Generation of Samples from
Indeed,
is a generative model.
In this book, we will continually discuss
the generative process of creating synthetic samples of
by drawing samples from
.
Generation of random samples from
has several purposes:
 Validation of feature information content.
Good reproducibility of the essential character
and intelligibility of the input data indicates
good feature extraction.
 The Use of PDF projection in sampling methods
(importance sampling, rejection sampling, MCMC methods).
 PDF sculpting (Section 12.3);
Generation of data from
is accomplished
using the following process ([5], Section 2.1)
 Draw a sample from
,
 Determine the manifold
,
which is the set of all points
that map to through transformation
:

(2.4) 
where is the set of valid input data samples
.
It is common to call
a manifold or level set
^{2.2}.
 draw a sample from
according to a distribution
proportional to
.
Note that drawing a sample from
according to a distribution
proportional to
can be regarded as a a posteriori
distribution of given . But, it is not a proper
distribution since all its probability mass exists on
which has zero volume, and so must have infinite value.
If we restrict our analysis just to the set
, we can
write down a representative distribution, called
the manifold distribution
,

(2.5) 
where
when
and is zero
otherwise. Clearly
Intuitively, the manifold distribution is just a distribution
on
that is proportional to
.
Subsections
Baggenstoss
20170519