8 Hidden Markov Models¶

8.1 The Markov Chain( Discrete-time Markov chain)¶

Let $\mathbf{X}=X_1X_2...X_n$ be a sequence of random variables from a finite dicrete alphabat $\mathbf{O}=o_1o_2...o_M$, then

$$P(X_1,X_2,...,X_n)=P(X_1)\prod_{i=2}^{n}P(X_i|X_1^{i-1}) \tag{8.1}$$

where $X_1^{i-1}=X_1X_2...X_{i-1}$.

• first-order Markov chain(first-order Markov assumption):

$$P(X_i|X_1^{i-1})=P(X_i|X_{i-1}) \tag{8.2}$$

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