  Typical dimensionality reduction methods focus on directly reducing the
number of random variables while retaining maximal variations in the data. In
this paper, we consider the dimensionality reduction in parameter spaces of
binary multivariate distributions. We propose a general
Confident-Information-First (CIF) principle to maximally preserve parameters
with confident estimates and rule out unreliable or noisy parameters. Formally,
the confidence of a parameter can be assessed by its Fisher information, which
establishes a connection with the inverse variance of any unbiased estimate for
the parameter via the Cram\'{e}r-Rao bound. We then revisit Boltzmann machines
(BM) and theoretically show that both single-layer BM without hidden units
(SBM) and restricted BM (RBM) can be solidly derived using the CIF principle.
This can not only help us uncover and formalize the essential parts of the
target density that SBM and RBM capture, but also suggest that the deep neural
network consisting of several layers of RBM can be seen as the layer-wise
application of CIF. Guided by the theoretical analysis, we develop a
sample-specific CIF-based contrastive divergence (CD-CIF) algorithm for SBM and
a CIF-based iterative projection procedure (IP) for RBM. Both CD-CIF and IP are
studied in a series of density estimation experiments.
