dy is 0.1, of tomorrow being rainy is 0.1, of tomorrow being sunny is 0.8 (2, p.50).Figure 2: Three-state Markov model of the weather (2, p.51). This example is an observable Markov model since we can check the state we are currently in (2, p.50). Nevertheless, speech recognition systems use hidden Markov models since the speech fragment is not observable by the speech recognition system (2, p.50). In hidden Markov models, a state can represent many outputs, therefore, a probability distribution of all possible outputs is associated with each state. A diagram of a three-state HMM is shown in figure 3 (4, p.44). This figure shows that each state has five possible outputs (A, B, C, D, and E) occurring with a probability according to b--1(s), b2(s), or b3(s). HMMs are doubly probabilistic since the transition from one state to the other and the output generated at that state are probabilistic (4, p.44). Therefore we notice that if we receive a sequence of outputs from an HMM, we are not able to retrace the sequence of states that the HMM passed by to get that sequence (4, p.44). Looking at figure 3, it is evident that an output sequence of A-B-C for example, can be achieved by any sequence of three states; however, each sequence of states has its own probability of occurrence. In speech recognition, each word is represented by a sequence of states (1, p.53), therefore, it is essential to find this sequence for any sequence of outputs. In fact, finding this sequence is equivalent to solving the speech recognition problem.Figure 3: Three-state hidden Markov model (4, p.44).The sequence of states is determined according to its probability. However, checking all the probabilities of all possible sequences can be very time consuming, especially in speech recognition HMMs that are much more complicated than our three-state example in figure 3. This problem was solved using an algorithm that utilises the fact that the probability of ...
