An important consideration in the development of aircraft collision avoidance systems is how to account for state uncertainty due to sensor limitations and noise. However, many collision avoidance systems simply use point estimates of the state instead of leveraging the full posterior state distribution. Recently, there has been work on applying decision-theoretic methods to collision avoidance, but the importance of accommodating state uncertainty has not yet been well studied. This paper presents a computationally efficient framework for accounting for state uncertainty based on dynamic programming. Examination of characteristic encounters and Monte Carlo simulations demonstrates that properly handling state uncertainty rather than simply using point estimates can significantly enhance safety and improve robustness to sensor error.