Using the Gaussian prior for the parameters of interest,

using an inverse covariance matrix, , of dimension by .

Neglecting all but the interesting parameters (for now) gives a
posterior of the form

When then the part of the posterior that depends on
can be written as

where , , and which is a residual forming matrix (no longer a projection matrix) and it also depends on and .

The full posterior is then

This form is extremely difficult (probably impossible) to integrate analytically with respect to and . Hence it is left in this semi-marginalised form.

- Marginalising over Null Parameters
- Marginalisation over Areas of No Interest
- Marginalisation over Partial Volume Parameters