  This paper provides conditions on the observation probability distribution in
Bayesian localization and optimal filtering so that the conditional mean
estimate satisfies convex stochastic dominance. Convex dominance allows us to
compare the unconditional mean square error between two optimal Bayesian state
estimators over arbitrary time horizons instead of using brute force
Monte-Carlo computations. The proof uses two key ideas from microeconomics,
namely, integral precision dominance and aggregation of single crossing.The
convex dominance result is then used to give sufficient conditions so that the
optimal policy of a controlled sensing two-state partially observed Markov
decision process (POMDP) is lower bounded by a myopic policy. Numerical
examples are presented where the Shannon capacity of the observation
distribution using one sensor dominates that of another, and convex dominance
holds but Blackwell dominance does not hold. These illustrate the usefulness of
the main result in localization, filtering and controlled sensing applications.
