{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Rank Estimation"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "%%capture\n",
    "# execute the creation & training notebook first\n",
    "%run \"02-01-creation_and_training.ipynb\"\n",
    "# execute the outlier detection notebook\n",
    "%run \"02-05-outlier_detection.ipynb\""
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the [outlier detection section](./02-05-outlier_detection.ipynb) we saw how to detect outliers in a test data set and how the outlier threshold influenced the detection results.\n",
    "\n",
    "In this section we take a look at the ``.estimate_ranks`` method. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The corresponding class ``RankEstimator`` is actually used by the ``OutlierDetector`` under the hood to estimate the typicality of individual data points. Let us apply the rank estimator to the modified test data set from the outlier detection example."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
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       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>(a)</th>\n",
       "      <th>(b|a)</th>\n",
       "      <th>(c|a,b)</th>\n",
       "      <th>graph</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>0.547</td>\n",
       "      <td>0.267</td>\n",
       "      <td>0.742</td>\n",
       "      <td>0.660</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>0.974</td>\n",
       "      <td>0.839</td>\n",
       "      <td>0.876</td>\n",
       "      <td>0.987</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>0.452</td>\n",
       "      <td>0.873</td>\n",
       "      <td>0.757</td>\n",
       "      <td>0.877</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>0.089</td>\n",
       "      <td>0.575</td>\n",
       "      <td>0.555</td>\n",
       "      <td>0.313</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>0.880</td>\n",
       "      <td>0.891</td>\n",
       "      <td>0.869</td>\n",
       "      <td>0.991</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>...</th>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "      <td>...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>95</th>\n",
       "      <td>0.171</td>\n",
       "      <td>0.840</td>\n",
       "      <td>0.540</td>\n",
       "      <td>0.506</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>96</th>\n",
       "      <td>0.660</td>\n",
       "      <td>0.576</td>\n",
       "      <td>0.800</td>\n",
       "      <td>0.897</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>97</th>\n",
