Empirical bayes - Chapter 6.ipynb

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    "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": 6,
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    "# 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\")\n",
    "# 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\"]\n",
    "# sort by eb_estimate\n",
    "batting = batting.sort_values(\"eb_estimate\", ascending=False)"
   ]
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       "  <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>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",
       "    </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",
       "    </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",
       "    </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",
       "    </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",
       "    </tr>\n",
       "  </tbody>\n",
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      "text/plain": [
       "       playerID    AB     H   average                  name  eb_estimate  \\\n",
       "4033  hornsro01  8173  2930  0.358497        Rogers Hornsby     0.354065   \n",
       "4216  jacksjo01  4981  1772  0.355752  Shoeless Joe Jackson     0.348884   \n",
       "2159  delahed01  7510  2597  0.345806          Ed Delahanty     0.341626   \n",
       "3559  hamilbi01  6283  2164  0.344421        Billy Hamilton     0.339554   \n",
       "4514  keelewi01  8591  2932  0.341287         Willie Keeler     0.337806   \n",
       "\n",
       "      alpha_1  beta_1  \n",
       "4033   3031.4  5530.3  \n",
       "4216   1873.4  3496.3  \n",
       "2159   2698.4  5200.3  \n",
       "3559   2265.4  4406.3  \n",
       "4514   3033.4  5946.3  "
      ]
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   "source": [
    "batting.head()"
   ]
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   "cell_type": "markdown",
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   "source": [
    "Let's take a look at Mike Piazza and Hank Aaron:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {},
   "outputs": [],
   "source": [
    "aaron = batting.loc[batting[\"name\"] == \"Hank Aaron\"]\n",
    "piazza = batting.loc[batting[\"name\"] == \"Mike Piazza\"]\n",
    "matsui = batting.loc[batting[\"name\"] == \"Hideki Matsui\"]\n",
    "melton = batting.loc[batting[\"name\"] == \"Bill Melton\"]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
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       "      <td>aaronha01</td>\n",
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       "      <td>3771</td>\n",
       "      <td>0.304998</td>\n",
       "      <td>Hank Aaron</td>\n",
       "      <td>0.303653</td>\n",
       "      <td>3872.4</td>\n",
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       "    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"
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       "      <th>6877</th>\n",
       "      <td>piazzmi01</td>\n",
       "      <td>6911</td>\n",
       "      <td>2127</td>\n",
       "      <td>0.30777</td>\n",
       "      <td>Mike Piazza</td>\n",
       "      <td>0.305273</td>\n",
       "      <td>2228.4</td>\n",
       "      <td>5071.3</td>\n",
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       "       playerID    AB     H  average         name  eb_estimate  alpha_1  \\\n",
       "6877  piazzmi01  6911  2127  0.30777  Mike Piazza     0.305273   2228.4   \n",
       "\n",
       "      beta_1  \n",
       "6877  5071.3  "
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       "      <td>Bill Melton</td>\n",
       "      <td>0.25355</td>\n",
       "      <td>1105.4</td>\n",
       "      <td>3254.3</td>\n",
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       "       playerID    AB     H   average         name  eb_estimate  alpha_1  \\\n",
       "5807  meltobi01  3971  1004  0.252833  Bill Melton      0.25355   1105.4   \n",
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "# plot the posterior distributions\n",
    "x = np.linspace(0.25,0.35,1000)\n",
    "\n",
    "for data, name, color in [[aaron, \"Hank Aaron\", \"green\"], \n",
    "                          [piazza, \"Mike Piazza\", \"red\"], \n",
