COMPAS_debias_experiments.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from sklearn import linear_model\n",
    "import pandas as pd\n",
    "import numpy as np\n",
    "import math\n",
    "import random\n",
    "import matplotlib.pyplot as plt\n",
    "import warnings\n",
    "warnings.filterwarnings(\"ignore\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# The Propublica/COMPAS Recidivism Data Set\n",
    "## Loading the Data\n",
    "Below, `allowed_factors` shows all of the data that the Propublica data set contains with 7214 people and 53 features. \n",
    "\n",
    "I have commented out the factors that are not relevant for predicting recidivism, not useful for debiasing, or not useful for measuring accuracy. \n",
    "\n",
    "If it is unclear what a variable is, I put a comment indicating what I think it is."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "(#people, #features) = (7214, 53)\n"
     ]
    }
   ],
   "source": [
    "raw_data = pd.read_csv(\"./compas-scores-two-years.csv\")\n",
    "print('(#people, #features) =', raw_data.shape)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "allowed_factors = [\n",
    "    #'id',\n",
    "    #'name',\n",
    "    #'first',\n",
    "    #'last',\n",
    "    #'compas_screening_date',\n",
    "    #'sex',\n",
    "    #'dob',\n",
    "    #'age_cat',\n",
    "    'race',\n",
    "    'decile_score', #COMPAS score\n",
    "    'age',\n",
    "    'juv_fel_count', \n",
    "    'juv_misd_count',\n",
    "    'juv_other_count',\n",
    "    'priors_count',\n",
    "    #'days_b_screening_arrest', propublica uses this to filter data\n",
    "    #  'c_jail_in', date\n",
    "    #  'c_jail_out', date\n",
    "    #  'c_case_number',\n",
    "    #  'c_offense_date',\n",
    "    #  'c_arrest_date',\n",
    "    #  'c_days_from_compas',\n",
    "    'c_charge_degree', #Misdemeanor or felony for original charge, need to make categorical\n",
    "    #'c_charge_desc',\n",
    "    'is_recid', \n",
    "    #'r_case_number',\n",
    "    'r_charge_degree',#Misdemeanor or felony of re-offense, need to make categorical\n",
    "    #  'r_days_from_arrest',\n",
    "    #  'r_offense_date',\n",
    "    #  'r_charge_desc',\n",
    "    #  'r_jail_in',\n",
    "    #  'r_jail_out',\n",
    "    #'violent_recid', #all are NAN\n",
    "    'is_violent_recid', #hey we can use this for groups!\n",
    "    #  'vr_case_number',\n",
    "    'vr_charge_degree', #maybe remove this from r_charge degree\n",
    "    #  'vr_offense_date',\n",
    "    #  'vr_charge_desc', #weird that vr_charge_desc and r_charge_desc do not overlap\n",
    "    #  'type_of_assessment',\n",
    "    #  'decile_score.1',#same as decile score?\n",
    "    'score_text',#low, medium, high risk based on COMPAS\n",
    "    #  'screening_date',\n",
    "    #  'v_type_of_assessment',\n",
    "    #  'v_decile_score',\n",
    "    #  'v_score_text', #low, medium high\n",
    "    #  'v_screening_date',\n",
    "    #  'in_custody',\n",
    "    #  'out_custody',\n",
    "    #'priors_count.1', #not sure what the dif is from priors_count. it is the same\n",
    "    #  'start',\n",
    "    #  'end',\n",
    "    # 'event', #no clue what this means?!?\n",
    "    #'two_year_recid' # same as is_recid\n",
    "] "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "predictive_features = [ # comment or uncomment any features to use in logistic regression training\n",
    "    'age',\n",
    "    'juv_fel_count',\n",
    "    'juv_misd_count',\n",
    "    'juv_other_count',\n",
    "    'priors_count',\n",
    "    #'c_charge_degree' #<-- not included in COMPAS algorithm\n",
    "    ]\n",
    "\n",
    "protected_features = ['race']"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Filter the data just as Propublica does [here](https://github.com/propublica/compas-analysis/blob/master/Compas%20Analysis.ipynb) and only include African Americans and Caucasians, leaving $\\sim$ 5000 people."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "the dimensions of final_data are (5278, 13)\n"
     ]
    }
   ],
   "source": [
    "filtered_data = raw_data.loc[(raw_data['is_recid'] != -1) & \n",
    "                             (raw_data['days_b_screening_arrest'] <=30) &\n",
    "                             (raw_data['days_b_screening_arrest'] >=-30) &\n",
    "                             (raw_data['c_charge_degree'] != 'O') &\n",
    "                             (raw_data['score_text'] != 'N/A') &\n",
    "                             (raw_data['race'].isin(['African-American', 'Caucasian']))\n",
    "                            ]\n",
    "final_data = filtered_data[allowed_factors]\n",
    "# reset index skewed by removing rows\n",
    "final_data = final_data.reset_index()\n",
    "del final_data[\"index\"]\n",
    "print('the dimensions of final_data are', final_data.shape)\n",
    "final_data[['r_charge_degree']] = final_data[['r_charge_degree']].fillna(0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Change current charge and race into 0/1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def coding(col, code_dict):\n",
    "    '''\n",
    "    col: column of df to change encoding\n",
    "    code_dict: {'key':value,...}\n",
    "    '''\n",
    "    col_coded = pd.Series(col, copy=True)\n",
    "    for key, value in code_dict.items():\n",
    "        col_coded.replace(key, value, inplace=True)\n",
    "    return col_coded"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
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       "    .dataframe tbody tr th {\n",
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       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>race</th>\n",
       "      <th>decile_score</th>\n",
       "      <th>age</th>\n",
       "      <th>juv_fel_count</th>\n",
       "      <th>juv_misd_count</th>\n",
       "      <th>juv_other_count</th>\n",
       "      <th>priors_count</th>\n",