       "      <td>0.684</td>\n",
       "      <td>0.761</td>\n",
       "      <td>0.850</td>\n",
       "      <td>0.946</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>98</th>\n",
       "      <td>0.903</td>\n",
       "      <td>0.830</td>\n",
       "      <td>0.863</td>\n",
       "      <td>0.985</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>99</th>\n",
       "      <td>0.879</td>\n",
       "      <td>0.369</td>\n",
       "      <td>0.879</td>\n",
       "      <td>0.853</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "<p>100 rows × 4 columns</p>\n",
       "</div>"
      ],
      "text/plain": [
       "      (a)  (b|a)  (c|a,b)  graph\n",
       "0   0.547  0.267    0.742  0.660\n",
       "1   0.974  0.839    0.876  0.987\n",
       "2   0.452  0.873    0.757  0.877\n",
       "3   0.089  0.575    0.555  0.313\n",
       "4   0.880  0.891    0.869  0.991\n",
       "..    ...    ...      ...    ...\n",
       "95  0.171  0.840    0.540  0.506\n",
       "96  0.660  0.576    0.800  0.897\n",
       "97  0.684  0.761    0.850  0.946\n",
       "98  0.903  0.830    0.863  0.985\n",
       "99  0.879  0.369    0.879  0.853\n",
       "\n",
       "[100 rows x 4 columns]"
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ranks = causal_structure.estimate_ranks(data=mod_test_data)\n",
    "ranks"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We see that the ranks are numbers between zero and one. These numbers describe the fraction of data points in a comparison data set with a lower probability that the data point in question. The comparison data set is generated from the trained graph under the hood.\n",
    "\n",
    "For example, a value of 0.8 means that 80% of the generated comparison data had a probability that was lower than the data point in question.\n",
    "\n",
    "This is the continuous analogon to the outlier detection. Consequently, we can apply the outlier treshold of 0.05 to the ranks."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
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       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>(a)</th>\n",
       "      <th>(b|a)</th>\n",
       "      <th>(c|a,b)</th>\n",
       "      <th>graph</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>25</th>\n",
       "      <td>0.814</td>\n",
       "      <td>0.030</td>\n",
       "      <td>0.794</td>\n",
       "      <td>0.173</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>31</th>\n",
       "      <td>0.045</td>\n",
       "      <td>0.769</td>\n",
       "      <td>0.507</td>\n",
