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
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Integrating with respect to
gives
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Substituting this back into the previous expression and integrating over
gives
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However, as shown above, if all the previous integrations are
performed first, then the remaining posterior takes the form
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Assuming that
and
both
have negligible off-diagonal terms gives
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