The existence of observables that are incompatible or not jointly measurable is a characteristic feature of quantum mechanics, which lies at the root of a number of nonclassical phenomena, such as uncertainty relations, wave-particle dual behavior, Bell-inequality violation, and contextuality. However, no simple criterion is available for determining the compatibility of even two (generalized) observables, despite the overarching importance of this problem and intensive efforts of many researchers over more than 80 years. Starting from the simple idea that every observable can only provide limited information and that information is monotonic under data processing, we introduce a powerful information theoretic paradigm together with an intuitive geometric picture for decoding incompatible observables. By virtue of quantum estimation theory, we introduce a family of universal criteria for detecting incompatible observables and a natural measure of incompatibility, which are applicable to arbitrary number of arbitrary observables. Based on this framework, we derive a family of universal measurement uncertainty relations, provide a simple information theoretic explanation of quantitative wave-particle duality, and offer new perspectives for understanding Bell nonlocality, contextuality, and quantum precision limit.