The authors consider the classical scalar linear Gaussian model for the state \(x\) and observations \(y\): \[ x_{k+1} =A_{k+1} x_k+ B_{k+1} w_{k+1}, \quad y_k= C_k x_k+ d_k v_k. \] They derive an exact finite-dimensional recursive filter for the vector-valued process \(s_k\): \[ s_{k+1} =F(x_k) s_k+ u_k, \] where \(F(x)\) is a polynomial matrix in \(x\), and \(u_k\) is a zero mean random vector independent of the processes \(x\), \(w\), and \(v\).