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Filtering hidden Markov models is analytically tractable only for a handful of models (e.g. Baum-Welch and Kalman filters).Recently, Papaspiliopoulos \& Ruggiero (2014) proposed another analytical approach exploiting a duality relation between the hidden process and an auxiliary process, called dual and related to the time reversal of the former. With this approach, the filtering distributions are obtained as a recursive series of finite mixtures.Here, we study the computational effort required to implement this strategy in the case of two hidden Markov models, the Cox-Ingersoll-Ross process and the $K$-dimensional Wright-Fisher process, and examine several natural and very efficient approximation strategies.