       "      <th>c_charge_degree</th>\n",
       "      <th>is_recid</th>\n",
       "      <th>r_charge_degree</th>\n",
       "      <th>is_violent_recid</th>\n",
       "      <th>vr_charge_degree</th>\n",
       "      <th>score_text</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
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       "      <td>1</td>\n",
       "      <td>(F3)</td>\n",
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       "      <td>(F3)</td>\n",
       "      <td>Low</td>\n",
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       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>(M1)</td>\n",
       "      <td>0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>Low</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>0</td>\n",
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       "      <td>0</td>\n",
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       "      <td>0</td>\n",
       "      <td>14</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>(F2)</td>\n",
       "      <td>0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>Medium</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>39</td>\n",
       "      <td>0</td>\n",
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       "      <td>0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>Low</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   race  decile_score  age  juv_fel_count  juv_misd_count  juv_other_count  \\\n",
       "0     1             3   34              0               0                0   \n",
       "1     1             4   24              0               0                1   \n",
       "2     0             6   41              0               0                0   \n",
       "3     0             1   39              0               0                0   \n",
       "4     0             4   27              0               0                0   \n",
       "\n",
       "   priors_count  c_charge_degree  is_recid r_charge_degree  is_violent_recid  \\\n",
       "0             0                1         1            (F3)                 1   \n",
       "1             4                1         1            (M1)                 0   \n",
       "2            14                1         1            (F2)                 0   \n",
       "3             0                0         0               0                 0   \n",
       "4             0                1         0               0                 0   \n",
       "\n",
       "  vr_charge_degree score_text  \n",
       "0             (F3)        Low  \n",
       "1              NaN        Low  \n",
       "2              NaN     Medium  \n",
       "3              NaN        Low  \n",
       "4              NaN        Low  "
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# for c_charge_degree, change M(misdomeaner)/F(felony) to 0/1\n",
    "final_data[\"c_charge_degree\"] = coding(final_data[\"c_charge_degree\"], {'M':0,'F':1})\n",
    "\n",
    "# for race, change Caucasian/African-American to 0/1\n",
    "final_data[\"race\"] = coding(final_data[\"race\"], {'Caucasian':0, 'African-American':1})\n",
    "\n",
    "final_data.head(5)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Now we need to shuffle the data and split it into 3 pieces:\n",
    "<ol>\n",
    "  <li>A debiasing train set we use to debias the data.</li>\n",
    "  <li>A training set to train logistic regression.</li>\n",
    "  <li>A test set to measure false positive rates, false negative rates, etc.</li>\n",
    "</ol>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from sklearn.model_selection import train_test_split\n",
    "def split_data(df, debias_train_size, standard_training_set_percentage):\n",
    "    '''\n",
    "    Splits data into testing, training, and debias estimator sets\n",
    "    \n",
    "    Args:\n",
    "        df: dataframe\n",
    "        debias_train_size: at least 10 * number of feautes\n",
    "        standard_training_set_percentage: percentage split for train/test of df\\debias_train_df\n",
    "        \n",
    "    Returns: \n",
    "        debias_train_df\n",
    "        train_df\n",
    "        test_df\n",
    "    '''\n",
    "    debias_train_index = random.sample(range(df.shape[0]), debias_train_size)\n",
    "    debias_train_df = df.loc[debias_train_index,:]\n",
    "    \n",
    "    df = df.drop(debias_train_index)\n",
    "    \n",
    "    train_df, test_df = train_test_split(df, train_size=standard_training_set_percentage, shuffle=True)\n",
    "    \n",
    "    test_df = test_df.reset_index()\n",
    "    del test_df[\"index\"]\n",
    "    \n",
    "    train_df = train_df.reset_index()\n",
    "    del train_df[\"index\"]\n",
    "    \n",
    "    debias_train_df = debias_train_df.reset_index()\n",
    "    del debias_train_df[\"index\"]\n",
    "    \n",
    "    return debias_train_df, train_df, test_df"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Debiasing Features"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We assume that the data $Y = X A^T + Z B^T + E$, where $X$ is protected attributes like gender, $Z$ are unknown but are \"discriminative\" attributes and $E$ is an error term. \n",
    "\n",
    "We partition the rows of $Y$ into multiple \"homogeneous groups\": \n",
    "<ol>\n",
    "  <li>Consider those who have been charged with a **non-violent** crime within 2 years. We split this set into multiple groups depending on whether or not someone has commmitted a $n$th degree felony or misdemeanor. </li>\n",
    "  <li>Consider those who have been charged with a **violent** crime within 2 years. We split this set into multiple groups depending on whether or not someone has commmitted a $n$th degree felony or misdemeanor. </li>\n",
    "  <li>Those who have **not** committed a crime after 2 years.</li>\n",
    "</ol>\n",
    "\n",
