Adaptive filtering is a powerful class of control theoretic concepts useful in extracting information from noisy data sets or performing forward prediction in time for a dynamic system. The broad utilization of the associated algorithms makes them attractive targets for similar problems in the quantum domain. To date, however, the construction of adaptive filters for quantum systems has typically been carried out in terms of stochastic differential equations for weak, continuous quantum measurements, as used in linear quantum systems such as optical cavities. Discretized measurement models are not as easily treated in this framework, but are frequently employed in quantum information systems leveraging projective measurements. This paper presents a detailed analysis of several technical innovations that enable classical filtering of discrete projective measurements, useful for adaptively learning system dynamics, noise properties, or hardware performance variations in classically correlated measurement data from quantum devices. In previous work we studied a specific case of this framework, in which noise and calibration errors on qubit arrays could be efficiently characterized in space; here, we present a generalized analysis of filtering in quantum systems and demonstrate that the traditional convergence properties of nonlinear classical filtering hold using single-shot projective measurements. These results are important early demonstrations indicating that a range of concepts and techniques from classical nonlinear filtering theory may be applied to the characterization of quantum systems involving discretized projective measurements, paving the way for broader adoption of control theoretic techniques in quantum technology.