle  =  $ $\displaystyle  \idotsint _{u_ i=-\infty }^{\infty } \Delta u_ i \Delta u_ j \mathrm{P}\left( \mathbf{u} | \left\{  \mathbf{x}_ i, f_ i, \sigma _ i \right\}  \right) \, \mathrm{d}^{n_\mathrm {u}}\mathbf{u}  $   (C.9)
  $\displaystyle  $ $\displaystyle  =  $ $\displaystyle  \frac{ \idotsint _{u_ i=-\infty }^{\infty } \Delta u_ i \Delta u_ j \exp (-Q) \, \mathrm{d}^{n_\mathrm {u}}\mathbf{u} }{ \idotsint _{u_ i=-\infty }^{\infty } \exp (-Q) \, \mathrm{d}^{n_\mathrm {u}}\mathbf{u} } \nonumber  $    

The normalisation factor in the denominator of this expression, which we denote as $Z$, the partition function, may be evaluated by $n_\mathrm {u}$-dimensional Gaussian integration, and is a standard result:

  $\displaystyle  Z  $ $\displaystyle  =  $ $\displaystyle  \idotsint _{u_ i=-\infty }^{\infty } \exp \left(\frac{1}{2} \Delta \mathbf{u}^\mathbf {T} \mathbf{A} \Delta \mathbf{u} \right) \, \mathrm{d}^{n_\mathrm {u}}\mathbf{u}  $   (C.10)
  $\displaystyle  $ $\displaystyle  =  $ $\displaystyle  \frac{(2\pi )^{n_\mathrm {u}/2}}{\mathrm{Det}(\mathbf{-A})} \nonumber  $    

Differentiating $\log _ e(Z)$ with respect of any given component of the Hessian matrix $A_{ij}$ yields:

  \begin{equation}  -2 \frac{\partial }{\partial A_{ij}} \left[ \log _ e(Z) \right] = \frac{1}{Z} \idotsint _{u_ i=-\infty }^{\infty } \Delta u_ i \Delta u_ j \exp (-Q) \, \mathrm{d}^{n_\mathrm {u}}\mathbf{u} \end{equation}   (C.11)

which we may identify as equalling $V_{ij}$:

  $\displaystyle  \label{eqa:v_ zrelate} V_{ij}  $ $\displaystyle  =  $ $\displaystyle  -2 \frac{\partial }{\partial A_{ij}} \left[ \log _ e(Z) \right]  $   (C.12)
  $\displaystyle  $ $\displaystyle  =  $ $\displaystyle  -2 \frac{\partial }{\partial A_{ij}} \left[ \log _ e((2\pi )^{n_\mathrm {u}/2}) - \log _ e(\mathrm{Det}(\mathbf{-A})) \right] \nonumber  $    
  $\displaystyle  $ $\displaystyle  =  $ $\displaystyle  2 \frac{\partial }{\partial A_{ij}} \left[ \log _ e(\mathrm{Det}(\mathbf{-A})) \right] \nonumber  $    

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:

  \begin{equation}  \frac{\partial }{\partial A_{ij}} \left[\mathrm{Det}(\mathbf{-A})\right] = -a_{ij} \end{equation}   (C.13)

where $a_{ij}$ is the cofactor of $A_{ij}$. Substituting this into Equation () yields:

  \begin{equation}  V_{ij} = \frac{-a_{ij}}{\mathrm{Det}(\mathbf{-A})} \end{equation}   (C.14)

Recalling that the adjoint $\mathbf{A}^\dagger $ of the Hessian matrix is the matrix of cofactors of its transpose, and that $\mathbf{A}$ is symmetric, we may write:

  \begin{equation}  V_{ij} = \frac{-\mathbf{A}^\dagger }{\mathrm{Det}(\mathbf{-A})} \equiv (-\mathbf{A})^{-1} \end{equation}   (C.15)

which proves the result stated earlier.