State estimation, a foundational problem in modern science and engineering, is the process of determining the internal state of a system when only indirect, noisy, or incomplete measurements are available. Nearly every intelligent technology depends on this capability, including autonomous vehicles, robotics, medical imaging, advanced manufacturing, energy networks and financial analytics. In artificial intelligence and data-driven decision systems, state estimation underlies the ability of machines to interpret data and make reliable predictions in real time. However, real-world systems often violate the ideal mathematical assumptions on which classical estimation methods are built. Measurements may contain frequent outliers or unexpected disturbances, and system dynamics may be nonlinear or poorly modeled. In these settings, existing approaches can become unreliable or computationally burdensome. This award supports research that develops a new mathematical framework that reformulates state estimation as a structured optimization problem, enabling more accurate and computationally efficient estimation even under complex and non-ideal conditions. By strengthening the reliability of intelligent technologies that depend on accurate state information, this research supports innovation, economic competitiveness, and public safety. The project will also contribute to education and workforce development by training graduate and undergraduate students, integrating research results into advanced coursework, and engaging students in hands-on research experiences. This research reformulates state estimation as a problem of maximum a posteriori optimization problem and develops a dynamic programming recursion analogous to that used in optimal control. In the classical linear Gaussian case, this framework recovers the standard Kalman filter, providing a unified perspective. For systems with non-Gaussian noise (Aim I), the research will develop new recursive estimators by lo