## 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:
1. Validation of feature information content. Good re-producibility of the essential character and intelligibility of the input data indicates good feature extraction.
2. The Use of PDF projection in sampling methods (importance sampling, rejection sampling, MCMC methods).
3. PDF sculpting (Section 12.3);
Generation of data from is accomplished using the following process ([5], Section 2.1)
1. Draw a sample from ,
2. 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.

3. 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 2017-05-19