causal_linear_program.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "# linear programming problem\n",
    "import pulp\n",
    "min_problem = pulp.LpProblem(\"min ATE\", pulp.LpMinimize)\n",
    "max_problem = pulp.LpProblem(\"max ATE\", pulp.LpMaximize)\n",
    "\n",
    "# our hidden variables are the the probability of being in one of the partitions\n",
    "partition_names = ['/'.join([compliance, response]) for compliance, response in partition_types]\n",
    "q = {partition: pulp.LpVariable(partition, lowBound=0) for partition in partition_names}"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "# since our hidden vars are probabilities the sum of them should all be under 1\n",
    "min_problem += sum([v for k,v in q.items()]) == 1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "# statements\n",
    "p_treatment_untreated_bad = q['never_taker/never_better'] + q['defier/never_better'] \\\n",
    "                             + q['never_taker/helped'] + q['defier/helped']\n",
    "\n",
    "p_treatment_untreated_good = q['never_taker/always_better'] + q['defier/always_better'] \\\n",
    "                             + q['never_taker/hurt'] + q['defier/hurt']\n",
    "\n",
    "p_treatment_treated_bad = q['always_taker/never_better'] + q['complier/never_better'] \\\n",
    "                             + q['always_taker/hurt'] + q['complier/hurt']\n",
    "\n",
    "p_treatment_treated_good = q['always_taker/always_better'] + q['complier/always_better'] \\\n",
    "                             + q['always_taker/helped'] + q['complier/helped']\n",
    "\n",
    "p_control_untreated_bad = q['never_taker/never_better'] + q['complier/never_better'] \\\n",
    "                             + q['never_taker/helped'] + q['complier/helped']\n",
    "\n",
    "p_control_untreated_good = q['never_taker/always_better'] + q['complier/never_better'] \\\n",
    "                             + q['never_taker/hurt'] + q['complier/hurt']\n",
    "\n",
    "p_control_treated_bad = q['always_taker/never_better'] + q['defier/never_better'] \\\n",
    "                             + q['always_taker/hurt'] + q['defier/hurt']\n",
    "\n",
    "p_control_treated_good = q['always_taker/always_better'] + q['defier/always_better'] \\\n",
    "                             + q['always_taker/helped'] + q['defier/helped']"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "# there's a natural mapping from probabilities we see to the hidden variables we have\n",
    "# we'll spell these out one by one.\n",
    "# there are probably smarter ways to express this as a vector operation,\n",
    "# but this is easier to understand\n",
    "min_problem += p_treatment_untreated_bad == p_states['treatment/untreated/bad']\n",
    "\n",
    "min_problem += p_treatment_untreated_good == p_states['treatment/untreated/good']\n",
    "\n",
    "min_problem += p_treatment_treated_bad == p_states['treatment/treated/bad']\n",
    "\n",
    "min_problem += p_control_untreated_bad == p_states['control/untreated/bad']\n",
    "\n",
    "min_problem += p_control_untreated_good == p_states['control/untreated/good']\n",
    "\n",
    "min_problem += p_control_treated_bad == p_states['control/treated/bad']\n",
    "\n",
    "#min_problem += p_treatment_treated_good == p_states['treatment/treated/good']\n",
    "\n",
    "#min_problem += p_control_treated_good == p_states['control/treated/good']"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "min_problem += q['complier/helped'] + q['defier/helped'] + q['always_taker/helped'] + q['never_taker/helped'] \\\n",
    "              - q['complier/hurt'] - q['defier/hurt'] - q['always_taker/hurt'] - q['never_taker/hurt']"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'Optimal'"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pulp.LpStatus[min_problem.solve()]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "{'always_taker/always_better': 0.174,\n",
       " 'always_taker/helped': 0.0,\n",
       " 'always_taker/hurt': 0.316,\n",
       " 'always_taker/never_better': 0.0,\n",
       " 'complier/always_better': 0.0,\n",
       " 'complier/helped': 0.03,\n",
       " 'complier/hurt': 0.0,\n",
       " 'complier/never_better': 0.028,\n",
       " 'defier/always_better': 0.02,\n",
       " 'defier/helped': 0.0,\n",
       " 'defier/hurt': 0.0,\n",
       " 'defier/never_better': 0.0,\n",
       " 'never_taker/always_better': 0.0,\n",
       " 'never_taker/helped': 0.0,\n",
       " 'never_taker/hurt': 0.154,\n",
       " 'never_taker/never_better': 0.278}"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "q_min = {partition:pulp.value(partition_p) for partition, partition_p in q.items()}\n",
    "q_min"
   ]
  }
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