    "We assume that when the text of `vr_charge_degree` and `r_charge_degree` do not match for a person, it is due to one incident resulting in different violent and non-violent charges. In the case when someone recidivates and this incident results in a non-violent and violent charge, we only consider the violent charge.\n",
    "\n",
    "To debias the data, we use least squares to estimate $A$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# encode race as dummmy variables for OLS estimate\n",
    "def race_dummy_encoding(df_race, white_black_ecoding):\n",
    "    white = white_black_ecoding[0]\n",
    "    black = white_black_ecoding[1]\n",
    "    white_dummy_encoding = [1 if x == white else 0 for x in df_race]\n",
    "    black_dummy_encoding = [1 if x == black else 0 for x in df_race]\n",
    "    return pd.DataFrame({\"white\":white_dummy_encoding, \"black\":black_dummy_encoding})"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def recid_no_recid_index(is_resid, group_elements):\n",
    "    group_indices = []\n",
    "    for i, element in enumerate (group_elements):\n",
    "        group_indices.append(np.where(col == element))\n",
    "        group_indices[i] = list(group_indices[i][0])\n",
    "    \n",
    "    return group_indices\n",
    "\n",
    "def violent_nonviolent_norecid_index(is_violent_recid, is_recid):\n",
    "    group_indices = [None]*3\n",
    "    violent_index = np.where(is_violent_recid == 1)[0]\n",
    "    group_indices[0] = violent_index\n",
    "\n",
    "    is_recid_dropped = is_recid.drop(violent_index, inplace=False)\n",
    "    \n",
    "    group_indices[1] = is_recid_dropped[is_recid_dropped == 1].index\n",
    "    \n",
    "    group_indices[2] = is_recid_dropped[is_recid_dropped == 0].index\n",
    "    \n",
    "    return group_indices\n",
    "\n",
    "def all_cat_index(vr_charge_degree, r_charge_degree): # <--current one in use\n",
    "\n",
    "    group_dict = {}\n",
    "    \n",
    "    # ['(F3)', '(M1)', '(F2)', '(F1)', '(F6)', '(F7)', '(M2)']\n",
    "    vr_charge = vr_charge_degree.dropna().unique()\n",
    "    \n",
    "    for charge in vr_charge:\n",
    "        group_dict[charge] = np.where(vr_charge_degree == charge)[0]\n",
    "        \n",
    "    # don't double count those already indexed by a violent crime\n",
    "    drop = np.concatenate(list(group_dict.values()))\n",
    "    r_charge_degree_dropped = r_charge_degree.drop(drop, inplace=False)\n",
    "    \n",
    "    # ['(M1)', 0 (no recid), '(M2)', '(F2)', '(F3)', '(F6)', '(F1)', '(F7)', '(MO3)']\n",
    "    r_charge = r_charge_degree.unique()\n",
    "\n",
    "    for charge in r_charge:\n",
    "        group_dict[(charge, 'nonV')] = r_charge_degree_dropped[r_charge_degree_dropped ==charge].index\n",
    "\n",
    "    return list(group_dict.values())\n",
    "    "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def center_by_homogeneous_subgroups(group_indices, Y):\n",
    "    '''\n",
    "    Args:\n",
    "        Y: dataframe to debias\n",
    "        group_indices: array whose elements are list of indices for the homogeneous subgroups\n",
    "    Returns:\n",
    "        centered Y\n",
    "    '''\n",
    "    G = pd.DataFrame(columns=list(Y))\n",
    "    for g in  group_indices:\n",
    "        G = G.append(Y.loc[g] - Y.loc[g].mean(0)) \n",
    "    centered_Y = G.sort_index(0)\n",
    "        \n",
    "    return centered_Y"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def estimate_A(centered_Y_train, X, fit_intercept):\n",
    "    ols = linear_model.LinearRegression(fit_intercept)\n",
    "    ols.fit(X, centered_Y_train)\n",
    "    A_hat = ols.coef_\n",
    "    b = ols.intercept_\n",
    "    return A_hat, b"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def biased_debiased_probs(data, debias_train_size, train_percentage_split):\n",
    "    # split into testing/training/debias estimate sets\n",
    "    debias_train_df, train_df, test_df = split_data(data, debias_train_size, train_percentage_split)\n",
    "\n",
    "    # encode dummy race variables for debias training set\n",
    "    X_dummy_encoding = race_dummy_encoding(debias_train_df[protected_features].race, [0,1])\n",
    "\n",
    "    # Y for debias training\n",
    "    Y = debias_train_df[predictive_features]\n",
    "\n",
    "    # get group indices\n",
    "    group_indices = all_cat_index(debias_train_df.vr_charge_degree, debias_train_df.r_charge_degree)\n",
    "    # center by group indices\n",
    "    centered_Y_train = center_by_homogeneous_subgroups(group_indices, Y)\n",
    "\n",
    "    # get estimate of A\n",
    "    A_hat, b = estimate_A(centered_Y_train, X_dummy_encoding, True)\n",
    "    \n",
    "    # ecode dummy race varialbes for training and testing set\n",
    "    train_race_dummy_encoding = race_dummy_encoding(train_df.race, [0,1])\n",
    "    test_race_dummy_encoding = race_dummy_encoding(test_df.race, [0,1])\n",
    "    \n",
    "    train = train_df[predictive_features]\n",
    "    train_debiased = train - np.matmul(train_race_dummy_encoding.values,A_hat.T)-b.T\n",
    "\n",
    "    test = test_df[predictive_features]\n",
    "    test_debiased = test - np.matmul(test_race_dummy_encoding.values,A_hat.T)-b.T\n",
    "    \n",
    "    logit = linear_model.LogisticRegression()\n",
    "    logit_debiased = linear_model.LogisticRegression()\n",
    "    logit.fit(train, train_df.is_recid)\n",
    "    logit_debiased.fit(train_debiased, train_df.is_recid)\n",
    "\n",
    "    probs = list(logit.predict_proba(test)[:,1])\n",
    "    probs_debiased = list(logit_debiased.predict_proba(test_debiased)[:,1])\n",
    "\n",
    "    return probs, probs_debiased, test_df.race, test_df.decile_score, test_df.is_recid"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def FPR(prediction, recid_label):\n",
    "    '''\n",
    "    false positive/(false positive + true negative)\n",
    "    '''\n",
    "    FP = sum([1 for x in range(len(prediction)) if prediction[x] == 1 and recid_label[x] ==0])\n",
