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Control Optim."],"published-print":{"date-parts":[[2023,8,31]]},"abstract":"<jats:p>Abstract.<\/jats:p>\n                  <jats:p>Nonlinear filtering is fundamental to many engineering problems, as it involves inferring the state of a system given complicated dynamics and noisy observations. Some approaches to nonlinear filtering use the analysis of the underlying PDE or stochastic PDE governing the evolution of the posterior probability distribution over time, one approach, in particular, being the Yau\u2013Yau method. In this paper, we give a maximum a posteriori (MAP) framework for the Yau\u2013Yau method. Furthermore, we propose convex filtering, intermediate between linear and nonlinear filtering, which gives criteria under which the posterior preserves log-concavity. The key tool from the PDE is a result from Korevaar, giving criteria under which a quasilinear parabolic PDE preserves convexity. A bridge between the MAP estimator and the structure of the posterior is explained. 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