Pure partial volume parameters can again be split into those associated with shapes of interest, , and those associated with areas of no interest, . The former have non-flat priors and the latter have flat priors. Integration of the former requires more care, as they interact with the multi-variate prior. In order to do this, rewrite the appropriate matrices as

Therefore the posterior integrations take the form

Integrating with respect to
gives

where and .

Substituting this back into the previous expression and integrating over
gives

where

However, as shown above, if all the previous integrations are
performed first, then the remaining posterior takes the form

where ; ; ; and

Assuming that
and
both
have negligible off-diagonal terms gives

where

;

;

is the th diagonal of ;

is the th diagonal of ;