Let the data vector be composed of two parts
and :

We have available training samples of , however
in the future, only will be available
from which we would like to compute estimates of .
We will shortly see that the GM density facilitates
the computation of the *conditional mean* or
*minimum mean square error* (MMSE)
estimator of .
The conditional mean estimator is the expected value of
conditioned on taking a specific (measured) value,
i.e.,

The maximum aposteriori (MAP) estimator is given by

Both the MAP and MMSE estimators are operations performed
on
. Which estimator is most appropriate
depends on the problem.
Suffice it to say that the distribution
expresses all the knowledge we have about after
having measured .

Baggenstoss
2017-05-19