  In previous work cite{Ha98:Towards} we presented a case-based approach to
eliciting and reasoning with preferences. A key issue in this approach is the
definition of similarity between user preferences. We introduced the
probabilistic distance as a measure of similarity on user preferences, and
provided an algorithm to compute the distance between two partially specified
{em value} functions. This is for the case of decision making under {em
certainty}. In this paper we address the more challenging issue of computing
the probabilistic distance in the case of decision making under{em
uncertainty}. We provide an algorithm to compute the probabilistic distance
between two partially specified {em utility} functions. We demonstrate the use
of this algorithm with a medical data set of partially specified patient
preferences,where none of the other existing distancemeasures appear definable.
Using this data set, we also demonstrate that the case-based approach to
preference elicitation isapplicable in domains with uncertainty. Finally, we
provide a comprehensive analytical comparison of the probabilistic distance
with some existing distance measures on preferences.
