You can use a recursive Bayes filtering (RBF) approach to do maximum a posteriori (MAP) estimation. While MMSE estimators look at the conditional mean, MAP estimators look at the conditional mode. See these notes for some details. In order to do RBF, you should look into particle filters / sequential monte carlo methods. There are plenty of books on this topic as well as review papers.
Alternatively, if you assume that your Kalman filter has white Gaussian noise independent for the process and measurement, and an independent Gaussian initial state, you can prove that the MAP estimator coincides with the Kalman filter. [ A Kalman filter is a RBF for the nice Gaussian case. ]
A good paper to look at is S. Godsill, A. Doucet, M. West's “Maximum a posteriori sequence estimation using Monte Carlo particle ﬁlters,” Ann. Inst. Stat. Math., vol. 52, no. 1, pp. 82–96, 2001.
The actual article is behind a paywall, but certainly, many articles on related topics have been written (You can just google Maximum A Posteriori estimation with particle filtering or recursive bayes filtering for details).