Log-concave Observers

In English The Kalman filter is the optimal state
observer in the case of linear dynamics and Gaussian noise.
In this paper, the observer problem
is studied when process noise and measurements
are generalized from Gaussian to log-concave. This
generalization is of interest for example in the case
where observations only give information that the
signal is in a given range. It turns out that the optimal
observer preserves log-concavity. The concept
of strong log-concavity is introduced and two new
theorems are derived to compute upper bounds on
optimal observer covariance in the log-concave case.
The theory is applied to a system with threshold
based measurements, which are log-concave but far
from Gaussian.