Abstract Forming the foundations of large-scale knowledge bases, probabilistic databases have been widely studied in the literature. In particular, probabilistic query evaluation has been investigated intensively as a central inference mechanism. However, despite its power, query evaluation alone cannot extract all the relevant information encompassed in large-scale knowledge bases. To exploit this potential, we study two inference tasks; namely finding the most probable database and the most probable hypothesis for a given query. As natural counterparts of most probable explanations (MPE) and maximum a posteriori hypotheses (MAP) in probabilistic graphical models, they can be used in a variety of applications that involve prediction or diagnosis tasks. We investigate these problems relative to a variety of query languages, ranging from conjunctive queries to ontology-mediated queries, and provide a detailed complexity analysis