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Markov models are a special case of hidden Markov models (HMM). In Markov models the state sequence is visible, whereas in a hidden Markov model the underlying state sequence is hidden and the sequence of observations is visible. Previous research on objective techniques for output-based speech quality (OBQ) showed that the state transition probability matrix A of a Markov model is capable of capturing speech quality information. On the other hand similar experiments using HMMs showed that the observation symbol probability matrix B is more effective at capturing the speech quality information. This shows that the speech quality information in A matrix of a Markov model shifts to the B matrix of an HMM. An HMM can have varying degrees of hiddenness, which can be intuitively guessed from the entries of its observation probability matrix B for the discrete models. In this paper, we propose a visibility measure to assess the hiddenness of a given HMM, and also a method to control the hiddenness of a discrete HMM. We test the advantage of implementing hiddenness control in output-based objective speech quality (OBQ) and isolated-word speech recognition. Our test results suggest that hiddenness control improves the performance of HMM-based OBQ and might be useful for speech-recognition as well.