MCMC-UMS for doubly-bounded ${\bf x}$

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

$${\bf x}\in \{ {\bf x}: \; 0 < x_i < 1, \;\;\; 1\leq i \leq N\}$$

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 $\lambda^L, \; \lambda^H$ the same as before, which are related to the lower bound on the input data. We also calculate two additional bounds $\tilde{\lambda}^L, \; \tilde{\lambda}^H$, related to the upper bound on the input data. We define the vector ${\bf d} = [d_{1}, d_{2}, \ldots d_{N}]$ calculated from ${\bf b}$ as $d_{i} = b_i/(1-x^0_i), \; \; 1\leq i \leq N.$ Then, $\tilde{\lambda}^L$ is equal to the reciprocal of the most negative value of ${\bf d}$, and $\tilde{\lambda}^H$ is the reciprocal of the largest positive value of ${\bf d}$. Then, the lower bound on $u_i$ is the largest of $\lambda^L$ and $\tilde{\lambda}^L$, and the upper bound on $u_i$ is the smallest of $\lambda^H$ and $\tilde{\lambda}^H$.