chapter-04/4.4-logistic-regression.ipynb

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    "# 4.4 Logistic Regression\n",
    "\n",
    "The model has the form (4.17):\n",
    "$$\n",
    "\\begin{equation}\n",
    "\\log \\cfrac{Pr(G = 1 | X = x)}{Pr(G = k | X =x)} = \\beta_{10} + \\beta_1^Tx\\\\\n",
    "\\log \\cfrac{Pr(G = 2 | X = x)}{Pr(G = k | X =x)} = \\beta_{20} + \\beta_2^Tx\\\\\n",
    "\\vdots\\\\\n",
    "\\log \\cfrac{Pr(G = K-1 | X = x)}{Pr(G = k | X =x)} = \\beta_{(K-1)0} + \\beta_{K-1}^Tx\\\\\n",
    "\\end{equation}\n",
    "$$\n",
    "\n",
    "The model is specified in terms of K - 1 log-odds or logit transformations. A simple calculation shows that (4.18):\n",
    "$$\n",
    "\\begin{equation}\n",
    "Pr(G=k|X=x) = \\cfrac{exp(\\beta_{k0} + \\beta_k^Tx)}{1+\\sum_{l=1}^{K-1}exp(\\beta_{l0} + \\beta_l^Tx)},\n",
    "k = 1,...,K - 1,\\\\\n",
    "Pr(G=K|X=x) = \\cfrac{1}{1+\\sum_{l=1}^{K-1}exp(\\beta_{l0} + \\beta_l^Tx)}\n",
    "\\end{equation}\n",
    "$$\n",
    "\n",
    "and they clearly sum to one."
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