    "    TN = sum([1 for x in range(len(prediction)) if prediction[x] == 0 and recid_label[x] ==0])\n",
    "    return FP/(FP+TN)\n",
    "\n",
    "def FNR(prediction, recid_label):\n",
    "    '''\n",
    "    false negative/(false negative + true positive)\n",
    "    '''\n",
    "    FN = sum([1 for x in range(len(prediction)) if prediction[x] == 0 and recid_label[x] ==1])\n",
    "    TP = sum([1 for x in range(len(prediction)) if prediction[x] == 1 and recid_label[x] ==1])\n",
    "    return FN/(FN+TP)\n",
    "\n",
    "def FPR_FNR_accuracy(recid_label, race, score, threshold=.5):\n",
    "    white_index = np.where(race == 0)[0]\n",
    "    black_index = np.where(race == 1)[0]\n",
    "    \n",
    "    prediction = np.array([1 if x >= threshold else 0 for x in score])\n",
    "    \n",
    "    pop_FPR = FPR(prediction, recid_label)\n",
    "    pop_FNR = FNR(prediction, recid_label)\n",
    "    \n",
    "    white_FPR = FPR([prediction[i] for i in white_index], [recid_label[i] for i in white_index])\n",
    "    white_FNR = FNR([prediction[i] for i in white_index], [recid_label[i] for i in white_index])\n",
    "    \n",
    "    black_FPR = FPR([prediction[i] for i in black_index], [recid_label[i] for i in black_index])\n",
    "    black_FNR = FNR([prediction[i] for i in black_index], [recid_label[i] for i in black_index])\n",
    "    \n",
    "    accuracy = sum(prediction == recid_label)/len(recid_label)\n",
    "    \n",
    "    return [pop_FPR, pop_FNR, white_FPR, white_FNR, black_FPR, black_FNR], accuracy"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Logistic Regression as a proxy to COMPAS"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We use logistic regression."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def get_logistic_threshold(scores, logistic_quantile=50):\n",
    "    return np.percentile(scores, logistic_quantile, interpolation='midpoint')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def plot_prob_hist(probs, race):\n",
    "    print('the avg is', np.mean(probs))\n",
    "    print('the median is', np.median(probs))\n",
    "    \n",
    "    probabilities_AA = [probs[i] for i in np.where(race == 1)[0]]\n",
    "    probabilities_white = [probs[i] for i in np.where(race == 0)[0]]\n",
    "    \n",
    "    plt.hist(probs, bins = 25)\n",
    "    plt.title(\"Histogram of Probabilities of Recidivism from Logistic Regression on Test Set\")\n",
    "    plt.xlabel(\"probability from logistic regression\")\n",
    "    plt.ylabel(\"Number of people\")\n",
    "    plt.ylim((0,200))\n",
    "    plt.show()\n",
    "\n",
    "    print('Notice the difference in scale on the y-axis!')\n",
    "    probabilities_AA = [probs[i] for i in np.where(race == 1)[0]]\n",
    "    probabilities_white = [probs[i] for i in np.where(race == 0)[0]]\n",
    "    \n",
    "    ax1 = plt.subplot(121)\n",
    "    ax1.set_ylim([0,140])\n",
    "    \n",
    "    plt.subplot(1,2,1)\n",
    "    plt.hist(probabilities_AA, bins = 25)\n",
    "    plt.title(\"Histogram of Probabilities \\n of Recidivism for African-Americans \\n under LR \\n on Test Set\")\n",
    "    plt.xlabel(\"probability from logistic regression\")\n",
    "    plt.ylabel(\"Number of African-Americans\")\n",
    "\n",
    "    plt.subplot(1,2,2)\n",
    "    plt.hist(probabilities_white, bins = 25)\n",
    "    plt.title(\"Histogram of Probabilities \\n of Recidivism for Caucasians \\n under LR \\n on Test Set\")\n",
    "\n",
    "    plt.xlabel(\"probability from logistic regression\")\n",
    "    plt.ylabel(\"Number of Caucasians\")\n",
    "    plt.tight_layout()\n",
    "    plt.ylim((0,140))\n",
    "    plt.show()    \n",
    "    \n",
    "    "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Quality of logistic regression trained on biased or debiased data & comparison to COMPAS\n",
    "\n",
    "First we look at the distribution of probabilities assigned by logistic regression on the whole population and on both the white and black population separately for one split on both the biased and debiased data set."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "the avg is 0.50438626285\n",
      "the median is 0.483707215381\n"
     ]
    },
    {
     "data": {
      "image/png": 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TGAMgaVHgCeBS4ADgpIj4ycLOw8zMbKBaJfm+GhE/a9K0twceiohHJTVpFmZm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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x1093ba7b8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Notice the difference in scale on the y-axis!\n"
     ]
    },
    {
     "data": {
      "image/png": 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zABFxB7D+QDNJOlnSM5LuLgybV9JVkh7Mf+fJwyXpGEkPSbpT0moly2jW6fYG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\nbAfkmU4AAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x108067b70>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "the avg is 0.504245925196\n",
      "the median is 0.482295001957\n"
     ]
    },
    {
     "data": {
      "image/png": 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InyzsMszMzPqrXZLvKxHxsxbNe3vggYh4WFKLFmFmZta8dvmp0cmSjpG0paRN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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10967df60>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Notice the difference in scale on the y-axis!\n"
     ]
    },
    {
     "data": {
      "image/png": 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AEXEXsMFAM0k6TdJzku4tDJtP0jWSHs5/583DJek4SY9IulvS6iXLaNbt9gPW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\n/wFfqYS65McKgQAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x109a2bac8>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "debias_train_size = 1000\n",
    "train_percentage_split = .6\n",
    "\n",
    "probs, probs_debiased, race, compas_decile_score, true_label = biased_debiased_probs(final_data, debias_train_size,  \n",
