Which conditional test should I use?

two_conditional_tests.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "from selection.constraints.affine import constraints, gibbs_test\n",
    "%load_ext rpy2.ipython"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(0.2895728486912888, 0.122875)"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def instance(mu, gamma):\n",
    "    mu = np.asarray(mu)\n",
    "    assert(mu.shape == (2,)) # this is just a 2D example!\n",
    "    while True:\n",
    "        Z = mu + np.random.standard_normal(2)\n",
    "        omega = gamma * np.random.standard_normal(2)\n",
    "        W = Z + omega\n",
    "        if W[0] > W[1]:\n",
    "            break\n",
    "            \n",
    "    # conditioning on Z[1]\n",
    "    \n",
    "    A1 = np.array([[-1,1,-1,1.]])\n",
    "    b1 = np.array([0])\n",
    "    con1 = constraints(A1, b1, covariance=np.diag([1, 1, gamma**2, gamma**2]))\n",
    "    feasible1 = np.array([Z[0], Z[1], omega[0], omega[1]])\n",
    "    assert(con1(feasible1)) # constraints should be satisfied\n",
    "    conditional_con1 = con1.conditional(np.array([0,1.,0,0]), Z[1])\n",
    "    eta1 = np.array([1.,0,0,0])\n",
    "\n",
    "    pval1 = gibbs_test(conditional_con1,\n",
    "                       feasible1,\n",
    "                       eta1,\n",
    "                       burnin=2000,\n",
    "                       ndraw=8000,\n",
    "                       UMPU=False,\n",
    "                       alternative='greater')[0]\n",
    "                       \n",
    "    \n",
    "    # conditioning on Z[1] + omega[1]\n",
    "           \n",
    "    A2 = np.array([[-1,-1,1.]])\n",
    "    b2 = np.array([0.])\n",
    "    con2 = constraints(A2, b2, covariance=np.diag([1,gamma**2,1.+gamma**2]))\n",
    "    feasible2 = np.array([Z[0], omega[0], Z[1] + omega[1]])\n",
    "    assert(con2(feasible2))\n",
    "    conditional_con2 = con2.conditional(np.array([0,0,1.]), Z[1] + omega[1])\n",
    "    eta2 = np.array([1.,0,0])\n",
    "    \n",
    "    pval2 = gibbs_test(conditional_con2,\n",
    "                       feasible2,\n",
    "                       eta2,\n",
    "                       burnin=2000,\n",
    "                       ndraw=8000,\n",
    "                       UMPU=False,\n",
    "                       alternative='greater')[0]\n",
    "      \n",
    "    return pval1, pval2\n",
    "\n",
    "instance([1,1], 1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Null\n",
    "\n",
    "## They're both valid saturated tests"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "P0 = []\n",
    "for _ in range(2000):\n",
    "    P0.append(instance([0,-1.], 0.5))\n",
    "P0 = np.array(P0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%%R -i P0\n",
    "plot(ecdf(P0[,1]), col='red')\n",
    "plot(ecdf(P0[,2]), col='blue', add=TRUE)\n",
    "abline(0,1)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Alternative\n",
    "\n",
    "## But they are not the same\n",
    "\n",
    "### And my guess was wrong..."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "PA = []\n",
    "for _ in range(2000):\n",
    "    PA.append(instance([1.5,-1], 0.9))\n",
    "PA = np.array(PA)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
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       "RK5CYII=\n"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%%R -i PA\n",
    "plot(ecdf(PA[,1]), col='red')\n",
    "plot(ecdf(PA[,2]), col='blue', add=TRUE)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 2",
   "name": "python2"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.8"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}