Initially consider the case of a flat prior on . In this case let the range of each individual be 0 to such that, then . When included, parameters associated with linear intensity gradients have the appropriate columns of scaled so that the range of is to , and is still true. Note that is a constant for all , and represents the inverse intensity range.
To start with, take the case where there are no uninteresting,
degenerate or partial volume parameters. Note that there may be null
parameters. In these conditions, the posterior can be simply
calculated using the integrals in appendix A, giving
Note that and both depend on the transformation . In fact, the dependence on is a form of normalisation for the number of degrees of freedom in the model. Also note that in all cases so that and that increasing the normalised residuals causes the posterior probability to decrease, as desired.
The above integrations use the approximation that
When these conditions are not true, the above approximation breaks down and the complimentary error function terms, as shown in equation 14 in appendix A, must be included. This is true for pure partial volume parameters, and will be treated in section 3.2.2 which will be the last parameters to be integrated over.