  Typical dimensionality reduction (DR) methods are often data-oriented,
focusing on directly reducing the number of random variables (features) while
retaining the maximal variations in the high-dimensional data. In unsupervised
situations, one of the main limitations of these methods lies in their
dependency on the scale of data features. This paper aims to address the
problem from a new perspective and considers model-oriented dimensionality
reduction in parameter spaces of binary multivariate distributions.
  Specifically, we propose a general parameter reduction criterion, called
Confident-Information-First (CIF) principle, to maximally preserve confident
parameters and rule out less confident parameters. Formally, the confidence of
each parameter can be assessed by its contribution to the expected Fisher
information distance within the geometric manifold over the neighbourhood of
the underlying real distribution.
  We then revisit Boltzmann machines (BM) from a model selection perspective
and theoretically show that both the fully visible BM (VBM) and the BM with
hidden units can be derived from the general binary multivariate distribution
using the CIF principle. This can help us uncover and formalize the essential
parts of the target density that BM aims to capture and the non-essential parts
that BM should discard. Guided by the theoretical analysis, we develop a
sample-specific CIF for model selection of BM that is adaptive to the observed
samples. The method is studied in a series of density estimation experiments
and has been shown effective in terms of the estimate accuracy.