       "      <td>0.193</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>50</th>\n",
       "      <td>0.366</td>\n",
       "      <td>0.000</td>\n",
       "      <td>0.000</td>\n",
       "      <td>0.000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>60</th>\n",
       "      <td>0.262</td>\n",
       "      <td>0.712</td>\n",
       "      <td>0.000</td>\n",
       "      <td>0.000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>74</th>\n",
       "      <td>0.009</td>\n",
       "      <td>0.523</td>\n",
       "      <td>0.403</td>\n",
       "      <td>0.045</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "      (a)  (b|a)  (c|a,b)  graph\n",
       "25  0.814  0.030    0.794  0.173\n",
       "31  0.045  0.769    0.507  0.193\n",
       "50  0.366  0.000    0.000  0.000\n",
       "60  0.262  0.712    0.000  0.000\n",
       "74  0.009  0.523    0.403  0.045"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ranks.loc[(ranks<=0.05).any(axis=1)]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We find pretty much the same entries as in the [outlier detection section](./02-05-outlier_detection.ipynb). Some ranks are close to the threshold and might sometimes lie above or below the threshold depending on sampling effects."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Rank estimation tells us how unusual certain data points are. We see that the modified data points with indices 50 and 60 are judged to be very unlikely. None of the 1000 comparison data points had a lower probability than those.\n",
    "\n",
    "Let's examine the ranks of the data points 50 and 60 more closely."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The ``.estimate_ranks`` method tells us that the values of '(a)' are not particularly unlikely in both of them.\n",
    "\n",
    "The value of '(b|a)' is judged to be common for data point 60, and next to impossible for data point 50.\n",
    "There is nothing particularly unusual about the value of '(b|a)' in data point 50 if we look at a 1-d histogram."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x20bcc86d640>"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "pl.hist(mod_test_data[\"(b|a)\"], density=True)\n",
    "pl.vlines(mod_test_data[\"(b|a)\"][[50,60]], [0,0], [0.15, 0.15],\n",
    "          colors=\"r\", label=\"data points 50 and 60\", ls=\"--\")\n",
    "pl.xlabel(\"(b|a)\")\n",
    "pl.ylabel(\"frequency\")\n",
