Another reason why HMMs are popular is because they can be trained automatically and are simple and computationally feasible to use. In speech recognition, the hidden Markov model would output a sequence of n -dimensional real-valued vectors (with n being a small integer, such as 10), outputting one of these every 10 milliseconds. The vectors would consist of cepstral coefficients, which are obtained by taking a Fourier transform of a short time window of speech and decorrelating the spectrum using a cosine transform , then taking the first (most significant) coefficients. The hidden Markov model will tend to have in each state a statistical distribution that is a mixture of diagonal covariance Gaussians, which will give a likelihood for each observed vector. Each word, or (for more general speech recognition systems), each phoneme , will have a different output distribution; a hidden Markov model for a sequence of words or phonemes is made by concatenating the individual trained hidden Markov models for the separate words and phonemes.
“Barack and I were raised with so many of the same values: that you work hard for what you want in life; that your word is your bond and you do what you say you’re going to do; that you treat people with dignity and respect, even if you don’t know them, and even if you don’t agree with them. And Barack and I set out to build lives guided by these values, and pass them on to the next generation . Because we want our children — and all children in this nation — to know that the only limit to the height of your achievements is the reach of your dreams and your willingness to work for them. ”