Empirical bayes - Chapter 5.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "import pandas as pd\n",
    "import scipy\n",
    "import numpy as np\n",
    "import scipy.stats as st\n",
    "import matplotlib.pyplot as plt\n",
    "import seaborn as sns\n",
    "sns.set(style=\"darkgrid\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Preparing the dataset\n",
    "\n",
    "Here, we will prepare the dataset with the data that we created in chapter three. For an explanation, see the notebook for chapter 3 as well as chapter 3 in the book."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "# create batting average dataset (sum over years)\n",
    "batting = pd.read_csv(\"~/baseball/core/Batting.csv\")\n",
    "pitching = pd.read_csv(\"~/baseball/core/Pitching.csv\")\n",
    "# filter pitchers\n",
    "batting_pitchers = batting.playerID.isin(pitching.playerID)\n",
    "batting = batting[~batting_pitchers]\n",
    "# sum over seasons\n",
    "batting = batting.loc[batting[\"AB\"] > 0].groupby(\"playerID\").agg({\"AB\": \"sum\", \"H\": \"sum\"})\n",
    "batting[\"average\"] = batting[\"H\"]/batting[\"AB\"]\n",
    "# add actual player name\n",
    "people = pd.read_csv(\"~/baseball/core/People.csv\")\n",
    "people[\"name\"] = people[\"nameFirst\"] + \" \" + people[\"nameLast\"]\n",
    "batting = batting.merge(people[[\"playerID\", \"name\"]], on=\"playerID\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "# I am using the book values here, not the values that we got in our last notebook. I think the difference comes\n",
    "# from the fact that we are using more data (data from more recent seasons).\n",
    "alpha_0 = 101.4\n",
    "beta_0 = 287.3\n",
    "# add empirical Bayes estimate\n",
    "# this is called a point estimate\n",
    "batting[\"eb_estimate\"] = (batting[\"H\"] + alpha_0) / (batting[\"AB\"] + alpha_0 + beta_0)\n",
    "batting[\"alpha_1\"] = alpha_0 + batting[\"H\"]\n",
    "batting[\"beta_1\"] = beta_0 + batting[\"AB\"] - batting[\"H\"]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Chapter 5\n",
    "\n",
    "Let's look at Hank Aaron:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\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>playerID</th>\n",
       "      <th>AB</th>\n",
       "      <th>H</th>\n",
       "      <th>average</th>\n",
       "      <th>name</th>\n",
       "      <th>eb_estimate</th>\n",
       "      <th>alpha_1</th>\n",
       "      <th>beta_1</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>aaronha01</td>\n",
       "      <td>12364</td>\n",
       "      <td>3771</td>\n",
       "      <td>0.304998</td>\n",
       "      <td>Hank Aaron</td>\n",
       "      <td>0.303653</td>\n",
       "      <td>3872.4</td>\n",
       "      <td>8880.3</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "    playerID     AB     H   average        name  eb_estimate  alpha_1  beta_1\n",
       "0  aaronha01  12364  3771  0.304998  Hank Aaron     0.303653   3872.4  8880.3"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "batting.loc[batting.name==\"Hank Aaron\"]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's plot his probability distribution:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[Text(0, 0.5, 'density'), Text(0.5, 0, 'Batting average')]"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "alpha_aaron = batting.loc[batting.name==\"Hank Aaron\", \"alpha_1\"]\n",
    "beta_aaron = batting.loc[batting.name==\"Hank Aaron\", \"beta_1\"]\n",
    "x = np.linspace(0.25,0.35,1000)\n",
    "pdf_aaron = st.beta.pdf(x,a=alpha_aaron,b=beta_aaron)\n",
    "ax = sns.lineplot(x,pdf_aaron, color=\"green\")\n",
    "plt.axvline(0.3,0,100, dashes=[2, 2], color=\"red\")\n",
    "ax.set(xlabel='Batting average', ylabel='density')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.18493608])"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "st.beta.cdf(0.3,a=alpha_aaron,b=beta_aaron)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [],
   "source": [
    "batting[\"PEP\"] = st.beta.cdf(0.3,a=batting[\"alpha_1\"],b=batting[\"beta_1\"])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can see that most players have a PEP of almost 1:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[Text(0, 0.5, 'density'), Text(0.5, 0, 'PEP')]"
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "ax = batting[\"PEP\"].hist(bins=100, density=True)\n",
    "ax.set(xlabel='PEP', ylabel='density')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 68,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.lines.Line2D at 0x1a22b4b6d8>"
      ]
     },
     "execution_count": 68,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "ax = sns.scatterplot(batting[\"eb_estimate\"], batting[\"PEP\"], hue=batting[\"AB\"])\n",
