### Estimation using Gaussian Mixtures

The GM representation of the density has the a remarkable property that can be computed in closed form. This is especially useful in visualization of information. For example, it is useful to show a human operator the distribution of likely after is measured. If desired, the MMSE can be computed in closed form as well. The MAP estimate can also be computed, but that requires a search over .

Let the GM approximation to the distribution be given by

 (13.4)

By Bayes rule,

where is the marginal distribution of . We now define as the marginal distributions of given that is a member of mode . These are, of course, Gaussian with means and covariances taken from the -partitions of the mode mean and covariance .

Then,

 (13.5)

where is the conditional density for given assuming that and are from that certain Gaussian sub-class . Fortunately, there is a closed-form equation for [55]. is Gaussian with mean

 (13.6)

and covariance

 (13.7)

Note that the conditional distribution is a Gaussian Mixture in its own right, with mode weights modified by which tends to select" the modes closest to . To reduce the number of modes in the conditioning process, one could easily remove those modes with a low value of (suggested by R. L. Streit).

This conditional distribution can be used for data visualization or, to easily calculate the conditional mean estimate, which is a by-product of equations (13.5),(13.6),(13.7):

 (13.8)

Baggenstoss 2017-05-19