    "pl.legend()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "However, given the values of '(a)' they certainly are unlikely as we can see from a 2-d scatter plot."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.legend.Legend at 0x20bcf1294c0>"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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XOAqRVhBnUkANuJOqZDIZenp6Co4lEglGRkbyzVnBJIRReYig2avSwEEFEpHWoRyFxObuDAwM5HMUExMT9PX15ffnzZuXf5hPJ3GuAXkiraVpNQozuxl4O/Ab4OfAe919PPfaeuB9wASQdPfvlLqPNE44IR5MAxLUHubNm8fGjRsnnRdOnAMVE+fuWmxJpNU0LUdhZucD/+buB8zsEwDufoOZnQr8A7ASOBa4C3iTu0+Uu59yFI0Tt1loqs1HQQ2iFostqQlLpLyWXrjI3b/r7gdyuz8CFuV+vgi4zd1fdfdfAE+QDRrSIuJOTFjtBIbh82qx2JKasERqo1VyFH8O/Gvu5+OAZ0Kv7cgdm8TMrjaz7Wa2fdeuXXUuojRK8EAPq3aMRbgJS3NKiUxTpf6z09nINhs9HLFdFDrnRuB2DjaDfRq4IvT654FLK72XxlHMDFHzRcWZP6rSvUBzSolEIcY4iroms919dbnXzewq4G3AebkCQ7YGcXzotEXAzvqUUFrNdBLhUfcaGhoqyHVovXCR6jUzmX0hMAiscvddoeNLgK9yMJl9N3CyK5k9q3gNktBew6S4yEzV0sls4FPAUcD3zGzMzD4L4O6PAP8IPAp8G/jLSkFCZp6pJsKDLz7hIKE5pUSmp2njKNz9pDKvfQz4WAOLIzNAsNRrUGOYN29ewUDAqTZhicx2msJDWl6cZiiPGKgXzHS7atWq/DVqdhKpngKFtLTiWkLQpNTZ2VkwHiJcYxgeHs4HjOKchIKESPVaZRyFyCThWkKcsRC1GqgnIoUUKKRlhRdXGh4epqOjI5+cjgoAQSAJU+JaZPoUKKSlxa0lqJeTSP0oUEhLi1tLqMVSryJSQqWh2+2yaQqPmWcq03lMd6lXkdmGZk/hITOf12AEdSlTmc5jqgP1RKQ0rZktUxa36+p0hYNP8Pdaau1tEalOq0/hIW3Mq+y6Oh1BIAjWlwiXQetLiNSfmp5kSuIOcKskbtNVODCBlkgVaSQ1Pcm0uDsdHQcrpplMJvYDu9qmq3CtJTDd2WDrmWMRaQdqepK6Ch7cYXHHLEyl6arWI6+1VKpITJW6RbXLpu6xjVWLleiqXYGulivW1XIlPZF2RozusU1/wNdqU6BovFQqVfBQDR62qVQq9j0ymUzBgz/8gC7+udYPdi2VKhIvUDRzhbubgbcDvwF+DrzX3cfN7A+BvwZek3vtenf/t0r3U46iOdyn3sbvPjnnkEwmueWWWwAm5Svq0R3Xfeo5FpGZIE6Oomk1AOB84JDcz58APpH7+feAY3M/LwV+Ged+qlG0l/C3+WQy6clkMv+tPrxf/A2/liOvVaMQaaOmJ+BiYFPEcQN2A4dVuocCRfsJN11lMpmCYFHvh7ZyFCJZcQJFS3SPNbM7ga+5+1eKjr8DuMbdV5e47mrgaoDFixf3PPXUU3Uvq9SWe+Go60Y2A4WbsgJBU1YqlVITlMwKcZqe6hoozOwuYGHESze6+x25c24EVgCXeKgwZrYE2AKc7+4/r/ReylG0N4/IV0x3jETx/aNyKe7Oxo0bSwYMdZWVma7p4yjcfbW7L43YgiBxFfA24PKiILEIuB24Mk6QkPYWDhL1WEui0niJ8HgOoG5TkQSK79kKtXqRcpo2hYeZXQjcAKxy95dDxzuBbwHr3f0HTSqeNNBUZomNy7381B/BMYg3FUlxU1lQ/uLXSmnURIoiNVUpiVGvDXgCeAYYy22fzR3/EPBS6PgYcEyl+ymZ3f7qtZZEnN5N5cZzBMLJ91Qqle+dlUqlYo0hUQJdWhHt0uupFpsChZRTaWBfnEAS9MqqpjtvVDnUJVdaiQKFiJd/OFf6lj8xMeHunq9BrF27tuA+4a23tzf21CWVai8ijRInUGhSQJnR3MsnyoGSa22PjY0xMDBAJpNhfHyckZER7r333pLv1dvbG7s8YbVI2IvUk9ajkBktTqI8nU7jfjARbWYMDg4yMDCQT24PDg7y/e9/n7GxsSmXpThohRPrML2ZcEXqKVagMLO5ZLuxngkcC+wDHga+5e6P1K94ItMXFQiKH8rFD+iOjo7I3lClJJNJRkZGIu8dfo969e4SqaeKA+7MLE128r57gFHgOWAu8CbgnNzP17r7g/UsaCUacCf14F44WryUZDIJQFdXV8VR3eGgFbUv0ki1GnD3Y3fvcfdr3f2r7n6Xu3/T3Qfd/e3A5WRnehWZUaLyCd3d3fmgAHD66aeTSCQYGRkBYMOGDRUXPyoOCvUOEsVfBpUPkWpVbHpy929VeP05srUMkbYW/mbv7vT39zMyMkJfXx/z5s1jy5YtjI2NMXfuXJLJJO7Otm3bGBsbI5FI0NnZmc9rBInzcG2kGTUHDfCTWoidzDazBWRHUp9KtrkJAHc/tw7lEmmo4gcqwNatW+nt7c0fS6VSDAwM5CcNDAQJ6SDRHQSWgYGBpj6g3cuPSleTl