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Typical dimensionality reduction (DR) methods are often data-oriented,focusing on directly reducing the number of random variables (features) whileretaining the maximal variations in the high-dimensional data. In unsupervisedsituations, one of the main limitations of these methods lies in theirdependency on the scale of data features. This paper aims to address theproblem from a new perspective and considers model-oriented dimensionalityreduction in parameter spaces of binary multivariate distributions.  Specifically, we propose a general parameter reduction criterion, calledConfident-Information-First (CIF) principle, to maximally preserve confidentparameters and rule out less confident parameters. Formally, the confidence ofeach parameter can be assessed by its contribution to the expected Fisherinformation distance within the geometric manifold over the neighbourhood ofthe underlying real distribution.  We then revisit Boltzmann machines (BM) from a model selection perspectiveand theoretically show that both the fully visible BM (VBM) and the BM withhidden units can be derived from the general binary multivariate distributionusing the CIF principle. This can help us uncover and formalize the essentialparts of the target density that BM aims to capture and the non-essential partsthat BM should discard. Guided by the theoretical analysis, we develop asample-specific CIF for model selection of BM that is adaptive to the observedsamples. The method is studied in a series of density estimation experimentsand has been shown effective in terms of the estimate accuracy.

A Confident Information First Principle for Parametric Reduction and Model Selection of Boltzmann Machines