AR_probabilistic.ipynb

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    "AR sequence generation can be formulated using the chain rule of probability. Let $x = (x_1, x_2, \\ldots, x_n)$ be the true sequence of length $n$.\n",
    "\n",
    "$$P(x) = P(x_1, x_2, ..., x_n)$$\n",
    "\n",
    "The joint probability of the sequence can be decomposed into a product of conditional probabilities:\n",
    "\n",
    "$$P(x) = P(x_1, x_2, ..., x_n) = P(x_1)P(x_2 | x_1)P(x_3 | x_1, x_2) \\dots P(x_n | x_1, x_2 \\dots, x_{n-1})$$\n",
    "\n",
    "Using prod notation this can be written as:\n",
    "\n",
    "$$P(x) = \\prod_{i=1}^nP(x_i|x_1,x_2,\\dots,x_{i-1})$$\n",
    "\n",
    "We can simplify the result to:\n",
    "\n",
    "$$P(x) = \\prod_{i=1}^nP(x_i|\\mathbf{x}_{<i})$$"
   ]
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