The discrete-time filtering problem can be seen as a dynamic generalization of the classical Bayesian inference problem. For practical applications it is important to identify filtering models that, analogously to the linear Gaussian model (Kalman filter), admit a finite-dimensional filter or, equivalently, a finite-dimensional family of filter-conjugate distributions. Our main purpose here is to give sufficient conditions for the existence of finite-dimensional filters. We use a method, based on the Laplace transform, which is also constructive.