asymarko_example.ipynb

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  "name": ""
 },
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  {
   "cells": [
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# confirm the asymptotic distribution of Theta\n",
      "# for scalar Gaussian case.\n",
      "from __future__ import division\n",
      "from sympy import *\n",
      "from sympy.physics.quantum import TensorProduct\n",
      "init_printing(use_unicode=False, wrap_line=False, \\\n",
      "              no_global=True)\n",
      "mu = symbols('\\mu')\n",
      "sg = symbols('\\sigma') \n",
      "# the elimination, duplication and U_{-1} matrices:\n",
      "Elim = Matrix(3,4,[1,0,0,0,  0,1,0,0,   0,0,0,1])\n",
      "Dupp = Matrix(4,3,[1,0,0, 0,1,0, 0,1,0, 0,0,1])\n",
      "Unun = Matrix(2,3,[0,1,0,  0,0,1])\n",
      "def Qform(A,x):\n",
      "    \"\"\"compute the quadratic form x'Ax\"\"\"\n",
      "    return x.transpose() * A * x"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 1
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "Theta = Matrix(2,2,[1,mu,mu,mu**2 + sg**2])\n",
      "Theta"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{matrix}1 & \\mu\\\\\\mu & \\mu^{2} + \\sigma^{2}\\end{matrix}\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAHIAAAAyBAMAAAByq5IiAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nvVl18V1RY9Fm/0Oa/6Ua/xlfA/3oioGwQ07xAAAAAElFTkSuQmCC\n",
       "prompt_number": 2,
       "text": [
        "[ 1        \\mu      ]\n",
        "[                   ]\n",
        "[        2         2]\n",
        "[\\mu  \\mu  + \\sigma ]"
       ]
      }
     ],
     "prompt_number": 2
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# compute tensor products and \n",
      "# the derivative d vech(Theta^-1) / d vech(Theta)\n",
      "# see also theoremref{inv_distribution}\n",
      "Theta_Theta = TensorProduct(Theta,Theta)\n",
      "iTheta_iTheta = TensorProduct(Theta.inv(),Theta.inv())\n",
      "theta_i_deriv = Elim * (iTheta_iTheta) * Dupp"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 3
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# towards theoremref{theta_asym_var_gaussian}\n",
      "DTTD = Qform(Theta_Theta,Dupp)\n",
      "D_DTTD_D = Qform(DTTD,theta_i_deriv)\n",
      "iOmega = Qform(D_DTTD_D,Unun.transpose())\n",
      "Omega = 2 * iOmega.inv()\n",
      "simplify(Omega)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{matrix}\\sigma^{2} & 2 \\mu \\sigma^{2}\\\\2 \\mu \\sigma^{2} & 2 \\sigma^{2} \\left(2 \\mu^{2} + \\sigma^{2}\\right)\\end{matrix}\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAANQAAAA0BAMAAAAAtAupAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nAAAAAElFTkSuQmCC\n",
       "prompt_number": 4,
       "text": [
        "[         2                        2        ]\n",
        "[   \\sigma             2*\\mu*\\sigma         ]\n",
        "[                                           ]\n",
        "[            2          2 /     2         2\\]\n",
        "[2*\\mu*\\sigma   2*\\sigma *\\2*\\mu  + \\sigma /]"
       ]
      }
     ],
     "prompt_number": 4
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# this matches the computation in eqnref{gauss_confirm_theta}\n",
      "# on to the inverse:\n",
      "# actually use theoremref{inv_distribution}\n",
      "theta_i_deriv_t = theta_i_deriv.transpose()\n",
      "theta_inv_var = Qform(Qform(Omega,Unun),theta_i_deriv_t)\n",
      "simplify(theta_inv_var)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{matrix}\\frac{2 \\mu^{2}}{\\sigma^{4}} \\left(\\mu^{2} + 2 \\sigma^{2}\\right) & - \\frac{2 \\mu}{\\sigma^{4}} \\left(\\mu^{2} + \\sigma^{2}\\right) & \\frac{2 \\mu^{2}}{\\sigma^{4}}\\\\- \\frac{2 \\mu}{\\sigma^{4}} \\left(\\mu^{2} + \\sigma^{2}\\right) & \\frac{1}{\\sigma^{4}} \\left(2 \\mu^{2} + \\sigma^{2}\\right) & - \\frac{2 \\mu}{\\sigma^{4}}\\\\\\frac{2 \\mu^{2}}{\\sigma^{4}} & - \\frac{2 \\mu}{\\sigma^{4}} & \\frac{2}{\\sigma^{4}}\\end{matrix}\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAWkAAABYCAMAAAAA9PbAAAAAP1BMVEX///8AAAAAAAAAAAAAAAAA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\ntp3p2cEG7wAAAABJRU5ErkJggg==\n",
       "prompt_number": 5,
       "text": [
        "[     2 /   2           2\\         /   2         2\\         2]\n",
        "[2*\\mu *\\\\mu  + 2*\\sigma /  -2*\\mu*\\\\mu  + \\sigma /    2*\\mu ]\n",