8RWqf9ssAHfBd4HPAasAv4fuTUkWmHTOAqZqnJjKZLJZNn1MNzdN2zYUDAu4sCBA55IJKZ8v3r9bsGmAX4SRi0H3AGjuX8fDB37ftzr670pUMh0VBqUV8rExEQ+KARbsD9//vyC4wsXLvSVK1dWXDq21lOZaICflBMnUFQz4G5/7t9fmdlbzez3gEVVVV9EWlS4q2og2C+VnHZ3BgYG8jmKwNjYGPPnz+f5558vOP/ZZ59l27Zt9Pf3F4ypGB8fJ5PJAAdnug32g/Oms9SrBvjJdFUTKD5qZvOAa4HrgL8D1pW/RKQ9eC55Hdbf309/f3/JKcfD4yJGR0cLXrvmmmsKggdAIpFg7dq1jIyM0NHRkc8VBPmMYAT48PAwPT09pFKpaU15Hg5G9Zi+XWaRSlWOdtnU9CRTVW79biLyCsUmJiYmNVsVN0eF7xXen5iYKJhQMKopazo5hfCMt+HftdwstzK7UIulUM3srJgx50l3f7q6MFU7GnAn01G8LGrcJVndS0/LUdz8lEgkeO6559i5c2f+WG9vL/v27ePwww9n69atke8xlSVh3ae3bobMHrUacPfemFtiOoUVaaZ0Ol2Qkwgr10wTNS3H4OAgiUSC559/Pt/kk0wmGRsbyweJZDJJMplk69atPPjgg+zbt69k2YpX5wuLKlfxin7BPYI8h4KEVK1SlaNdNjU9yXTFXVgoqldS8bENGzZMuqa3t9dXrlw5qfnpmGOOmdRElUgkCpq0+vr6Iu9Z3IykxZGkWtR6PQrgEmAQ+L/An1RzbcS9bgYeBx4kuz52Z9Hri4EXgevi3E+BQmqhUpt+NW3+pQJKcXfV4sARHoMRBIsgSMQJABo7IdWoaaAA/pbsoLugqenbwKfjXh9xv/OBQ3I/f4KiwXvAPwFfV6CQRis1jqEW39ajHuKvf/3rC/bXrl2bX2K11PtXCgAaOyFx1TpQPEJuttncfgfwSNzrK9z7YmBTaP9PcjWOtAKFtJLpfFuPWj0vqD10d3cXNDWV6mkVJwBUKmO91iaX9hQnUFQzjuJnueagwPFkm41q4c+BfwUwsyPJzim1sdJFZna1mW03s+27du2qUVFESis1MC9Ogrg48d3R0cGaNWtIJBJcdNFF+TUwkskkXV1dk+7pXnnwXHBOqbETwdiM4JrgfE0QKOVUDBRmdqeZbQGOBh4zs3vM7N/Jzvm0oMK1d5nZwxHbRaFzbgQOAJtyhzYCQ+7+YqWyufut7r7C3VcsWFC2KCJTEn4IQ7ar6nRGOge9q8KLKJ155pkVH9SVAkDw/qUWRwoG9u3du7fgmukM6JNZpFKVAziX7CSAkVvuHKt0nxL3vgq4HzgidOw+4MncNg78D/C/K91LTU9Sa8WJ63BT0XR6FE015zHdRHrxe1Jl05nMTNQiR0F2Zbu1wOKi468BzgO+BLyn0n0i7nsh8CiwoMw5aZSjkCYo9xAPuq6Gz4s70rlc8Inz4K5FfkGJbgmrVaCYC3wA+AGwM/dw/2/gKeBzQKLSPUrc9wngGWAst3024hwFCmmaUt++gyARPi/OA7xc8GnUg1s1CilWk0BRcDIcCryBojEPrbApUEg9xPn2XW2TUPGDupZzO1X6XTQYT4rFCRTV9HrC3fe7+6/cfbya60TakXu8XkbBjK9xEsRRvabGxsYaMrtruUR3Z2enpvaQ0ipFknbZVKOQWqrm23e1A+GiahSlch71GPOgcRQSRq2bnlp5U6CQWqu2SYkKTVTVziVV6f2Lg1XxvkgccQLFIU2oxIi0hXQ6jbsXjHmIGlznHt1EFXVuuOkHDq6iF276MbOCJq3gvPA4ilQqxd69exkaGmLjxo3s2bMHgK6urvygus7OTg2kk9qoFEnaZVONQpohbi0hXDsItqB2UKrXVLleV8Hx4oWWwvtKUEscqOlJpP7iNBGVCyblpg8v1aRVqmttOKAoSEgcChQiDVI8tiJqrEWp2kFxDSDYj1qWtTiglAoUccd2iMQJFFV1jxWRydLpNAMDA9lvXmS/fA0MDBTkB0pNJnjTTTcB2RXvhoeH6ejoYGRkhJUrVwIwMjIS2XU2as6psDPOOIP+/v6CMmnyP5kqBQqRaXCPN44iOB7W39/Pnj17GBkZmXTft7zlLWUn9xsYGGB4eDi/pGogvMTqyMhIPlho8j+ZlkpVjnbZ1PQkzRJn/YdSOYqo5qXgeLnmo3BeJJVK5e8T5DWSyaT39vZWnH5EhBhNT+Yz5NvFihUrfPv27c0uhsxS7k5Hx8EKeiaTKegam06nGR8fz9cOPPctf968eYyPjxfUKpLJZL7JqdxaF+4Hu+4G/x+H99PpdL5pC2BiYoKBgYGqu82G3ydqX9qbmY26+4py52gchcg0BQ/9sOJxFKXGZEA2n1AsmUxWnFYj/FrU2I4tW7YUHOvp6clPFxL1sI8KCBs3bowMcBqjMctUqnK0y6amJ2mG6Uy0V9wEFbU/3TJ1d3cXND8FS64WK+6iG/TGCpqvNIngzIVGZovUV6mJ9oBYNYKpXhunTL29vfT29vLAAw/kX5s7dy433XRTQW0glUqxZcsWxsbGABgcHMzXPpLJJL29vQwPD+dHiVdqEpMZqFIkaZdNNQpppumMWajXxH9RSXKKaivFCzKFzwsmK9RCRzMbrTyOwsxuNrPHzexBM7vdzDpDry03s/vN7BEze8jM5jarnCJxFH+7rubb9nSuzf5/Xno/znsH3W6DGkVgdHQUM5vWGuEyMzRzHMX3gKXuvhz4T2A9gJkdAnwFuMbdlwBnA/ubVUiRVpVOpwse2u4HB9WZGV1dXQVjLCCbJO/q6pqUCB8cHJx0/3Xr1tHf35+fiLDe62VI62paoHD377r7gdzuj4BFuZ/PBx509wdy5+1294lmlFGkFUTVGtwrD/RLpVKR9ys+nslk6OnpKTiWSCQYGRlh69atJJPJfE5icHBQCx3NRpXaphqxAXcCV+R+7ge+DHwH+AnwwTLXXQ1sB7YvXrx4uk11Ii2n3ISD5Qb6VbP2RThHEZ6ZduHChb5y5cpJiypt2LChaZ+H1B7NzlGY2V1m9nDEdlHonBuBA8Cm3KFDgD8ALs/9e7GZnRd1f3e/1d1XuPuKBQsW1PNXEWk4r1BrACLnjzKz2Muehs8bHR2lo6ODoaEhkskkixcvZtu2bfl5rIL33rt3r5qdZptKkaSeG3AVcD9wROjYZcAXQ/sfBq6vdC/1epKZKG6tofi18PXF9yv1PsX71SzxKu2LVp5mHLgQeBRYUHS8i2yT0xFkaxd3AW+tdD8FCpmporqnTmegX/G9K+0Xv3fUedK+4gSKZvZ6+hRwFPA9Mxszs88CuPseYBD4MTAG/MTdv9W0Uoo0kXv09CBArKalcsr1mir33sEU5608hUfwO5XalypViiTtsqlGITNNnFrDVAfrVbp3OKkdDMQr/rdVm6EqrTgohWjlpqdabwoUMhPV86FXKQcRvHc4aESd10pq1SQ3m8QJFJpmXKTFuddvmm/38tOjB+8VdR5UN4p8OmWs5vd3P9hDK6D5qUqLM824VrgTaXHTmeKjnKjlVIMcRPi9MpnMpKnQ+/v76e/vL5mnKP4CGnwzLXdOlEp5lCillp1VkJg6BQppOx56kEXtS2WpVIqenp789BwTExMkEgmGh4fp6enJB4tg1PbWrVuBg0utjoyMMDIywp49eyY98KMe7meccQZnnHFGVQ9893jLzEZdp/mpaqxS21S7bMpRzA7/vmqV35NIeCYYLTwx4fckEv7vq1Y1t2BtpNxo7Pnz50e273d3d/vatWsL8hS9vb2RyfTinEB4FtvidTcq5Q02bNgQOattqdHhylFUDyWzZSYJgoJDPlgU70s8pRLZBw4cmHS8t7fX165d6xMTEwXH4zyswwGieNrzSg/uqPvEuVa9nqoTJ1AomS1txTMZ7u3pYVVoSuzvJxKcNTqKdagltRru0Yns4uMrV65k27ZtdHd3FyyCtHDhQn75y18WnFvu3kDZxHmUoOkrPAV6IpHITzdS7nfTOt/xKJktM451dHDW6GjBMQWJ6rmXH0wX9uqrrwIUBAmAZ599NrLtP+reQfK7+P2iri2+T/E6GWNjY/n5p0qpVweAWatSlaNdNjU9zQ7h5qZgU7NTdcq14xcPqgtyFkcccURB08/atWs9mUxOas6ZTo4iqslo5cqVvnDhwtg5CqkeylHITKIcRe2UasdftWpVPldRnEQOb8lkMj/9eJx79/b2FiS/i/MGUSPDq02C12NJ2dkgTqBQjkLayj1nn43t3ZtvbgpyFj5vHmffc0+zi9dW3KPb8YN/M5kMc+bMKXl9MpnklltuiWzWibo3UHD/qH+LB8r19vbS29sLQFdXF6lUinXr1tHZ2VnQtTadTjM+Pp4fLxHcq/g8mSxOjqLpNYFabapRzB7FNQfVJGqvVI+jAwcO5L/pR3WPraRSj6TinlXhmkW5+a3UJXbqUNOTiFQrKmcRbIlEIh8squ1uWs1EhMVbqQd+1Ep94WtKNY/JQQoUIjIl4YFuxTmL6TyAyz3Qi4NGcc2iWHHt5MMf/vCkazR+orI4gUJ9CkVkko0bN7JmzZr8ZHpz5sxhdHQ0v95FuTEM5ZSah6mjoyO/vsbg4CADAwMF5xR3h3UvnN5jYmKCz3zmMwXXBFOUlJvuQ2KqFEnaZVONQqT2at2TqNLU5lE1i1L5hqh7Bd15gy1oKpPSaOUahZndbGaPm9mDZna7mXXmjh9qZl8ys4fM7DEzW9+sMorMdrUcuOZ+sFdTX18fmUyGvr6+gkn/wjWLoKZRauW+qNrJNddcU7C/Zs0arr32WvV8mqZDmvje3wPWu/sBM/sEsB64AfhT4DB3X2ZmRwCPmtk/uPuTTSyriFTgXn7aDDOLXL4VKAgC6XS64NrgvOIgFQSesM9+9rMF+1u2bGFsbIy+vr5J5ZEqVKpyNGIDLgY25X7+M+BOskHsaOA/gddVuoeankSap5qJ+GrRnBXVY6p4NDmh5if1fiqNVm56KvLnwL/mft4MvAT8Cnga+KS7/0/URWZ2tZltN7Ptu3btakxJRaSAhxLL/f39BU1MUetV1KI5q7h20tHRwZo1a0gkEnzgAx8oOHfNmjX55HtxWSSeuo7MNrO7gIURL93o7nfkzrkRWAFc4u5uZr8PfAB4D9AF3Af8kbv/d7n30shskeZxzy5OFCxwBNmR25AdUV2vHIEXNSdNTExw7bXXFozuDkaQAxqtHSHOyOy65ijcfXW5183sKuBtwHl+MGK9C/i2u+8HnjOzH5ANJGUDhYg0V29vb0GgABgZGalrfqA4JxIEiSBIBSvxNao8M1XTktlmdiHZ5PUqd3859NLTwLlm9hXgCOAtwC2NL6GITMfIyAjJZLJh61UXN0eFyxEEi3AiXeJr2qSAZvYEcBiwO3foR+5+jZm9FvgCcCpgwBfc/eZK91PTk0hzhHMSwXragXITB9azPMH7uUcvztRMxbWZZtdumt70VI67n1Ti+Itku8iKSBsIvskHzT1hxU1RjSoPlF6cqZk1inad5bZVej2JSBtLpVLAwRxAJpMhmUyydevWyJXsqlF8bZx7hWs5pQb3NVq4d1hQhqCMrT7NSDMH3InIDGFmdHV1FeQAgian4hHV1ZjqN/C4g/saKVyG4eHhfM+stsibVBpo0S6bBtyJNF8t54aKu85EufdsxVXvMplMwYDAZpeJNhpwJyIzQC3nhgq+gQdNRh0dHfmmpOAbeDqdLmhK8lyNI6ht1LI8tRCUL6xZTWHVUKAQkZZValryoBmqndr8w+VrlbxJXMpRiEjLKvUNvDjv0A5t/q2YN4mtUttUu2zKUYjMLNXkKGhwm/908iKtljchRo5CNQoRaUlxvoF7hRpHPZTriQVU7KXVanmTWCpFknbZVKMQmZlKfQOPW+OodVlKvWcymfRkMtnQ8tQCqlGISLsr9Q28GW3+lfIiwTntkDOpRtPmeqo1zfUkMju5N37uJPfSc0iVe60VxZnrSd1jRaStNbrN30vkRYJmmnYcJ1GJAoWISExBIIgaC9Hf309/f39bjpOoRDkKEZGYKuVFgPYcJ1GBchQiIlUqlxdpRs5kOlo6R2FmHzGzB81szMy+a2bHhl5bb2ZPmNnPzOyCZpVRRNpf8ZfhWnw5LpcXactxEhU0M0dxs7svd/cE8E1gA4CZnQpcBiwBLgT+1szmNK2UItK2Kk0a2Gj1CFqN0LRA4e6/Du0eSXb4PcBFwG3u/qq7/wJ4AljZ6PKJSHtzb61JA1staFWjqclsM/sYcCWwFzgnd/g44Eeh03bkjomIxNZKkwaGgxZkZ8AN955q+TxGPaOqmd0FLIx46UZ3vyN03npgrrunzOzTwP3u/pXca58H/sXd/yni/lcDVwMsXry456mnnqrHryEibaxVBsCFazSBVhi13fRktruvdvelEdsdRad+Fbg09/MO4PjQa4uAnSXuf6u7r3D3FQsWLKj9LyAiba2VBsCVW1uj1TWz19PJod01wOO5n7cAl5nZYWZ2InAysK3R5ROR9lZucFwzgkUrBa1qNTNH8ddmdgqQAZ4CrgFw90fM7B+BR4EDwF+6+0Tziiki7aiVFgoqDlrhHAW0fs1CA+5EZEZrlQFw5daxaGbPpzg5CgUKEZEGaZWgFdb0ZLaIiBzUrqO2FShERKQsBQoRESlLgUJEpI7adX6nMAUKEZE6CeZ3ymQywMFusqlUqsklq44ChYhIjQXLogbzO5122mlkMpn82IktW7a0VbDQCnciIjUUHi8xODjIPffcwwMPPMCcOdnVEhKJBGNjY6xataolusfGoRqFiEiNFE9tbmasWrWq4JyxsbGaTAbYyNyHBtyJiNRQ1CyxxSYmJgpmtK1WLUd5a8CdiEiDRc0S293dXbDf09OTT3BXqxkLMilQiIjUUNQssQ888ADJZJKJiYl8jmJgYGBKD/UgEAUz4XZ0dBRMNliPnIcChYhIjURNbd7b25t/3cwYHR2lr69vWjPYNnptC/V6EhGpkaipze+//376+/vp6urCzPIP+ekmsqPWtqhbsAj6+7b71tPT4yIirSCTyZTdn+69+/r6HPC+vr7I/WoA273C81U1ChGRGqvnLLHNWJCpad1jzewjwEVkV7h7DniPu+80sz8E/hp4DfAb4Hp3/7dK91P3WBGZTbxGa1u0evfYm919ubsngG8CG3LHnwfe7u7LgKuALzepfCIiLauRa1s0renJ3X8d2j0S8Nzxn4aOPwLMNbPD3P3VRpZPRESympqjMLOPAVcCe4FzIk65FPhpqSBhZlcDVwMsXry4XsUUEZnV6pqjMLO7gIURL93o7neEzlsPzHX3VOjYEmALcL67/7zSeylHISJSvTg5irrWKNx9dcxTvwp8C0gBmNki4HbgyjhBQkRE6qdpTU9mdrK7/1dudw3weO54J9mgsd7dfxD3fqOjo8+b2VM1L2hrmU822S9Z+jwK6fMopM+jUKnP442VLmxm99h/Ak4h2z32KeAad/+lmX0IWA/8V+j08939uSYUs6WY2fZKVcTZRJ9HIX0ehfR5FJrO59HMXk+Xljj+UeCjDS6OiIiUoEkBRUSkLAWK9nJrswvQYvR5FNLnUUifR6Epfx4zZoU7ERGpD9UoRESkLAUKEREpS4GiRZnZHDP7qZl9M+K1eWZ2p5k9YGaPmNl7m1HGRqrweXSZ2e1m9qCZbTOzpc0oY6OY2ZNm9pCZjZnZpOkILGvEzJ7IfSanNaOcjRLj8/hdM7vfzF41s+uaUcZGivF5XJ77u3jQzH5oZt1R9wnTehStqw94DPitiNf+EnjU3d9uZguAn5nZJnf/TUNL2FjlPo//A4y5+8Vm9rvAp4HzGlm4JjjH3UsNJvsj4OTc1gt8JvfvTFbu8/gfIAn8SeOK03TlPo9fAKvcfY+Z/RHZJHfZvw/VKFpQbgqTtwJ/V+IUB46y7LzCryX7P8KBBhWv4WJ8HqcCdwO4++PACWb2+gYVrxVdBPx9bgGzHwGdZvaGZheqWdz9OXf/MbC/2WVpBe7+Q3ffk9v9EbCo0jUKFK3pFuCDZEetR/kU8GZgJ/AQ0Ofupc6dCW6h/OfxAHAJgJmtJDslQcU//jbmwHfNbDQ3g3Kx44BnQvs7csdmqkqfx2xTzefxPuBfK91QTU8txszeBjzn7qNmdnaJ0y4AxoBzgd8Bvmdm9xWt8TEjxPw8/hoYNrMxsoHzp8zgGhbw+7nVII8h+9/+cXe/N/R61Ao2M7kffKXPY7aJ9XmY2TlkA8UfVLqhahSt5/eBNWb2JHAbcK6ZfaXonPcC/5xrWniCbJvj7za2mA1T8fNw91+7+3tzqyVeCSwg+5nMSO6+M/fvc2RnWV5ZdMoO4PjQ/iKytc8ZKcbnMavE+TzMbDnZptyL3H13pXsqULQYd1/v7ovc/QTgMuDf3P2KotOeJpeszbXFnwL8d0ML2iBxPg8z6zSz1+R2/wK4dybWrgDM7EgzOyr4GTgfeLjotC3AlbneT28B9rr7rxpc1IaI+XnMGnE+DzNbDPwz8G53/88491XTU5sws2sA3P2zwEeAL5rZQ2SbGW4o08NhRir6PN4M/L2ZTQCPkq1Oz1SvB27PrY98CPBVd/920efxL8AfA08AL5Otgc5UFT8PM1sIbCfbYy5jZv3AqTP0y0Scv48NwNHA3+bOO1BpVllN4SEiImWp6UlERMpSoBARkbIUKEREpCwFChERKUuBQkREylKgEKkhMzvczL5vZnNKvP4aM7vXzNQ1XdqGAoVIbf052VHzE1Ev5mb4vRt4Z0NLJTINChQitXU5cIeZvdbM7jazn+TWBrgodM43cueJtAUNuBOpkdw0Ik+7+8Jc09IR7v5rM5tPdjrnk93dc81Sz7r7gqYWWCQmtZOK1M58YDz3swEfN7OzyE6PfhzZ6RWedfcJM/uNmR3l7i80p6gi8SlQiNTOPmBu7ufLyc5i2+Pu+3Oz384NnXsY8EpjiycyNcpRiNRIbtWwOWY2F5hHdh2N/bl5/98YnGdmRwO73F0rrklbUI1CpLa+S3YhmE3AnbnF7ceAx0PnnEN2hleRtqBktkgNmdnvAQPu/u4y5/wzsN7df9a4kolMnZqeRGrI3X8K/Hu5AXfANxQkpJ2oRiEiImWpRiEiImUpUIiISFkKFCIiUpYChYiIlKVAISIiZf1/rsRdedVjDdEAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "pl.scatter(mod_test_data[\"(a)\"], mod_test_data[\"(b|a)\"],\n",
    "           marker=\"x\", color=\"k\", label=\"data points\")\n",
    "pl.scatter(mod_test_data[\"(a)\"].loc[[50]],\n",
    "            mod_test_data[\"(b|a)\"].loc[[50]],\n",
    "            marker=\"x\", color=\"r\", label=\"data point 50\")\n",
    "pl.xlabel(\"(a)\")\n",
    "pl.ylabel(\"(b|a)\")\n",
    "pl.legend()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The unlikelyness of the combination of the values of '(a)' and '(b|a)' is attributed to '(b|a)' along the causal directions. If '(a)' causes '(b|a)' and the value of '(a)' is common by itself, then the unusual behavior must have occured in the generation of '(b|a)'."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For further details about the ``RankEstimator`` see the [corresponding section](../02_objectives/05_rank_estimator.ipynb) in the core-documentation."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Internally the ``RankEstimator`` uses the ``ProbabilityEstimator`` which is explained in the [next section](./02-08-probability_estimation.ipynb)."
   ]
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.8.8"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}