In this paper we generalize the estimation-control duality that exists in the
linear-quadratic-Gaussian setting. We extend this duality to maximum a
posteriori estimation of the system's state, where the measurement and
dynamical system noise are independent log-concave random variables. More
generally, we show that a problem which induces a convex penalty on noise terms
will have a dual control problem. We provide conditions for strong duality to
hold, and then prove relaxed conditions for the piecewise linear-quadratic
case. The results have applications in estimation problems with nonsmooth
densities, such as log-concave maximum likelihood densities. We conclude with
an example reconstructing optimal estimates from solutions to the dual control
problem, which has implications for sharing solution methods between the two
types of problems
