John Hawkins 01-22-14 Valuable Boxes.ipynb

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      "# Here we proceed with the assumption that the number of good boxes is 3\n",
      "\n",
      "# Assume as a prior that there are 3 of 13 good boxes\n",
      "priorG = 3.0/13.0\n",
      "priorB = 1.0-priorG\n",
      "print 'P( HG ) = ',priorG,' '*7,\n",
      "print 'P( HB ) = ',priorB\n",
      "\n",
      "# Calculate posteriors that box in hand is good box\n",
      "piG = 9.0/10.0 * priorG\n",
      "piB = 2.0/10.0 * priorB\n",
      "posteriorG = piG/(piG + piB)\n",
      "posteriorB = piB/(piG + piB)\n",
      "print 'P( HG | data ) = ',posteriorG,' '*7,\n",
      "print 'P( HB | data ) = ',posteriorB\n",
      "print\n",
      "\n",
      "# Calculate probability next item is valuable\n",
      "print 'P( V | data ) = ',(8.0/9.0 * posteriorG + 1.0/9.0 * posteriorB) "
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        "P( HG ) =  0.230769230769         P( HB ) =  0.769230769231\n",
        "P( HG | data ) =  0.574468085106         P( HB | data ) =  0.425531914894\n",
        "\n",
        "P( V | data ) =  0.557919621749\n"
       ]
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      "# Here we allow for a distribution of priors for the number of good boxes\n",
      "\n",
      "def p_Hi_given_data(priors):\n",
      "    pi = [(9.0/10.0*float(i)/13.0 + 2.0/10.0*(13.0-float(i))/13.0)*priors[i]\\\n",
      "              for i in range(len(priors))]\n",
      "    return [p/sum(pi) for p in pi]\n",
      "\n",
      "def p_v_given_Hi_data(i):\n",
      "    pi_HG_given_Hi_data = 9.0/10.0*float(i)/13.0\n",
      "    pi_HB_given_Hi_data = 2.0/10.0*(13.0-float(i))/13.0\n",
      "    p_HG_given_Hi_data = pi_HG_given_Hi_data/(pi_HG_given_Hi_data + pi_HB_given_Hi_data)\n",
      "    p_HB_given_Hi_data = pi_HB_given_Hi_data/(pi_HG_given_Hi_data + pi_HB_given_Hi_data)\n",
      "    \n",
      "    return 8.0/9.0*p_HG_given_Hi_data + 1.0/9.0*p_HB_given_Hi_data\n",
      "    \n",
      "    \n",
      "def p_next_valuable(posteriors):\n",
      "    return sum( p_v_given_Hi_data(i)*posteriors[i] for i in range(len(posteriors)))\n",
      "\n",
      "# First we consider the same case as above, only allowing for 3 good boxes.\n",
      "priors = [0.0, 0.0, 0.0, 1.0] + [0.0] * 9\n",
      "posteriors = p_Hi_given_data(priors)\n",
      "print 'Assuming 3 good boxes:'\n",
      "print 'Priors:    ',priors\n",
      "print 'Posteriors:',posteriors\n",
      "print 'P( V | data ) = ',p_next_valuable(posteriors)\n",
      "print \n",
      "\n",
      "# Finally, we consider the distribution Bill gave in class\n",
      "print 'Assuming following prior distribution:'\n",
      "priors = [0.0, 0.0, .2, .3, .3, .1, .1] + [0.0] * 7\n",
      "posteriors = p_Hi_given_data(priors)\n",
      "print 'Priors:    ',priors\n",
      "print 'Posteriors:',posteriors\n",
      "print 'P( V | data ) = ',p_next_valuable(posteriors)\n"
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        "Assuming 3 good boxes:\n",
        "Priors:     [0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]\n",
        "Posteriors: [0.0, 0.0, 0.0, 1.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]\n",
        "P( V | data ) =  0.557919621749\n",
        "\n",
        "Assuming following prior distribution:\n",
        "Priors:     [0.0, 0.0, 0.2, 0.3, 0.3, 0.1, 0.1, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]\n",
        "Posteriors: [0.0, 0.0, 0.15625, 0.275390625, 0.31640624999999994, 0.119140625, 0.1328125, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0]\n",
        "P( V | data ) =  0.603298611111\n"
       ]
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     "source": [
      "We see, happily, that assuming exactly 3 good boxes gives the same result with both methods. Furthermore, the distributed prior gives an even higher probability that the next choice will be valuable."
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