    "ax.set(xlabel='Batting average', ylabel='PEP')\n",
    "plt.axvline(0.3,0,100, dashes=[2, 2], color=\"green\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [],
   "source": [
    "top_players = batting.sort_values(\"PEP\").head(100)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The false discovery rate (FDR) when including the first 100 players would be 5.93%:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5.9297100246041605"
      ]
     },
     "execution_count": 37,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "top_players.PEP.sum()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.0592971002460416"
      ]
     },
     "execution_count": 38,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "top_players.PEP.mean()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can quickly print some FDRs for other values of included players:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 46,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "10 0.000\n",
      "50 0.002\n",
      "100 0.059\n",
      "150 0.164\n",
      "200 0.272\n"
     ]
    }
   ],
   "source": [
    "for n in [10, 50, 100, 150, 200]:\n",
    "    temp = batting.sort_values(\"PEP\").head(n)\n",
    "    print(n, \"%.3f\" % temp.PEP.mean())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can make this a little bit more formal and automatic: We will sort the whole data frame by PEP and use the `expanding` functionality from Pandas and calculate the cumulative mean of the `PEP`-column. This is called the `q-value`. This q-value can be used to control the false discovery rate (FDR)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {},
   "outputs": [],
   "source": [
    "batting = batting.sort_values(\"PEP\")\n",
    "batting[\"cum_pep_mean\"] = batting[\"PEP\"].expanding(1).mean()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 69,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\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>playerID</th>\n",
       "      <th>AB</th>\n",
       "      <th>H</th>\n",
       "      <th>average</th>\n",
       "      <th>name</th>\n",
       "      <th>eb_estimate</th>\n",
       "      <th>alpha_1</th>\n",
       "      <th>beta_1</th>\n",
       "      <th>PEP</th>\n",
       "      <th>cum_pep_mean</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>4033</th>\n",
       "      <td>hornsro01</td>\n",
       "      <td>8173</td>\n",
       "      <td>2930</td>\n",
       "      <td>0.358497</td>\n",
       "      <td>Rogers Hornsby</td>\n",
       "      <td>0.354065</td>\n",
       "      <td>3031.4</td>\n",
       "      <td>5530.3</td>\n",
       "      <td>3.100352e-27</td>\n",
       "      <td>3.100352e-27</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2159</th>\n",
       "      <td>delahed01</td>\n",
       "      <td>7510</td>\n",
       "      <td>2597</td>\n",
       "      <td>0.345806</td>\n",
       "      <td>Ed Delahanty</td>\n",
       "      <td>0.341626</td>\n",
       "      <td>2698.4</td>\n",
       "      <td>5200.3</td>\n",
       "      <td>7.806414e-16</td>\n",
       "      <td>3.903207e-16</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4514</th>\n",
       "      <td>keelewi01</td>\n",
       "      <td>8591</td>\n",
       "      <td>2932</td>\n",
       "      <td>0.341287</td>\n",
       "      <td>Willie Keeler</td>\n",
       "      <td>0.337806</td>\n",
       "      <td>3033.4</td>\n",
       "      <td>5946.3</td>\n",
       "      <td>5.453547e-15</td>\n",
       "      <td>2.078063e-15</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4216</th>\n",
       "      <td>jacksjo01</td>\n",
       "      <td>4981</td>\n",
       "      <td>1772</td>\n",
       "      <td>0.355752</td>\n",
       "      <td>Shoeless Joe Jackson</td>\n",
       "      <td>0.348884</td>\n",
       "      <td>1873.4</td>\n",
       "      <td>3496.3</td>\n",
       "      <td>6.620512e-15</td>\n",
       "      <td>3.213675e-15</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4827</th>\n",
       "      <td>lajoina01</td>\n",
       "      <td>9590</td>\n",
       "      <td>3243</td>\n",
       "      <td>0.338165</td>\n",
       "      <td>Nap Lajoie</td>\n",
       "      <td>0.335154</td>\n",
       "      <td>3344.4</td>\n",
       "      <td>6634.3</td>\n",
       "      <td>1.717115e-14</td>\n",
       "      <td>6.005171e-15</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3489</th>\n",
       "      <td>gwynnto01</td>\n",
       "      <td>9288</td>\n",
       "      <td>3141</td>\n",
       "      <td>0.338178</td>\n",
       "      <td>Tony Gwynn</td>\n",
       "      <td>0.335073</td>\n",
       "      <td>3242.4</td>\n",
       "      <td>6434.3</td>\n",
       "      <td>4.734851e-14</td>\n",