    "                                                                                     train_percentage_split)\n",
    "plot_prob_hist(probs, race)\n",
    "plot_prob_hist(probs_debiased, race)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def logistic_histogram_before_after_AA(probs, probs_debiased, race):\n",
    "    probabilities_AA = [probs[i] for i in np.where(race == 1)[0]]\n",
    "    probabilities_AA_debias = [probs_debiased[i] for i in np.where(race == 1)[0]]\n",
    "    \n",
    "    #plt.subplot(1,2,1)\n",
    "    plt.hist(probabilities_AA, bins = 25)\n",
    "    plt.title(\"Histogram of Raw Logistic Regression \\n Scores of African-Americans\")\n",
    "    plt.xlabel(\"probability from logistic regression\")\n",
    "    plt.ylabel(\"Number of African-Americans\")\n",
    "\n",
    "    #plt.subplot(1,2,2)\n",
    "    plt.show()\n",
    "    plt.hist(probabilities_AA_debias, bins = 25)\n",
    "    plt.title(\"Histogram of Debiased Logistic Regression \\n Scores of African-Americans\")\n",
    "\n",
    "    plt.xlabel(\"probability from logistic regression\")\n",
    "    plt.ylabel(\"Number of African-Americans\")\n",
    "    plt.tight_layout()\n",
    "    plt.show()\n",
    "\n",
    "\n",
    "    ax1 = plt.subplot(121)\n",
    "    ax1.set_ylim([0,140])\n",
    "    \n",
    "    plt.subplot(1,2,1)\n",
    "    plt.hist(probabilities_AA, bins = 25)\n",
    "    plt.title(\"Raw\")\n",
    "    plt.xlabel(\"probability from logistic regression\")\n",
    "    plt.ylabel(\"Number of African-Americans\")\n",
    "\n",
    "    plt.subplot(1,2,2)\n",
    "    #plt.show()\n",
    "    plt.hist(probabilities_AA_debias, bins = 25)\n",
    "    plt.title(\"Debiased\")\n",
    "\n",
    "    plt.xlabel(\"probability from logistic regression\")\n",
    "    plt.ylabel(\"Number of African-Americans\")\n",
    "    plt.tight_layout()\n",
    "    plt.savefig('histo.pdf', format='pdf', dpi=1000)\n",
    "    plt.ylim((0,140))\n",
    "    plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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lje6jeMzMrE1U6aReDrhf0q3Aqx0TI2J8T3YsaSgwBXgqIraTtDJwLrAUcAew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\ntHCet22DbVwNfK0Q99j8d9W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      "text/plain": [
       "<matplotlib.figure.Figure at 0x1093e0710>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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gNeD4qnnusNDMzJqq3nNQFwEXSfpuRBzTizGZmZk1riQREcdI2hXYOk+aEhFX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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10c318240>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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AiUO8dxNgSf65cZ7euFN1DVr+ENLJ/Mq/s/z5OwLbA4uHef3NwI9I1xq9Ariq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\nrkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10c4cdcc0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "logistic_histogram_before_after_AA(probs, probs_debiased, race)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we look at ROC also under the same scenarios above."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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zofyNKVdJifLl6t3c8e4yikuUmIZRXDywHZcMbMtQG00laFiCqgWy8gq4+e2l\nLNqyj0mnd+aB83pR15r0jDlKTn4h8zdm8O26vczbkE567mH6tWvKpDM6c0G/NtYMHoQsQYW4LekH\nuPGNJHbuP8TTlw/g0sHtA10lY4JGTn4hT365jnVpuSTvyKKoRGlcP5Ize7RkRM9WXDigDfUiIwJd\nTVMBS1AhbP6mDG55eymREXWYdtNQhsS1CHSVjAkay3dkcc/7yWzJOMiA9s2YfGYXRvRqxaAOzexs\nKURYggpR0xZv45FPV9O1ZUNeuX4IHVrYFOzGHCoo5rPlu5i2eBvLU7Np0TCKdyYNI7GrXVcKRZag\nQkxxifKXz9fw2vwUhvdsyb/GD7Jp2I0BFm/JZNzURQB0b9WIP18UzyWD2tE02uIjVFmCCiG5+YXc\n8e4y5q1PZ+JpcTx0fm9rqjAGSM89zO3vLqNFwygeOr83/3dyO5uTqRawBBUiduzL48Y3lrA5/SB/\n/VVfrh7aKdBVMiYoFBaXcM/7yeQcKmTGbafRu02Tql9kQoIlqBCQlLKPyW8tpai4hDdvOMWmyDDG\nNXfdXh77fA1b0g/y90v7WXKqZSxBBbmPf07lwY9W0q55NC9fn0DXlo0CXSVjAm7T3lwem7WW7zak\n0zm2Ia9cn8DZvW3UlNrGElSQKilR/vH1ev49bzOJXWJ48ZqTadYgKtDVMibg/rdmD79+K4mGUZE8\nfEFvrkuMIyrSrsXWRpagglBeQRH3vLecL1fvZvwpHXj04r42MoQxwLLt+3nyy3VERdZh3n3DiWlU\nL9BVMj5kCSrI7M7OZ9KbS1i9K4eHL+jNjad3tt5IJuwVlyi//3gl7yXtoGXjeky5cpAlpzBgCSqI\nrEzNZtKbSziQX8Qr19tI5MaUemHuJt5L2sFNZ3TmN6N62FTrYcK+5SAxe2Ua97yfTEzDenx066n0\nOsl6IxkDsGBzBv+cs4FLBrbl9+f3thaFMGIJKsBUlRfmbuIfX2/g5I7NmHpdArHWdGEM4Nz/d+e7\ny4iLbchff9XPklOYsQQVQPmFxTz40QpmJO/ikoFteeLS/tSvayMrGwPOdaerX15MxoEC3p40lIbW\nrBd27BsPkNz8Qia9kcTirfu495we3Daimx0dGuMqKCph0ptJbN+Xx1k9WlqTd5iyBBUAWXkFXP/q\nT6zelWPTSxtThqry0Ccr+X5DOo9eHM9Vp3QMdJVMgFiC8rP03MNc+8pitmQc5KVrBjOqj/XUM6ZU\nSYny6Kw1fLA0ld+c3Z3rEuMCXSUTQJag/Ghn1iGueXkxu7PzeW3CEBtTz5gyfv/JSqYv2cENp3Xm\nrlHdA10dE2A+HZ5ARMaIyHoYnyO5AAAf1UlEQVQR2SQiD1awzRUiskZEVovIO76sTyClZBzkipcW\nknHgMG9PsgFfTfWFQxwVlyi/fiuJ6Ut28H8nt+MPY607ufHhGZSIRAAvAKOBVGCJiMxU1TUe23QH\nfgecpqr7RaSVr+oTSOt353LNK4spLlHevWkYfds1DXSVTIgIlzhanprFV6v3cNXQjjxk9zoZly/P\noE4BNqnqFlUtAKYDF5fZ5ibgBVXdD6Cqe31Yn4BYkZrFuKkLEeC9yZacTI3V+jgqKVE+W74LgN+O\n7mHdyc0RvkxQ7YAdHs9T3WWeegA9RGS+iCwSkTE+rI/fLUnZx1X/XUyjepF8ePOpdG/dONBVMqGn\nVsfRtsyDnD/lB16bn8Ko3q1tfD1zFF8eqpR3jq7llN8dGA60B34Qkb6qmnXUjkQmA5MBOnYMjS6n\nP2xM56Y3k2jbLJppk4bSpml0oKtkQpPX4giCL5bufHcZ63bn8tglfbnaupObMnx5BpUKdPB43h7Y\nVc42n6pqoapuBdbjBNpRVHWqqiaoakLLli19VmFv+Wr1bm58PYnOsY14/9eJlpzMifBaHEFwxdKO\nfXksT82mffNorh3WiTp17LqTOZovE9QSoLuIdBaRKOBKYGaZbWYAIwBEJBanqWKLD+vkc58m7+TW\naT/Tp20Tpt80zMbVMyeqVsZRcYny9NfriagjfHBzYqCrY4JUjROUiESIyNVVbaeqRcDtwFfAWuB9\nVV0tIo+KyEXuZl8BmSKyBpgL3KeqmTWtU7B4Z/F27novmSFxzXl70lCaNqgb6CqZIFadWKqNcbRg\nUwbxf/ySGcm7uG14V2thMBUS1bLN2e4KkSbAbTgXZGcC/8MJlHuBZFUt25PILxISEjQpKSkQRVfq\n5R+28JfP1zKiZ0tevGawDfpaC4jIUlVN8MJ+LJZch4uKSfzbt+w7WMCk0zvz8Ng+fi3fBMbxxlJl\nnSTeAvYDC4FJwH1AFHCxqiYfVy1rIVXluW828s85Gzm/30n8c9wgoiJtenZzFIsl16OfrWHfwQIb\ng9JUS2UJqouq9gMQkZeBDKCjqub6pWYhQFV5fPZa/vvDVi49uT1/v7QfkRGWnMwxLJaAn7buY9ri\n7dx0RmdLTqZaKktQhaX/UdViEdkabgFVmZIS5eFPV/HO4u1cl9iJP10Yb72QTEXCPpZSMg5y4xtL\naNcsmrtH9wh0dUyIqCxBDRCRHH65DyPa47mqalhP0PLorDW8s3g7twzvyv3n9rShWUxlwj6WftiY\nTm5+EX+/tD8NomykCFM9Ff5SVNWu8ldgwaYMXl+QwoRT43hgTK9AV8cEOYslSMvOp47AiJ4hN0yg\nCaAKE5SI1AduBroBK4BX3S6vYe3g4SLu/2gFnWMbWnIy1RLusXSooJj3luzgjO4tiY4K+1xtaqCy\nc+03cNrOfwDOB+KB3/ijUsHsiS/WsTPrEO//OtGCzVRX2MbSgs0ZXPXfxQDcNqJbgGtjQk1lCaqP\nR8+jV4Cf/FOl4LVgcwZvLdrGDad1Zkhci0BXx4SOsIyl7LxC7vtgBQAPnteLUzpbzJiaqW4vvqJw\n7wRw8HARD3y0griYBtx3bs9AV8eElrCMpe83prMz6xBv3ziU07vbBJ2m5ipLUAPdnkbg9DYKu55H\nnp78ch2p+w/x3mRr2jM1FnaxpKq8tXAbAL3b2DQz5vhUlqCWq+ogv9UkiC3akskbC7cx8bQ4a6Yw\nxyOsYklVue2dn/kpZR89WjeiSbSNSWmOT2UJqvxB+sJMXkER93+4gk7WtGeOX1jF0rz16cxeuZtb\nhnfl3nN6EmE3sJvjVFmCaiUi91S0UlWf8UF9gs6TX65n+7483ps8zG4wNMcrrGLprveSadW4HneP\n6mHJyZyQyv7iRgCNKH9Gz7CwaEvmkRtyh3aJCXR1TOgKm1jalnmQ7EOF9DqpsQ2abE5YZQkqTVUf\n9VtNgkxegdNrr2OLBtw/xpr2zAkJi1jKyS/krKfmAXBaN+u1Z05cZQmq1h/tVeapr9azLTOP6da0\nZ05cWMTSve8vB+DOkd245xw7qDMnrrJz8LP9Vosg89PWfby+IIXrEzsxzJr2zImr9bH0+Yo0vl6z\nB4A7zu4e4NqY2qLCBKWq+/xZkWBxqKCY+z5cTvvm0dxvY+0ZL6jtsZSUso/b3vmZJvUj+eH+EdS1\nOdGMl1jbVRmlTXvv3jSMhvXs4zGmMtl5hVz20kIA/nNtAh1aNAhwjUxtYoc6Hpak7OO1BVu5LrET\niV2tac+YyuQXFnPqE98AcPNZXS1mjNdZgnKpKn+bvZa2TaNtGg1jquGalxdzsKCY2Eb1ePA8ixnj\nfZagXEnb9vPz9ixuPquLNe0ZUw2l15oWPDgywDUxtZVPE5SIjBGR9SKySUQerGS7y0RERSTBl/Wp\nzEvzNtOiYRSXDe4QqCoYU6Fgi6Xn5mxk4ZZMzugeazfkGp/x2S9LRCKAF4DzgD7AeBHpU852jYE7\ngcW+qktVNuzJ5Zt1e7k+Mc5GKjdBJxhj6dk5GwAY27+Nr4syYcyXhz6nAJtUdYuqFgDTgYvL2e4x\n4Ekg34d1qdTU77cQXTeC6xI7BaoKxlQmqGJp/8ECAB4Y04txQzr6sigT5nyZoNoBOzyep7rLjhCR\nQUAHVZ3lw3pUKi37EJ8m72TckA40bxgVqGoYU5mgiqXnvtkIwOBOzX1dlAlzvkxQ5Q3vcmTaARGp\nAzwL/LbKHYlMFpEkEUlKT0/3YhXhtfkplCjceHpnr+7XGC8KmljanH6A1xek0K9dUxIsQRkf82WC\nSgU8exy0B3Z5PG8M9AXmiUgKMAyYWd7FXVWdqqoJqprQsmVLr1Uw+1Ah7yzeztj+bewGQxPMgiaW\nNu45AMDkM7tQx6bSMD7mywS1BOguIp1FJAq4EphZulJVs1U1VlXjVDUOWARcpKpJPqzTUaYt3saB\nw0VMPrOLv4o05ngETSw98ukqGtWLtNHKjV/4LEGpahFwO/AVsBZ4X1VXi8ijInKRr8qtrvzCYl6b\nn8IZ3WOJb9s00NUxpkLBFEtZhwqJbRRFC7tea/zAp3ekqupsYHaZZY9UsO1wX9alrBnLdpKee5jn\nxg30Z7HGHJdgiKVVO7MpKCphdJ/Wvti9MccIyzvsSkqUqd9voV+7pjZ+mDHV9M3avQBMPM06FBn/\nCMsE9fWaPWzJOMivz+qCiF3oNaYqBw8X8eJ3m2jfPJq2zaIDXR0TJsIuQakqL323mY4tGjAm/qRA\nV8eYkJCWfYj8whJO7mhdy43/hF2CWpKyn+QdWdx0RmcibWI1Y6qluMT5164/GX8Ku7/Q//luMzEN\no7g8wQaFNaa6tu/LA6BhPRur0vhPWCWoI4PCnhpH/boWaMZU17bMgwB0tBvajR+FVYJ6a+E26tet\nw7XDbFBYY6qrqLiEv3y+FoCOLRoGuDYmnIRNglJV5qzdw1k9WtqgsMbUQIF7ASqhU3Ob+8n4Vdj8\n2tbtziUtO5+RvVoFuirGhJQvVu4G4OKBbQNcExNuwiZBfbvOuclwRE9LUMbUxMfLUgH41cntA1wT\nE27CKkH1a9eUVk3qB7oqxoSU+Zsy6dKyIY3q+XRkNGOOERYJav/BApZt388Ia94zpkZmr0wD4GyL\nHRMAYZGgvtuQTolakBlTU//6dhMA1w6LC2xFTFgKiwT1zbq9xDaqR792Nq2GMdWlqqxNy6F+3Tp0\njLH7n4z/1foEVVRcwnfr9zK8Z0ubAdSYGrjn/eUA3H9urwDXxISrWp+gft6eRU5+kTXvGVMDqson\ny3YCcOUpNiyYCYxan6C+WbeHuhHC6d1timpjqutgQTEAA9o3pUGU9d4zgVHrE9TcdXsZEteCxvXr\nBroqxoSMQ26CGtvfbs41gVOrE9SOfXls2HPARo8wpobeWJACQHy7JoGtiAlrtTpBzV3vjB5hCcqY\nmilNUAPaNwtsRUxYq9UJ6tt1e4mLaUCXlo0CXRVjQsbu7HxyDxfxq0HtaGijR5gAqrUJKq+giAWb\nMxnZy2YANaYmVu3MBmBYlxYBrokJdz49PBKRMcBzQATwsqo+UWb9PcAkoAhIB25Q1W3eKHvBpkwK\nikoC3rxXWFhIamoq+fn5Aa2HqVj9+vVp3749desGZ0caf8dRijs5Yc+Tguv6k8VS8PN2LPksQYlI\nBPACMBpIBZaIyExVXeOx2TIgQVXzROQW4ElgnDfK/3b9XhpGRXBK58AeBaamptK4cWPi4uIQsRuF\ng42qkpmZSWpqKp07dw50dY4RiDg6XOTM/9S2aXANrGyxFNx8EUu+bOI7BdikqltUtQCYDlzsuYGq\nzlXVPPfpIsAr4/mrKnPX7eWM7i0DPsFafn4+MTExFlBBSkSIiYkJ5qNyv8dR6U+1SXRwnVFaLAU3\nX8SSL/96twN2eDxPdZdV5EbgC28UvDYtuCYntIAKbkH+/fg9jnZnB22yDvbvKux5+/vxZYIqr6Za\n7oYi1wAJwFMVrJ8sIkkikpSenl5lwV+t3o0IDO/Vsib1rbUiIiIYOHAgffv25cILLyQrK+vIutWr\nVzNy5Eh69OhB9+7deeyxx1D95Wv64osvSEhIoHfv3vTq1Yt77703EG8hnHktjtxtqoylJSn76duu\nCfXrRhxPfWs1iyX/8mWCSgU8B/FqD+wqu5GIjAIeAi5S1cPl7UhVp6pqgqomtGxZedJRVT5N3smp\nXWNo1Ti42tADJTo6muTkZFatWkWLFi144YUXADh06BAXXXQRDz74IBs2bGD58uUsWLCAf//73wCs\nWrWK22+/nbfffpu1a9eyatUqunTp4tW6FRUVeXV/tZDX4giqjiVVZXf2Ido1iz7xmtdCFkv+5csE\ntQToLiKdRSQKuBKY6bmBiAwC/oMTVHu9UWjyjixSMvO4ZGBlrSDhKzExkZ07nUFA33nnHU477TTO\nOeccABo0aMDzzz/PE084ncSefPJJHnroIXr1ckazjoyM5NZbbz1mnwcOHGDixIn069eP/v3789FH\nHwHQqNEv9599+OGHTJgwAYAJEyZwzz33MGLECO677z7i4uKOOhLt1q0be/bsIT09nUsvvZQhQ4Yw\nZMgQ5s+f7/0PJPj5NY76/elr9ucVEhVpZ09VsVjyPZ/14lPVIhG5HfgKp3vsq6q6WkQeBZJUdSZO\nU0Qj4AO37XK7ql50IuXOWLaTepF1GNP3pBN8B7VPcXEx33zzDTfeeCPgNEkMHjz4qG26du3KgQMH\nyMnJYdWqVfz2t7+tcr+PPfYYTZs2ZeXKlQDs37+/ytds2LCBOXPmEBERQUlJCZ988gkTJ05k8eLF\nxMXF0bp1a6666iruvvtuTj/9dLZv3865557L2rVrj+Odhy5/xlF+YTEHDjtH4Q+M6emtt1ArWSz5\nh0/vg1LV2cDsMsse8fj/KG+WV1hcwmcr0hjVp7UNDuvh0KFDDBw4kJSUFAYPHszo0aMBpzmnooua\nNbnYOWfOHKZPn37kefPmzat8zeWXX05EhHOUPm7cOB599FEmTpzI9OnTGTdu3JH9rlnzS2/qnJwc\ncnNzady4cbXrVhv4K45+2JgBwB0ju9G+uU1QWB6LJf+qVSNJ/LAxnX0HC/iVNe8dpbTdfNu2bRQU\nFBxpN4+PjycpKemobbds2UKjRo1o3Lgx8fHxLF26tMr9VxScnsvKdj1t2LDhkf8nJiayadMm0tPT\nmTFjBv/3f/8HQElJCQsXLiQ5OZnk5GR27twZ9AEVymavTAPgkkEWPxWxWPKvWpWgPlm2i+YN6nJm\nD+u9V56mTZsyZcoU/vGPf1BYWMjVV1/Njz/+yJw5cwDn6PDOO+/k/vvvB+C+++7j8ccfZ8OGDYDz\nI3/mmWeO2e8555zD888/f+R5abNE69atWbt27ZFmh4qICL/61a+455576N27NzExMeXuNzk5+QQ/\nAVOZ5TuyiKwjdLWxK6tkseQftSZB5eYX8vXq3Yzt3zbgN+cGs0GDBjFgwACmT59OdHQ0n376KX/5\ny1/o2bMn/fr1Y8iQIdx+++0A9O/fn3/+85+MHz+e3r1707dvX9LS0o7Z58MPP8z+/fvp27cvAwYM\nYO7cuQA88cQTjB07lpEjR9KmTZtK6zVu3DjefvvtI00SAFOmTCEpKYn+/fvTp08fXnrpJS9+Eqas\nLRkH6dKyYdUbGsBiyR/Es59+KEhISNCyp9IAHy5N5d4PlvPRLacyuFPV7bb+snbtWnr37h3oapgq\nlPc9ichSVU0IUJV8zjOWkndkcckL84lv24TP7zwjwDUrn8VSaPBmLNWaU40Zy3bSsUUDTu5o89cY\nU1NXTl0IwK3DuwW4Jsb8olYkqD05+czfnMElg9rZUCjG1NDm9APkFzoDxF7Qv/LmI2P8qVbMRjYz\neReqcMnAtoGuSqX+/Nlq1uzK8eo++7Rtwh8vjPfqPk14ueOdZQD8dnSPANek+iyWwkOtOIP6ZNlO\nBnRoZjPnVoOqMnLkSHJyvBvc3rR06VL69etHt27duPPOOynvOulTTz3FwIEDj4yLFhERwb59+wC4\n4YYbaNWqFX379j3qNffeey/ffvutX95DKIltXA+A20da815NhEMsgXNT8qBBgxg7duyRZVdeeSUb\nN270/RtQ1ZB6DB48WD2tS8vRTg/M0td+3KLBaM2aNYGuwlFmzZqld911V41eU1RU5KPalG/IkCG6\nYMECLSkp0TFjxujs2bMr3X7mzJk6YsSII8+/++47Xbp0qcbHxx+1XUpKio4ePbrcfZT3PeGM1BDw\n37yvHqWxdNV/F+pF//qh3M8lmFgs1dyJxpKq6tNPP63jx4/XCy644MiyefPm6aRJk8rdhzdjKeTP\noGYk7ySijjB2QHA37wWLadOmcfHFv0wndMkllzB48GDi4+OZOnXqkeWNGjXikUceYejQoSxcuJCl\nS5dy1llnMXjwYM4999wjXWT/+9//MmTIEAYMGMCll15KXl7eMWXWRFpaGjk5OSQmJiIiXHfddcyY\nMaPS17z77ruMHz/+yPMzzzyTFi2OnaiyU6dOZGZmsnv37hOqY20zf1PmkUkKTfWFQyylpqby+eef\nM2nSpKO2O+OMM5gzZ47vB6g9nqwWyIfnGVRxcYkmPj5HJ7