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    "Intro\n",
    "\n",
    "- definition: classification problem\n",
    "  - $(x,c) \\in X \\times C$, $X = R^d$, $C = Z_r$\n",
    "  - joint probability $P(x,c)$\n",
    "- classifier\n",
    "- training set $T = \\\\{(x_1,c_1),...,(x_N,c_N)\\\\}$\n",
    "- test set\n",
    "- error rate\n",
    "- empirical error rate\n",
    "\n",
    "Bayes\n",
    "\n",
    "- Bayes formula\n",
    "- Bayes risk\n",
    "- zero-one loss function, posterior\n",
    "\n",
    "Classifiers\n",
    "\n",
    "- classifier: $c = C(X)$\n",
    "- discriminant functions: $c = \\arg\\max_c g(x)$\n",
    "- posterior probabilities: $c = \\arg\\max_c P(c|x)$\n",
    "\n",
    "Nearest Neighbor Classifier\n",
    "\n",
    "- $C_{NN}(x) = c_k$ where $k = \\arg\\min_i ||x_i-x||$ for $(x_i,c_i)$ in training set\n",
    "- $C_{NN}$ has at most twice the Bayes error rate (why?)\n",
    "- k-NN classifier: use a vote among the $k$ nearest neighbors\n",
    "- remember $||x-y||^2 = x^2+y^2-2x\\cdot y$"
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