The posterior distributions of the sources are effectively
approximated by mixtures of Gaussians which means that the above
update rules are not directly applicable for them. The new values for
discrete posterior probabilities of the source indices
and the posterior means
and
variances
of the Gaussians corresponding to different
source are most easily solved by making a quadratic approximation for
*C*_{f} based on the derivatives
and
,
where *C*_{f} denotes the sum
of terms of the form (27).

The usefulness of this approximation is based on the fact that the cost function tries to minimise the misfit between the approximation and the actual posterior. If we can solve the actual posterior and show that it can be described by our parametric approximation, we know that the cost function will be minimised by setting the parameters of the approximation to values corresponding to the actual posterior.

In this case, making a second order approximation for *C*_{f} is
equivalent to approximating
*p*(*X* | *s*_{i}(*t*)) by an unnormalised
Gaussian distribution. Since the prior *p*(*s*_{i}(*t*)) is a mixture of
Gaussians and the posterior will be given by
*p*(*s*_{i}(*t*) | *X*) = *p*(*X* |
*s*_{i}(*t*)) *p*(*s*_{i}(*t*)) / *p*(*X*), we notice that the posterior is also a
mixture of Gaussians since a Gaussian multiplied with an unnormalised
Gaussian will produce another unnormalised Gaussian and the
normalising factor *p*(*X*) will then make sure that the posterior is a
normalised mixture of Gaussians. From the resulting mixture of
Gaussians one can then determine the values for
,
and
.
In [1], a
similar method without approximations was used for a linear model.
Due to the linearity of the mapping,
is quadratic
and no approximations are needed.