    "                          [matsui, \"Hideki Matsui\", \"blue\"]]:\n",
    "    alpha = data[\"alpha_1\"]\n",
    "    beta = data[\"beta_1\"]\n",
    "    pdf = st.beta.pdf(x,a=alpha,b=beta)\n",
    "    ax = sns.lineplot(x,pdf, color=color, label=name)\n",
    "ax.set(xlabel='Batting average', ylabel='density');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, if we were to ask, who had a better batting average, Hank Aaron or Mike Piazza, there are multiple possibilities of doing that:\n",
    "\n",
    "1. Simulation of posterior draws\n",
    "2. Integration\n",
    "3. Closed-form solution\n",
    "4. Closed-form approximation\n",
    "\n",
    "Let's start with the first. We sample both distributions 1 million times and calculate the ratio that Piazza's draw was larger than Aaron's:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {},
   "outputs": [],
   "source": [
    "# random draw\n",
    "alpha_aaron = batting.loc[batting.name==\"Hank Aaron\", \"alpha_1\"]\n",
    "beta_aaron = batting.loc[batting.name==\"Hank Aaron\", \"beta_1\"]\n",
    "aaron_simulation = st.beta.rvs(alpha_aaron, beta_aaron, size=1000000)\n",
    "alpha_piazza = batting.loc[batting.name==\"Mike Piazza\", \"alpha_1\"]\n",
    "beta_piazza = batting.loc[batting.name==\"Mike Piazza\", \"beta_1\"]\n",
    "piazza_simulation = st.beta.rvs(alpha_piazza, beta_piazza, size=1000000)\n",
    "alpha_melton = batting.loc[batting.name==\"Bill Melton\", \"alpha_1\"]\n",
    "beta_melton = batting.loc[batting.name==\"Bill Melton\", \"beta_1\"]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.59436"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# compare\n",
    "comparison = np.where(piazza_simulation > aaron_simulation, 1, 0)\n",
    "np.mean(comparison)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that this answer for our hypothesis test is quite different from the frequentist outcome. If we were to use a frequentist hypothesis test, our p-value to reject the null hypothesis probably wouldn't be significant and we wouldn't be able to say anything except \"we cannot reject the null hypothesis\" and our test would be inconclusive. Here, we can now say that with a probability of 0.594 Piazza had a higher batting average than Aaron.\n",
    "\n",
    "Now, our next possibility of calculating this is numerical integration:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [],
   "source": [
    "from scipy.integrate import dblquad\n",
    "area = dblquad(lambda x, y: st.beta.pdf(x,a=alpha_aaron.values[0],b=beta_aaron.values[0])*st.beta.pdf(y,a=alpha_piazza.values[0],b=beta_piazza.values[0]), 0, 1, lambda x: x, lambda x: 1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.5943243597039876"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "1 - area[0] #since I switched around the order"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Next, we can use an approximation: For large alpha and beta, the beta distribution converges to the normal distribution. We are working with decently large numbers, so the approximation is pretty good. We can now calculate the probability that one normal variable is larger than another, which is a very easy calculation:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "metadata": {},
   "outputs": [],
   "source": [
    "def credile_interval_approximation(alpha_1, beta_1, alpha_2, beta_2):\n",
    "    \"\"\"Calculate the probability \n",
    "    \n",
    "    Parameters\n",
    "    ----------\n",
    "    alpha_1: float\n",
    "        Alpha parameter of the beta posterior distribution of player 1/test A\n",
    "    beta_1: float\n",
    "        Beta parameter of the beta posterior distribution of player 1/test A\n",
    "    alpha_2: float\n",
    "         Alpha parameter of the beta posterior distribution of player 2/test B \n",
    "    beta_2: float\n",
    "         Beta parameter of the beta posterior distribution of player 2/test B\n",
    "       \n",
    "    Returns\n",
    "    -------\n",
    "    float\n",
    "        Probability that test A has a higher probability than B\n",
    "    float\n",
    "        Difference in posterior average (mu_2 - mu_1)\n",
    "    float\n",
    "        Lower boundary of 95% credible interval\n",
    "    float\n",
    "        Upper boundary of 95% credible interval\n",
