In general density estimation the predictive density can only be calculated approximately, e.g. in maximum a posteriori approximation or by Monte Carlo methods. For Gaussian regression, however the predictive density of mixture models can be calculated exactly for given (and ). This provides us with the opportunity to compare the simultaneous maximum posterior approximation with respect to and with an analytical -integration followed by a maximum posterior approximation with respect to .
Maximising the posterior
(with respect to , , and
possibly )
is equivalent to minimising
the mixture energy
(regularised error functional
[13,17,15,16])
(20) |
In a direct saddle point approximation with respect
to and
stationarity equations
are obtained by setting the (functional)
derivatives with respect
to and to zero,
(24) |
(26) |