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In this paper, we propose a method for retrieving promising candidate solutions in case-based problem solving. Our method, referred to as credible case-based inference, makes use of so-called similarity profiles as a formal model of the key hypothesis underlying case-based reasoning (CBR), namely, the assumption that similar problems have similar solutions. Proceeding from this formalization, it becomes possible to derive theoretical properties of the corresponding inference scheme in a rigorous way. In particular, it can be shown that, under mild technical conditions, a set of candidates covers the true solution with high probability. Thus, the approach supports an important subtask in CBR, namely, to generate potential solutions for a new target problem in a sound manner and hence contributes to the methodical foundations of CBR. Due to its generality, it can be employed for different types of performance tasks and can easily be integrated in existing CBR systems.