This paper outlines a method for improving the precision of atmospheric neutrino oscillation measurements. One experimental signature for these oscillations is an observed deficit in the rate of $ u_{mu}$ charged-current interactions with an oscillatory dependence on $L_{ u}/E_{ u}$, where $L_{ u}$ is the neutrino propagation distance, and $E_{ u}$ is the neutrino energy. For contained-vertex atmospheric neutrino interactions, the $L_{ u}/E_{ u}$ resolution varies significantly from event to event. The precision of the oscillation measurement can be improved by incorporating information on $L_{ u}/E_{ u}$ resolution into the oscillation analysis. In the analysis presented here, a Bayesian technique is used to estimate the $L_{ u}/E_{ u}$ resolution of observed atmospheric neutrinos on an event-by-event basis. By separating the events into bins of $L_{ u}/E_{ u}$ resolution in the oscillation analysis, a significant improvement in oscillation sensitivity can be achieved.
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