This paper explores the idea of knowledge-based security policies, which are used to decide whether to answer queries over secret data based on an estimation of the querier's (possibly increased) knowledge given the results. Limiting knowledge is the goal of existing information release policies that employ mechanisms such as noising, anonymization, and redaction. Knowledge-based policies are more general: they increase flexibility by not fixing the means to restrict information flow. We enforce a knowledge-based policy by explicitly tracking a model of a querier's belief about secret data, represented as a probability distribution, and denying any query that could increase knowledge above a given threshold. We implement query analysis and belief tracking via abstract interpretation, which allows us to trade off precision and performance through the use of abstraction. We have developed an approach to augment standard abstract domains to include probabilities, and thus define distributions. We focus on developing probabilistic polyhedra in particular, to support numeric programs. While probabilistic abstract interpretation has been considered before, our domain is the first whose design supports sound conditioning, which is required to ensure that estimates of a querier's knowledge are accurate. Experiments with our implementation show that several useful queries can be handled efficiently, particularly compared to exact (i.e., sound) inference involving sampling. We also show that, for our benchmarks, restricting constraints to octagons or intervals, rather than full polyhedra, can dramatically improve performance while incurring little to no loss in precision. © 2013 IOS Press and the authors.