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**Time-inhomogeneous hidden Bernoulli model (TI-HBM)** is an alternative to hidden Markov model (HMM) for automatic speech recognition. Contrary to HMM, the state transition process in TI-HBM is not a Markov-dependent process, rather it is a generalized Bernoulli (an independent) process. This difference leads to elimination of dynamic programming at state-level in TI-HBM decoding process. Thus, the computational complexity of TI-HBM for probability evaluation and state estimation is (instead of in the HMM case, where and are number of states and observation sequence length respectively). The TI-HBM is able to model acoustic-unit duration (e.g. phone/word duration) by using a built-in parameter named survival probability. The TI-HBM is simpler and faster than HMM in a phoneme recognition task, but its performance is comparable to HMM.

In probability theory and statistics, the **Bernoulli distribution**, named after Swiss mathematician Jacob Bernoulli, is the discrete probability distribution of a random variable which takes the value 1 with probability and the value 0 with probability that is, the probability distribution of any single experiment that asks a yes–no question; the question results in a boolean-valued outcome, a single bit of information whose value is success/yes/true/one with probability *p* and failure/no/false/zero with probability *q*. It can be used to represent a coin toss where 1 and 0 would represent "heads" and "tails", respectively, and *p* would be the probability of the coin landing on heads or tails, respectively. In particular, unfair coins would have

**Hidden Markov Model** (**HMM**) is a statistical Markov model in which the system being modeled is assumed to be a Markov process with unobserved states.

In mathematics, the **Bernoulli scheme** or **Bernoulli shift** is a generalization of the Bernoulli process to more than two possible outcomes. Bernoulli schemes are important in the study of dynamical systems, as most such systems exhibit a repellor that is the product of the Cantor set and a smooth manifold, and the dynamics on the Cantor set are isomorphic to that of the Bernoulli shift. This is essentially the Markov partition. The term *shift* is in reference to the shift operator, which may be used to study Bernoulli schemes. The Ornstein isomorphism theorem shows that Bernoulli shifts are isomorphic when their entropy is equal.

[1] Jahanshah Kabudian, M. Mehdi Homayounpour, S. Mohammad Ahadi, "Bernoulli versus Markov: Investigation of state transition regime in switching-state acoustic models," *Signal Processing*, vol. 89, no. 4, pp. 662–668, April 2009.

[2] Jahanshah Kabudian, M. Mehdi Homayounpour, S. Mohammad Ahadi, "Time-inhomogeneous hidden Bernoulli model: An alternative to hidden Markov model for automatic speech recognition," *Proceedings of the IEEE **International Conference on Acoustics, Speech and Signal Processing (ICASSP)*, pp. 4101–4104, Las Vegas, Nevada, USA, March 2008.

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