       "      <td>1.289573e-14</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3754</th>\n",
       "      <td>heilmha01</td>\n",
       "      <td>7787</td>\n",
       "      <td>2660</td>\n",
       "      <td>0.341595</td>\n",
       "      <td>Harry Heilmann</td>\n",
       "      <td>0.337757</td>\n",
       "      <td>2761.4</td>\n",
       "      <td>5414.3</td>\n",
       "      <td>8.901185e-14</td>\n",
       "      <td>2.376946e-14</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3079</th>\n",
       "      <td>gehrilo01</td>\n",
       "      <td>8001</td>\n",
       "      <td>2721</td>\n",
       "      <td>0.340082</td>\n",
       "      <td>Lou Gehrig</td>\n",
       "      <td>0.336413</td>\n",
       "      <td>2822.4</td>\n",
       "      <td>5567.3</td>\n",
       "      <td>3.075781e-13</td>\n",
       "      <td>5.924553e-14</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3559</th>\n",
       "      <td>hamilbi01</td>\n",
       "      <td>6283</td>\n",
       "      <td>2164</td>\n",
       "      <td>0.344421</td>\n",
       "      <td>Billy Hamilton</td>\n",
       "      <td>0.339554</td>\n",
       "      <td>2265.4</td>\n",
       "      <td>4406.3</td>\n",
       "      <td>1.638891e-12</td>\n",
       "      <td>2.347617e-13</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1680</th>\n",
       "      <td>collied01</td>\n",
       "      <td>9949</td>\n",
       "      <td>3315</td>\n",
       "      <td>0.333199</td>\n",
       "      <td>Eddie Collins</td>\n",
       "      <td>0.330480</td>\n",
       "      <td>3416.4</td>\n",
       "      <td>6921.3</td>\n",
       "      <td>1.052787e-11</td>\n",
       "      <td>1.264073e-12</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "       playerID    AB     H   average                  name  eb_estimate  \\\n",
       "4033  hornsro01  8173  2930  0.358497        Rogers Hornsby     0.354065   \n",
       "2159  delahed01  7510  2597  0.345806          Ed Delahanty     0.341626   \n",
       "4514  keelewi01  8591  2932  0.341287         Willie Keeler     0.337806   \n",
       "4216  jacksjo01  4981  1772  0.355752  Shoeless Joe Jackson     0.348884   \n",
       "4827  lajoina01  9590  3243  0.338165            Nap Lajoie     0.335154   \n",
       "3489  gwynnto01  9288  3141  0.338178            Tony Gwynn     0.335073   \n",
       "3754  heilmha01  7787  2660  0.341595        Harry Heilmann     0.337757   \n",
       "3079  gehrilo01  8001  2721  0.340082            Lou Gehrig     0.336413   \n",
       "3559  hamilbi01  6283  2164  0.344421        Billy Hamilton     0.339554   \n",
       "1680  collied01  9949  3315  0.333199         Eddie Collins     0.330480   \n",
       "\n",
       "      alpha_1  beta_1           PEP  cum_pep_mean  \n",
       "4033   3031.4  5530.3  3.100352e-27  3.100352e-27  \n",
       "2159   2698.4  5200.3  7.806414e-16  3.903207e-16  \n",
       "4514   3033.4  5946.3  5.453547e-15  2.078063e-15  \n",
       "4216   1873.4  3496.3  6.620512e-15  3.213675e-15  \n",
       "4827   3344.4  6634.3  1.717115e-14  6.005171e-15  \n",
       "3489   3242.4  6434.3  4.734851e-14  1.289573e-14  \n",
       "3754   2761.4  5414.3  8.901185e-14  2.376946e-14  \n",
       "3079   2822.4  5567.3  3.075781e-13  5.924553e-14  \n",
       "3559   2265.4  4406.3  1.638891e-12  2.347617e-13  \n",
       "1680   3416.4  6921.3  1.052787e-11  1.264073e-12  "
      ]
     },
     "execution_count": 69,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "batting.head(10)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [],
   "source": [
    "hall_of_fame = batting.loc[batting[\"cum_pep_mean\"]< 0.05]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(94, 10)"
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "hall_of_fame.shape"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can plot the number of players for each q-value threshold:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 70,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[Text(0, 0.5, 'Number of included players'), Text(0.5, 0, 'q-value threshold')]"
      ]
     },
     "execution_count": 70,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": "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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "value_list = []\n",
    "for perc in np.linspace(0, 0.3, 200):\n",
    "    value_list.append(batting.loc[batting[\"cum_pep_mean\"]< perc].shape[0])\n",
    "    \n",
    "ax = sns.lineplot(np.linspace(0, 0.3, 200), value_list)\n",
    "ax.set(xlabel='q-value threshold', ylabel='Number of included players')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "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": "3.6.7"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}