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This paper focuses on the development of decision making criteria for autonomous vehicles where the tasks to be performed are competing under limited vehicle and sensory resources. More specifically, we are interested in the search and characterization of multiple objects given a limited number of autonomous sensor vehicles. In this case, search and characterization are two competing demands since an autonomous vehicle in the system can perform either the search task or the characterization task, but not both at the same time. This is a very critical decision as choosing one option over the other may mean missing other, more important objects not yet found, or missing the opportunity to satisfactorily characterize a found critical object. Building on previous deterministic-based work by the authors, in this paper we develop Bayesian-based search versus characterization decision making criteria that result in guaranteed detection and characterization of all objects in the domain.