  This paper presents a novel optimization framework for beamforming design in
integrated sensing and communication systems where a base station seeks to
minimize the Bayesian Cram\'er-Rao bound of a sensing problem while satisfying
quality of service constraints for the communication users. Prior approaches
formulate the design problem as a semidefinite program for which acquiring a
beamforming solution is computationally expensive. In this work, we show that
the computational burden can be considerably alleviated. To achieve this, we
transform the design problem to a tractable form that not only provides a new
understanding of Cram\'er-Rao bound optimization, but also allows for an
uplink-downlink duality relation to be developed. Such a duality result gives
rise to an efficient algorithm that enables the beamforming design problem to
be solved at a much lower complexity as compared to the-state-of-the-art
methods.
