In the previous section we saw how the class-conditional density functions can be estimated in a one-dimensional normal case. Here we'll generalize the ideas and develop the formulas for a multi-dimensional case. The details will be skipped, because the arguments and proofs are quite similar to those in the one-dimensional case.

As before, we'll assume that the *p(x|X) *is distributed normally and
the covariance matrix
of the distribution
is known. That is the only parameter to be estimated is the mean vector
. As in the one-dimensional case, we assume
that the unknown parameter
is distributed normally with the mean
and the covariance matrix
. In other words,

To remind you, we are dealing with the multi-dimensional case, so the means and are vectors.

The probability density function *p(
|X)*
can be computed by using Bayes rule in the following manner:

where *n* is the number of samples,
is the mean vector and
is the covariance matrix. Therefore,
*p(
|X)* is also distributed normally:

By equating coefficients it's easy to obtain:

where as before

As for the estimate of the class-conditional density function, it can be computed as before by means of integration:

where

It follows that

In order to illustrate the formulas above, we'll have a look at the two-dimensional case.

Suppose that

The figure below
presents a set of *100* samples.

Based on the samples above, we plotted the density function
*
p(
|X)* as
number of samples increased from *20* to *100*. The figure below
illustrates how the estimated density function becomes sharper as the number of
samples increases.