Filtering hidden Markov models, which can be seen as performing sequential Bayesian inference on the hidden state of a latent signal, is an analytically tractable problem only for a handful of models. Among these are models with finite-dimensional state space and linear Gaussian systems, which give rise to the celebrated Baum-Welch and Kalman filters. Recently, Papaspiliopoulos and Ruggiero (2014) and Papaspiliopoulos et al. (2016) proposed a principled approach for extending the realm of analytically tractable models, exploiting a duality relation between the hidden process of interest and an auxiliary process, called dual and related to the time reversal of the former. When such a dual process is available and has certain characteristics, the solution of the filtering problem is available analytically and takes the form of a finite mixture of distributions, which can be evaluated by means of a recursion similar to the Baum-Welch filter. Here, we study the computational effort required to implements the above strategy in the case of two hidden Markov models given respectively by the Cox-Ingersoll-Ross process with Poisson observations and the K-dimensional Wright-Fisher process with multinomial observations. In both cases, the number of components involved in the filtering distributions increases polynomially with the number of observations, yielding a so-called computable filter. This behaviour could render the algorithm impractical for large dimensional hidden spaces or very long observation sequences and undermine its practical relevance. However, the mathematical form of the filtering distributions suggest that, in certain regimes of separation between observation times and speed of the underlying signal, the number of components which contribute most of the mixture mass remains small. This in turn suggests several natural and very efficient approximation strategies. In this contribution, we assess the performance of these strategies in terms of accuracy and speed, which we can benchmark against the exact solution. Abstract Abstract in Italian