ABSTRACT

In this chapter, we reviewed the aspects of probability theory, statistics, and random process theory relevant to statistical speech processing. We first introduced the concept of random variable and the associated concepts of probability distribution and summary statistics. We then discussed conditional probability, conditional independence, conditional expectation, and Bayes' rule. We then turned to the general characterization of (discrete-time) random sequences, with a special focus on the Markov sequence as the most commonly used class of the general random sequence. Central to the Markov sequence is the concept of state, which is itself a random variable. When the state of the Markov sequence is confined to be discrete, we have a Markov chain, where all possible values taken by the discrete state variable constitutes the (discrete) state space. When each discrete state value is generalized to be a new random variable (discrete or continuous), the Markov chain is then generalized to the (discrete or continuous) HMM. Some basic properties of the HMM have been discussed in this chapter, where the discussion made use of several more basic concepts introduced earlier in the chapter.