  Duality between estimation and control is a foundational concept in Control
Theory. Most students learn about the elementary duality -- between
observability and controllability -- in their first graduate course in linear
systems theory. Therefore, it comes as a surprise that for a more general class
of nonlinear stochastic systems (hidden Markov models or HMMs), duality is
incomplete.
  Our objective in writing this article is two-fold: (i) To describe the
difficulty in extending duality to HMMs; and (ii) To discuss its recent
resolution by the authors. A key message is that the main difficulty in
extending duality comes from time reversal in going from estimation to control.
The reason for time reversal is explained with the aid of the familiar linear
deterministic and linear Gaussian models. The explanation is used to motivate
the difference between the linear and the nonlinear models. Once the difference
is understood, duality for HMMs is described based on our recent work. The
article also includes a comparison and discussion of the different types of
duality considered in literature.
