  The interactions between parameters, model structure, and outputs can
determine what inferences, predictions, and control strategies are possible for
a given system. Parameter space reduction and parameter estimation---and, more
generally, understanding the shape of the information contained in models with
observational structure---are thus essential for many questions in mathematical
modeling and uncertainty quantification. As such, different disciplines have
developed methods in parallel for approaching the questions in their field.
Many of these approaches, including identifiability, sloppiness, and active
subspaces, use related ideas to address questions of parameter dimension
reduction, parameter estimation, and robustness of inferences and quantities of
interest.
  In this paper, we show that active subspace methods have intrinsic
connections to methods from sensitivity analysis and identifiability, and
indeed that it is possible to frame each approach in a unified framework. A
particular form of the Fisher information matrix (FIM), which we denote the
sensitivity FIM, is fundamental to all three approaches---active subspaces,
identifiability, and sloppiness. Through a series of examples and case studies,
we illustrate the properties of the sensitivity FIM in several contexts. These
initial examples show that the interplay between local and global and linear
and non-linear strongly impact the insights each approach can generate. These
observations underline that one's approach to parameter dimension reduction
should be driven by the scientific question and also open the door to using
tools from the other approaches to generate useful insights.