    "    \"\"\"\n",
    "    mu_1 = alpha_1 / (alpha_1 + beta_1)\n",
    "    mu_2 = alpha_2 / (alpha_2 + beta_2)\n",
    "    variance_1 = alpha_1 * beta_1 / ((alpha_1 + beta_1)**2 * (alpha_1 + beta_1 + 1))\n",
    "    variance_2 = alpha_2 * beta_2 / ((alpha_2 + beta_2)**2 * (alpha_2 + beta_2 + 1))\n",
    "    mu_diff = mu_2 - mu_1\n",
    "    sd_diff = np.sqrt(variance_1 + variance_2)\n",
    "    \n",
    "    return st.norm(mu_diff, sd_diff).cdf(0), mu_diff, st.norm(mu_diff, sd_diff).ppf(0.025), st.norm(mu_diff, sd_diff).ppf(0.975)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.00675490900223421\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "(0.5947371612228725,\n",
       " -0.0016194754331937955,\n",
       " -0.014858853796418242,\n",
       " 0.011619902930030648)"
      ]
     },
     "execution_count": 53,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "credile_interval_approximation(alpha_piazza.values[0], beta_piazza.values[0], alpha_aaron.values[0], beta_aaron.values[0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 76,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.008511829366870331\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "(0.999999999386339,\n",
       " -0.05172326326881427,\n",
       " -0.06840614227043049,\n",
       " -0.03504038426719805)"
      ]
     },
     "execution_count": 76,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "credile_interval_approximation(alpha_piazza.values[0], beta_piazza.values[0], alpha_melton.values[0], beta_melton.values[0])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A normal chi squared hypothesis test yields:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 82,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(20.598817323604425,\n",
       " 5.663108815999413e-06,\n",
       " 1,\n",
       " array([[1111.59102262, 3863.40897738],\n",
       "        [2019.40897738, 7018.59102262]]))"
      ]
     },
     "execution_count": 82,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "observed = np.array([[1004, 3971], [2127, 6911]])\n",
    "st.chi2_contingency(observed)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If we wanted to calculate the confidence interval like R does:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 89,
   "metadata": {},
   "outputs": [],
   "source": [
    "def diffprop(contingency_table):\n",
    "    \"\"\"\n",
    "    Parameters\n",
    "    ----------\n",
    "    contingency_table: np.array\n",
    "        must be a 2x2 numpy array \n",
    "\n",
    "    Returns\n",
    "    -------\n",
    "    delta: float\n",
    "        The difference in proportions\n",
    "    ci: (float, float)\n",
    "        The Wald 95% confidence interval for delta\n",
    "    corrected_ci: (float, float)\n",
    "        Yates continuity correction for the 95% confidence interval of delta.\n",
    "    \"\"\"\n",
    "    n1, n2 = contingency_table[0,1], contingency_table[1,1]\n",
    "    prop1 = contingency_table[0,0] / n1\n",
    "    prop2 = contingency_table[1,0] / n2\n",
    "    delta = prop1 - prop2\n",
    "\n",
    "    # Wald 95% confidence interval for delta\n",
    "    se = np.sqrt(prop1*(1 - prop1)/n1 + prop2*(1 - prop2)/n2)\n",
    "    ci = (delta - 1.96*se, delta + 1.96*se)\n",
    "\n",
    "    # Yates continuity correction for confidence interval of delta\n",
    "    correction = 0.5*(1/n1 + 1/n2)\n",
    "    corrected_ci = (ci[0] - correction, ci[1] + correction)\n",
    "\n",
    "    return delta, ci, corrected_ci"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 90,
   "metadata": {},
   "outputs": [],
   "source": [
    "delta, ci, corrected_ci = diffprop(observed)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 91,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "-0.05493718184955526"
      ]
     },
     "execution_count": 91,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "delta"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 92,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(-0.07229169032276282, -0.0375826733763477)"
      ]
     },
     "execution_count": 92,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ci"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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