        "[-------------------------  ------------------------  -------]\n",
        "[               4                         4                 4]\n",
        "[         \\sigma                    \\sigma            \\sigma ]\n",
        "[                                                            ]\n",
        "[       /   2         2\\             2         2             ]\n",
        "[-2*\\mu*\\\\mu  + \\sigma /        2*\\mu  + \\sigma       -2*\\mu ]\n",
        "[------------------------       ----------------      -------]\n",
        "[              4                          4                 4]\n",
        "[        \\sigma                     \\sigma            \\sigma ]\n",
        "[                                                            ]\n",
        "[               2                                            ]\n",
        "[          2*\\mu                    -2*\\mu               2   ]\n",
        "[         -------                   -------           -------]\n",
        "[               4                         4                 4]\n",
        "[         \\sigma                    \\sigma            \\sigma ]"
       ]
      }
     ],
     "prompt_number": 5
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# this matches the computation in eqnref{gauss_confirm_itheta}\n",
      "# now check conjectureref{theta_asym_var_gaussian}\n",
      "conjec = Qform(Theta_Theta,Dupp)\n",
      "e1 = Matrix(3,1,[1,0,0])\n",
      "convar = 2 * (conjec.inv() - e1 * e1.transpose())\n",
      "simplify(convar)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{matrix}\\frac{2 \\mu^{2}}{\\sigma^{4}} \\left(\\mu^{2} + 2 \\sigma^{2}\\right) & - \\frac{2 \\mu}{\\sigma^{4}} \\left(\\mu^{2} + \\sigma^{2}\\right) & \\frac{2 \\mu^{2}}{\\sigma^{4}}\\\\- \\frac{2 \\mu}{\\sigma^{4}} \\left(\\mu^{2} + \\sigma^{2}\\right) & \\frac{1}{\\sigma^{4}} \\left(2 \\mu^{2} + \\sigma^{2}\\right) & - \\frac{2 \\mu}{\\sigma^{4}}\\\\\\frac{2 \\mu^{2}}{\\sigma^{4}} & - \\frac{2 \\mu}{\\sigma^{4}} & \\frac{2}{\\sigma^{4}}\\end{matrix}\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
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       "prompt_number": 6,
       "text": [
        "[     2 /   2           2\\         /   2         2\\         2]\n",
        "[2*\\mu *\\\\mu  + 2*\\sigma /  -2*\\mu*\\\\mu  + \\sigma /    2*\\mu ]\n",
        "[-------------------------  ------------------------  -------]\n",
        "[               4                         4                 4]\n",
        "[         \\sigma                    \\sigma            \\sigma ]\n",
        "[                                                            ]\n",
        "[       /   2         2\\             2         2             ]\n",
        "[-2*\\mu*\\\\mu  + \\sigma /        2*\\mu  + \\sigma       -2*\\mu ]\n",
        "[------------------------       ----------------      -------]\n",
        "[              4                          4                 4]\n",
        "[        \\sigma                     \\sigma            \\sigma ]\n",
        "[                                                            ]\n",
        "[               2                                            ]\n",
        "[          2*\\mu                    -2*\\mu               2   ]\n",
        "[         -------                   -------           -------]\n",
        "[               4                         4                 4]\n",
        "[         \\sigma                    \\sigma            \\sigma ]"
       ]
      }
     ],
     "prompt_number": 6
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "# are they the same?\n",
      "simplify(theta_inv_var - convar)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "latex": [
        "$$\\left[\\begin{matrix}0 & 0 & 0\\\\0 & 0 & 0\\\\0 & 0 & 0\\end{matrix}\\right]$$"
       ],
       "metadata": {},
       "output_type": "pyout",
       "png": "iVBORw0KGgoAAAANSUhEUgAAAFgAAABLCAMAAADDCbAzAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA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\nGckr4SSG0EcAAAAASUVORK5CYII=\n",
       "prompt_number": 7,
       "text": [
        "[0  0  0]\n",
        "[       ]\n",
        "[0  0  0]\n",
        "[       ]\n",
        "[0  0  0]"
       ]
      }
     ],
     "prompt_number": 7
    }
   ],
   "metadata": {}
  }
 ]
}