y6uNxMHgyC7aivY8eOmpOTc+R5Zmam\nqqrm5eVpfHy8ZmRkqKoqoO+9956qqhYUFGhiYqLu3btXVVWnT5+uEydOVFU9sr2q6kMPPaRTpkw5\npsxvv/1WBwwYcMwjMTHxmG2XLFmiZ5999pHn33///VFHbmUdPHhQmzdvfuR9lNq6desxZ1CqqpMm\nTdIPP/zwmOXhegZ1IL9QOz0wS8/75/flfLrBxWLJ/7F06aWXalJSks6dO/eY144aNUqTkpKO2Y83\nYymkO0n8lLKPXdn5PHBer0BXJWTs27fvqCFOpkyZcuTO9B07drBx40ZiYmKIiIjg0ksvBWD9+vWs\nWrXqyLhjxcXFR24WXLVqFQ8//DBZWVkcOHCAc88995gyR4wYUe07153f8tEq65n52Wefcdppp5V7\nxlSeVq1asWvXMbNVhK2VO7MBOLVrTIBrEnpqeyzNmjWLVq1aMXjwYObNm3fM9qWxVHaQXG8K6QT1\nafIuGkRFMLpP60BXJWRERkZSUlJCnTp1mDdvHnPmzGHhwoU0aNCA4cOHHxnnq379+kcGoFRV4uPj\nWbhw4TH7mzBhAjNmzGDAgAG8/vrr5f6Q586dy913333M8gYNGrBgwYKjlrVv357U1NQjz1NTU2nb\ntuLm2+nTpx/VJFGV/Px8oqNtrqNS17y8GIBRFkM1Vttjaf78+cycOZPZs2eTn59PTk4O11xzDW+/\n/Tbgn1gK2WtQJSXKnLV7GNGrFQ2iQjrP+lXPnj3ZsmULANnZ2TRv3pwGDRqwbt06Fi1aVOFr0tPT\njwRVYWEhq1evBiA3N5c2bdpQWFjItGnTyn196VFf2UfZgAJo06YNjRs3ZtGiRagqb7755lHt/J6y\ns7P57rvvKlxfng0bNhzTuy+cFZU4R9lD4qp3Bmp+Udtj6W9/+xupqamkpKQwffp0Ro4ceSQ5gRNL\n8fG+7ZYfsgkqOTWL9NzDnGNHfjVywQUXHDkyGzNmDEVFRfTv358//OEPDBs2rNzXREVF8eGHH/LA\nAw8wYMAABg4ceCQgHnvsMYYOHcro0aOPTMZ2ol588UUmTZpEt27d6Nq1K+eddx4AL7300lFjiH3y\nySecc845R43mDDB+/HgSExNZv3497du355VXXgGcPwabNm0iIaHWjl5UI2nZzhH+ad1iiKhjN7jX\nVDjEUkX27NlDdHR0leMCnrDjuXAVyEdpJ4knvlirXX/3uWblFRxzQS6YBNuF3V27dumoUaMCXY2A\n+Pjjj/Xhhx8ud104dpKIOqmbdnpglu4/eLjan2EgWSwFj2eeeUZffvnlctdZN3Pg69W7GdqlBU2j\nbWLCmmjTpg033XRTUN9c6CtFRUXVmtU0nMQ0jKJZg6hAVyMkhXMsNWvWjOuvv97n5YTkxZst6QfY\nnH6Qa4d1CnRVQtIVV1wR6CoExOWXXx7oKgSd+HZNA12FkBausTRx4kS/lBOSZ1D/W7MHsJ5Hxpyo\na4Z2DHQVjKlQyCao+LZNaN+8QaCrUi1azv0IJniE8/cTau88nL+rUODt7yfkElRRibJ0+/6Qufep\nfv36ZGZmWmAFKVUlMzOT+vXrB7oqATGqd2jEEVgsBTtfxFLIXYPKPVSIKiGToEpvlktPTw90VUwF\n6tevT/v27QNdDb9rEBURUt3LLZaCn7djyacJSkTGAM8BEcDLqvpEmfX1gDeBwUAmME5VUyrbZ05+\nET2aRdOnTRPfVNrL6tatS+fOnQNdDRPifBFLocZiKfz4rIlPRCKAF4DzgD7AeBHpU2azG4H9qtoN\neBb4e1X7zc0vZHSf1jZzrgkbvoolaykzwc6X16BOATap6hZVLQCmA2XH2bgYeMP9/4fA2VJF5lFC\np3nPGC/xSSyVWIYyQc6XCaodsMPjeaq7rNxtVLUIyAYqHVY5QoRTOtu4YSas+CSWYhraDbomuPny\nGlR5R29lD9mqsw0iMhmY7D49HBUZseoE63a8YoEMKztsyu4ZoHLL8lksiYjFkpXtD8cVS75MUKlA\nB4/n7YGyE/GUbpMqIpFAU2Bf2R2p6lRgKoCIJKlqQEb7tLLDr+xAlFsOiyUrO+TLPp7X+bKJbwnQ\nXUQ6i0gUcCUws8w2M4HSAZ0uA75Vu8nBmLIslkxY8tkZlKoWicjtwFc4XWNfVdXVIvIozsi2M4FX\ngLdEZBPO0d6VvqqPMaHKYsmEK5/eB6Wqs4HZZZY94vH/fKCmI3hO9ULVjpeVbWUHhMWSlR2OZYu1\nAhhjjAlGITcWnzHGmPAQtAlKRMaIyHoR2SQiD5azvp6IvOeuXywicX4s+x4RWSMiK0TkGxHx2sRU\nVZXtsd1lIqIi4rVeOdUpW0SucN/7ahF5x19li0hHEZkrIsvcz/18L5X7qojsrai7tTimuPVaISIn\ne6Ncf7E4sjgqs94nceTu2/uxdDzT8Pr6gXMheDPQBYgClgN9ymxzK/CS+/8rgff8WPYIoIH7/1v8\nWba7XWPge2ARkODH990dWAY0d5+38mPZU4Fb3P/3AVK8VPaZwMnAqgrWnw98gXOf0TBgsbd+575+\nWBxZHPkrjtz9eT2WgvUMyidDu3irbFWdq6p57tNFOPeleEN13jfAY8CTQL6Xyq1u2TcBL6jqfgBV\n3evHshUoHSG4KcfeB3RcVPV7yrlfyMPFwJvqWAQ0E5E23ijbDyyOLI78Ekfgm1gK1gTlk6FdvFi2\npxtxjgq8ocqyRWQQ0EFVZ3mpzGqXDfQAeojIfBFZJM4I2/4q+0/ANSKSitOb7Q4vlV2Vmv4egonF\nUQVlWxz5PY7gOGIpWOeD8trQLj4q29lQ5BogATjLC+VWWbaI1MEZqXqCl8qrdtmuSJzmieE4R7s/\niEhfVc3yQ9njgddV9WkRScS556evqpacYNneqFuwsjgqp2yLo4DEERzHby1Yz6BqMrQLUsnQLj4q\nGxEZBTwEXKSqh71QbnXKbgz0BeaJSApOO+5ML13gre5n/qmqFqrqVmA9TqD5o+wbgfcBVHUhUB9n\nbDFfq9bvIUhZHJVftsURfo8jOJ5Y8tYFMm8+cI4wtgCd+eViX3yZbW7j6Iu77/ux7EE4FyO7+/t9\nl9l+Ht67uFud9z0GeMP9fyzO6XqMn8r+Apjg/r+3+8MWL733OCq+sHsBR1/Y/cmb37kvHxZHFkf+\njCN3n16NJZ8Ehpfe6PnABvcH/JC77FGcIy1wMv8HwCbgJ6CLH8ueA+wBkt3HTH+VXWZbrwVWNd+3\nAM8Aa4CVwJV+LLsPMN8NumTgHC+V+y6QBhTiHOHdCNwM3Ozxnl9w67XSm5+3Px4WRxZH/ogjd99e\njyUbScIYY0xQCtZrUMYYY8KcJShjjDFByRKUMcaYoGQJyhhjTFCyBGWMMSYoWYKqBUSkWESSPR5x\nIjJcRLLdUYvXisgf3W09l68TkX8Euv7GBAuLpeASrEMdmZo5pKoDPRe40yb8oKpjRaQhkCwipeOO\nlS6PBpaJyCeqOt+/VTYmKFksBRE7gwoDqnoQWAp0LbP8EM7NeqEy+KkxAWWx5F+WoGqHaI8miU/K\nrhSRGJyhRVaXWd4cZwyw7/1TTWOCnsVSELEmvtrhmGYJ1xkisgwoAZ5Q1dUiMtxdvgLo6S7f7ce6\nGhPMLJaCiCWo2u0HVR1b0XIR6QH86LabJ/u7csaEEIulALAmvjCmqhuAvwEPBLouxoQyiyXfsARl\nXgLOFJHOga6IMSHOYsnLbDRzY4wxQcnOoIwxxgQlS1DGGGOCkiUoY4wxQckSlDHGmKBkCcoYY0xQ\nsgRljDEmKFmCMsYYE5QsQRljjAlK/w8TfBEBh8bTBAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x101967630>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "data": {
      "image/png": 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7Xx7pzwJSVDVVVQuABcCUMpabCzwC5PswlgptzzjKhMe/AmBMnzb+CsOYOuHL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\nxFjSGxNiLOmNCTGW9MaEmP8PDJVbzb2SR0UAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10c486630>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from sklearn.metrics import roc_curve, auc\n",
    "from sklearn.preprocessing import label_binarize\n",
    "\n",
    "probs, probs_debiased, race, compas_decile_score, true_label = biased_debiased_probs(final_data, debias_train_size,  \n",
    "                                                                                     train_percentage_split)\n",
    "\n",
    "compasFpr, compasTpr, _ = roc_curve(true_label, compas_decile_score)\n",
    "compasRocAuc = auc(compasFpr, compasTpr)\n",
    "\n",
    "logisticFpr, logisticTpr, _ = roc_curve(true_label, probs)\n",
    "logisticRocAuc = auc(logisticFpr, logisticTpr)\n",
    "\n",
    "logisticFpr_debiased, logisticTpr_debiased, _ = roc_curve(true_label, probs_debiased)\n",
    "logisticRocAuc_debiased = auc(logisticFpr_debiased, logisticTpr_debiased)\n",
    "\n",
    "plt.figure()\n",
    "plt.subplot(1,2,1)\n",
    "plt.plot(compasFpr, compasTpr, label='ROC curve \\n (area = %0.2f)' % compasRocAuc)\n",
    "plt.xlim([0.0, 1.0])\n",
    "plt.ylim([0.0, 1.05])\n",
    "plt.xlabel('FPR')\n",
    "plt.ylabel('TPR')\n",
    "plt.title('COMPAS scores ROC')\n",
    "plt.legend(loc=\"lower right\")\n",
    "plt.gca().set_aspect('equal', adjustable='box')\n",
    "#plt.show()\n",
    "\n",
    "plt.subplot(1,2,2)\n",
    "plt.plot(logisticFpr, logisticTpr, label='ROC curve \\n (area = %0.2f)' % logisticRocAuc)\n",
    "plt.xlim([0.0, 1.0])\n",
    "plt.ylim([0.0, 1.05])\n",
    "plt.xlabel('FPR')\n",
    "plt.ylabel('TPR')\n",
    "plt.title('Logistic Regression ROC \\n on Biased Data')\n",
    "plt.legend(loc=\"lower right\")\n",
    "plt.tight_layout()\n",
