  We derive the forward and backward filtering equations for a class of
degenerate partially observable diffusions, satisfying the weak H\"ormander
condition. Our approach is based on the H\"older theory for degenerate SPDEs
that allows to pursue the direct approaches proposed by N. V. Krylov and A.
Zatezalo, and A. Yu. Veretennikov, avoiding the use of general results from
filtering theory. As a by-product we also provide existence, regularity and
estimates for the filtering density.
