Optimal projection filters with information geometry

Abstract We review the introduction of several types projection filters. Projection structures coming from information geometry are used to obtain a finite dimensional filter in form stochastic differential equation (SDE), starting exact infinite-dimensional partial (SPDE) for optimal filter. start with Stratonovich filters based on Hellinger distance as introduced and developed Brigo et al. (IEEE Trans Autom Control 43(2):247–252, 1998, Bernoulli 5(3):495–534, 1999), where SPDE is put before projection, hence term “Stratonovich projection”. The correction step filtering algorithm can be made by choosing suitable exponential family manifold, there equivalence assumed density numerical examples have been studied. Other authors further these we present brief literature review. A second type was Armstrong (Math Signals Syst 28(1):1–33, 2016) direct $$L^2$$ L 2 metric projection. Projecting mixtures densities manifold coincides Galerkin methods. All above lack optimality, single vector fields projected optimally but solution whole not approximated SDE according clear criterion. This led (Proc Lond Math Soc 119(1):176–213, 2019, SDEs onto submanifolds. “Information Geometry”, 2023 special issue half century geometry, 2018), Ito jet projections, mean square distances between sought approximations minimized, examples. After reviewing developments, conclude remaining challenges.