Frameworks and libraries provide application programming interfaces (APIs) that serve as building blocks in modern software development. As APIs present the opportunity of increased productivity, it also calls for correct use to avoid buggy code. The usage-based specification mining technique has shown great promise in solving this problem through a data-driven approach. These techniques leverage the use of the API in large corpora to understand the recurring usages of the APIs and infer behavioral specifications (preconditions and postconditions) from such usages. A challenge for such technique is thus inference in the presence of insufficient usages, in terms of both frequency and richness. We refer to this as a "sparse usage problem." This paper presents the first technique to solve the sparse usage problem in usage-based precondition mining. Our key insight is to leverage implicit beliefs to overcome sparse usage. An implicit belief (IB) is the knowledge implicitly derived from the fact about the code. An IB about a program is known implicitly to a programmer via the language's constructs and semantics, and thus not explicitly written or specified in the code. The technical underpinnings of our new precondition mining approach include a technique to analyze the data and control flow in the program leading to API calls to infer preconditions that are implicitly present in the code corpus, a catalog of 35 code elements in total that can be used to derive implicit beliefs from a program, and empirical evaluation of all of these ideas. We have analyzed over 350 millions lines of code and 7 libraries that suffer from the sparse usage problem. Our approach realizes 6 implicit beliefs and we have observed that addition of single-level context sensitivity can further improve the result of usage based precondition mining. The result shows that we achieve overall 60% in precision and 69% in recall and the accuracy is relatively improved by 32% in precision and 78% in recall compared to base usage-based mining approach for these libraries.