  Without unrealistic continuity and smoothness assumptions on a distributional
density of one dimensional dataset, constructing an authentic possibly-gapped
histogram becomes rather complex. The candidate ensemble is described via a
two-layer Ising model, and its size is shown to grow exponentially. This
exponential complexity makes any exhaustive search in-feasible and all boundary
parameters local. For data compression via Uniformity, the decoding error
criterion is nearly independent of sample size. These characteristics nullify
statistical model selection techniques, such as Minimum Description Length
(MDL). Nonetheless practical and nearly optimal solutions are algorithmically
computable. A data-driven algorithm is devised to construct such histograms
along the branching hierarchy of a Hierarchical Clustering tree. Such resultant
histograms naturally manifest data's physical information contents:
deterministic structures of bin-boundaries coupled with stochastic structures
of Uniformity within each bin. Without enforcing unrealistic Normality and
constant variance assumptions, an application of possibly-gapped histogram is
devised, called analysis of Histogram (ANOHT), to replace Analysis of Variance
(ANOVA). Its potential applications are foreseen in digital re-normalization
schemes and associative pattern extraction among features of heterogeneous data
types. Thus constructing possibly-gapped histograms becomes a prerequisite for
knowledge discovery, via exploratory data analysis and unsupervised Machine
Learning.
