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    "# 8 Hidden Markov Models\n",
    "\n",
    "## 8.1 The Markov Chain( Discrete-time Markov chain)\n",
    "\n",
    "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\n",
    "\n",
    "$$ P(X_1,X_2,...,X_n)=P(X_1)\\prod_{i=2}^{n}P(X_i|X_1^{i-1}) \\tag{8.1}$$\n",
    "\n",
    "where \\\\(X_1^{i-1}=X_1X_2...X_{i-1}\\\\).\n",
    "\n",
    "* **first-order Markov chain(first-order Markov assumption):**\n",
    "\n",
    "$$P(X_i|X_1^{i-1})=P(X_i|X_{i-1}) \\tag{8.2}$$"
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    "## Building ..."
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