  Set-Membership Filter (SMF) has been extensively studied for state estimation
in the presence of bounded noises with unknown statistics. Since it was first
introduced in the 1960s, the studies on SMF have used the set-based description
as its mathematical framework. One important issue that has been overlooked is
the optimality of SMF. In this work, we put forward a new mathematical
framework for SMF using concepts of uncertain variables. We first establish two
basic properties of uncertain variables, namely, the law of total range (a
non-stochastic version of the law of total probability) and the equivalent
Bayes' rule. This enables us to put forward a general SMFing framework with
established optimality. Furthermore, we obtain the optimal SMF under a
non-stochastic Markov condition, which is shown to be fundamentally equivalent
to the Bayes filter. Note that the classical SMF in the literature is only
equivalent to the optimal SMF we obtained under the non-stochastic Markov
condition. When this condition is violated, we show that the classical SMF is
not optimal and it only gives an outer bound on the optimal estimation.
