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.