    "plt.gca().set_aspect('equal', adjustable='box')\n",
    "plt.show()\n",
    "\n",
    "plt.figure()\n",
    "plt.plot(logisticFpr_debiased, logisticTpr_debiased, label='ROC curve \\n (area = %0.2f)' % logisticRocAuc_debiased)\n",
    "plt.xlim([0.0, 1.0])\n",
    "plt.ylim([0.0, 1.05])\n",
    "plt.xlabel('FPR')\n",
    "plt.ylabel('TPR')\n",
    "plt.title('Logistic Regression ROC \\n on Debiased Data')\n",
    "plt.legend(loc=\"lower right\")\n",
    "plt.tight_layout()\n",
    "plt.gca().set_aspect('equal', adjustable='box')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "                 COMPAS       LR raw                  \n",
      "                    FPR   FNR    FPR    SE   FNR    SE\n",
      "Population         0.09  0.69   0.08  0.01  0.68  0.01\n",
      "Caucasian          0.05  0.81   0.05  0.01  0.80  0.01\n",
      "African-American   0.14  0.63   0.11  0.01  0.62  0.01\n",
      "----------------\n",
      "                 COMPAS       LR debiased                  \n",
      "                    FPR   FNR         FPR    SE   FNR    SE\n",
      "Population         0.09  0.69        0.09  0.01  0.69  0.01\n",
      "Caucasian          0.05  0.81        0.09  0.01  0.71  0.03\n",
      "African-American   0.14  0.63        0.09  0.01  0.68  0.01\n",
      "-----------\n",
      "Average biased logistic regression accuracy: 0.62 SE: 0.01\n",
      "Average debiased logistic regression accuracy: 0.61 SE: 0.01\n",
      "Average COMPAS accuracy: 0.61 SE: 0.01\n",
      "-----------\n",
      "Average logistic regression biased threshold: 0.65\n",
      "Average logistic regression debiased threshold: 0.64\n"
     ]
    }
   ],
   "source": [
    "num_splits = 30\n",
    "compas_score_threshold = 8 # 5 for medium and high, 8 for high\n",
    "logistic_quantile = 80\n",
    "\n",
    "biased_accuracy = np.zeros(num_splits)\n",
    "debiased_accuracy = np.zeros(num_splits)\n",
    "compas_accuracy = np.zeros(num_splits)\n",
    "\n",
    "biased_logistic_regression_threshold = np.zeros(num_splits)\n",
    "debiased_logistic_regression_threshold = np.zeros(num_splits)\n",
    "\n",
    "# [pop_FPR, pop_FNR, white_FPR, white_FNR, black_FPR, black_FNR]\n",
    "rates_biased = np.zeros((num_splits,6))\n",
    "rates_debiased = np.zeros((num_splits,6))\n",
    "\n",
    "for i in range(num_splits):\n",
    "\n",
    "    probs, probs_debiased, race, compas_decile_score, recid_label = biased_debiased_probs(final_data, 1000, .6)\n",
    "    \n",
    "    risk_threshold_biased = get_logistic_threshold(probs, logistic_quantile)\n",
    "    biased_logistic_regression_threshold[i] = risk_threshold_biased\n",
    "    rates_biased[i,], biased_accuracy[i] = FPR_FNR_accuracy(recid_label, race, probs, risk_threshold_biased)\n",
    "    \n",
    "    risk_threshold_debiased = get_logistic_threshold(probs_debiased, logistic_quantile)\n",
    "    debiased_logistic_regression_threshold[i] = risk_threshold_debiased\n",
    "    rates_debiased[i,], debiased_accuracy[i] = FPR_FNR_accuracy(recid_label, \n",
    "                                                                race, \n",
    "                                                                probs_debiased, \n",
    "                                                                risk_threshold_debiased)\n",
    "    \n",
    "    compas_prediction = np.array([1 if x >= compas_score_threshold else 0 for x in compas_decile_score])\n",
    "    compas_accuracy[i] = sum(compas_prediction == recid_label) / len(recid_label)\n",
    "    \n",
    "compas_FPR_FNR, _ = FPR_FNR_accuracy(recid_label, race, compas_decile_score, compas_score_threshold)\n",
    "\n",
    "rate_debiased = np.mean(rates_debiased, axis=0)\n",
    "rate_biased = np.mean(rates_biased, axis=0)\n",
    "se_debiased = np.std(rates_debiased, axis=0)\n",
    "se_biased = np.std(rates_biased, axis=0)\n",
    "\n",
    "dfMatBiased = np.zeros((3,6))\n",
    "dfMatBiased[:,2:6:2] = np.reshape(rate_biased,(3,2)).round(2)\n",
    "dfMatBiased[:,3:7:2] = np.reshape(se_biased,(3,2)).round(2)\n",
    "dfMatBiased[:, 0:2] = np.reshape(compas_FPR_FNR, (3,2)).round(2)\n",
    "raw = dfMatBiased\n",
    "\n",
    "labels = [np.array(['COMPAS', 'COMPAS', 'LR raw', 'LR raw', 'LR raw', 'LR raw']),\n",
    "          np.array(['FPR', 'FNR', 'FPR', 'SE', 'FNR', 'SE'])]\n",
    "raw_df = pd.DataFrame(raw, index=['Population', 'Caucasian', 'African-American'], columns=labels)\n",
    "\n",
    "dfMatDebiased = np.zeros((3,6))\n",
    "dfMatDebiased[:,2:6:2] = np.reshape(rate_debiased,(3,2)).round(2)\n",
    "dfMatDebiased[:,3:7:2] = np.reshape(se_debiased,(3,2)).round(2)\n",
    "dfMatDebiased[:, 0:2] = np.reshape(compas_FPR_FNR, (3,2)).round(2)\n",
    "debiased = dfMatDebiased\n",
    "\n",
    "labels = [np.array(['COMPAS', 'COMPAS', 'LR debiased', 'LR debiased', 'LR debiased', 'LR debiased']),\n",
    "          np.array(['FPR', 'FNR', 'FPR', 'SE', 'FNR', 'SE'])]\n",
    "debiased_df = pd.DataFrame(debiased, index=['Population', 'Caucasian', 'African-American'], columns=labels)\n",
    "\n",
    "print(raw_df)\n",
    "print('----------------')\n",
    "print(debiased_df)\n",
    "print('-----------')\n",
    "print('Average biased logistic regression accuracy:', np.mean(biased_accuracy).round(2),\n",
    "      'SE:', np.std(biased_accuracy).round(2))\n",
    "print('Average debiased logistic regression accuracy:', np.mean(debiased_accuracy).round(2),\n",
    "      'SE:', np.std(debiased_accuracy).round(2))\n",
    "print('Average COMPAS accuracy:', np.mean(compas_accuracy).round(2), 'SE:', np.std(compas_accuracy).round(2))\n",
    "print('-----------')\n",
    "print('Average logistic regression biased threshold:', np.mean(biased_logistic_regression_threshold).round(2))\n",
    "print('Average logistic regression debiased threshold:', np.mean(debiased_logistic_regression_threshold).round(2))"
   ]
  }
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