We now consider the inversion problem when
the input data is bounded both above and below
(doubly-bounded) by adapting the methods of Section 5.
Our results in Section 5 were closely
related to classical methods of spectral estimation.
But now as we apply our sampling approach to the doubly-bounded problem,
we will obtain a novel method that has no classical equivalent.
Specifically, we will show that the entropy measure differs from (5.16).
With no loss of generality,
data is assumed to be in the range
MCMC-UMS for doubly-bounded
and the feature transformation is given by (5.1).
Applications of this transformation
include principal component analysis and linear filtering of
data with hard upper and lower
bounds including some types of images such as found
in optical character recognition (OCR).
The sampling procedure is very similar to the previous
example, section 5.
Generating data by MCMC-UMS is affected by the double bound on the input data
and the procedure detailed in Section 5.3.1
is easily adapted. We calculate the bounds
the same as before,
which are related to the lower bound on the input data.
We also calculate two additional bounds
related to the upper bound on the input data.
We define the vector
calculated from as
is equal to the reciprocal of
the most negative value of ,
is the reciprocal of
the largest positive value of .
Then, the lower bound on is
the largest of and
and the upper bound on is
the smallest of and