alinaselega/stirling_approx.ipynb

## stirling_approx.ipynb
{
  "nbformat": 4,
  "nbformat_minor": 0,
  "metadata": {
    "colab": {
      "name": "stirling.ipynb",
      "provenance": [],
      "collapsed_sections": []
    },
    "kernelspec": {
      "name": "python3",
      "display_name": "Python 3"
    }
  },
  "cells": [
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "tro3nhSgFLNi",
        "colab_type": "text"
      },
      "source": [
        "This is a quick notebook to check the consistency of Stirling's approximation to the log-factorial for different $n$."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "fy7-PIFbE9GJ",
        "colab_type": "code",
        "outputId": "fce5aa43-5002-48b5-c8e5-3af1cd5e3284",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 929
        }
      },
      "source": [
        "import numpy as np\n",
        "from scipy.special import binom\n",
        "from scipy import special as special\n",
        "import matplotlib.pyplot as plt\n",
        "\n",
        "n = np.linspace(1, 10000, 1000)\n",
        "log_fact = np.log(special.factorial(n))\n",
        "stirling = n * np.log(n) - n\n",
        "log_fact\n",
        "\n",
        "print(log_fact[0:17])\n",
        "print(stirling[0:17])\n",
        "n[0:17]\n",
        "\n",
        "# OK, values until 150 can be done as factorial\n",
        "\n",
        "plt.plot(log_fact[0:17], stirling[0:17])\n",
        "plt.show()\n",
        "\n",
        "plt.plot(n, stirling)\n",
        "plt.show()"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        },
        {
          "output_type": "stream",
          "text": [
            "[  0.          17.52431733  45.4354283   78.18547919 114.16848752\n",
            " 152.58716172 192.96172229 234.97101196 278.38476633 323.02972509\n",
            " 368.77075792 415.49950668 463.12713656 511.5794903  560.79372487\n",
            " 610.7159037  661.29922726]\n",
            "[ -1.          15.39845435  42.98983518  75.54642558 111.39029298\n",
            " 149.70023642 189.98553906 231.91911692 275.26713064 319.85399657\n",
            " 365.54298899 412.22460677 459.80916814 508.22187073 557.39936885\n",
            " 607.2873266  657.83862195]\n"
          ],
          "name": "stdout"
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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+VTO7CdgEXOltvoTAqZW5BE6vvKERMoeMgr1lzJifyXvZBST1jOLFm0cxqIeG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\nkTCnohcRCXMqehGRMKeiFxEJc/8fe4tNJ0EICT4AAAAASUVORK5CYII=\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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zV/PF2p0k1K/BC8O6c0nX5lTRdZBEJMQpHCrA5t0Hef7TdUxdlk2darE8OLg9\n1/dL1ofYRCRsKBzK0ba8w/x17jomLNpKTBVj1FmtufncNtSvqYvjiUh4UTiUg9yDBbz6xXre/XoT\nxSWO4b2TuPW8tpqBJCJhS+FwCvYfLuSN+Rt5818bOVhQxOU9Erjz/HYknVbT79JERE6JwuEkHDhS\nxHvfbOL1LzewJ7+QQZ2acveF7TQtVUQihsLhBOw7XMi7X23iza82sje/kHPaNeLuAe3opmsgiUiE\nUTiUwd78At76ahNvf7WR/YeLuKBDY249L0UXxhORiKVw+Ak79h/mna828d43mzlwpIiBnZpw23kp\ndE6o53dpIiIVSuHwA9Zt388b8zfy4dJsCktKGNylGbed15b2Tev6XZqISKUImXAws0HAC0AM8IZz\n7k+V+f7OOb5Zv5vX529g3pqdVK9ahavPSGTkz1rTqmGtyixFRMR3IREOZhYDvAQMALKARWY2zTmX\nUdHvvedgAVP+ncW4b7ewfudBGtaO4+4B7fhl35Y0qKUPr4lIdAqJcAB6A5nOuQ0AZjYeGAJUSDgc\nKijmi7U7mL4ih08ytlNQVEKPpPo8O7Qrl3RrrstciEjUC5VwSAC2Bq1nAX3K+00Kikq46tWvWZWz\nn4LiEuJrVmXYGS0Y3juJDs00niAiclSohEOZmNkoYBRAUlLSCT8/LrYKrRrWok/r0zinXSP6tGpA\nbEyV8i5TRCTshUo4ZAMtgtYTvbbvcc6NBcYCpKamupN5o+eH9TiZp4mIRJVQ+bN5EZBiZq3MLA4Y\nBkzzuSYRkagVEkcOzrkiM7sVmE1gKutbzrl0n8sSEYlaIREOAM65mcBMv+sQEZHQOa0kIiIhROEg\nIiKlKBxERKQUhYOIiJSicBARkVLMuZP6LJnvzGwnsPkkn94Q2FWO5YQDbXPki7btBW3ziWrpnGtU\nlo5hGw6nwswWO+dS/a6jMmmbI1+0bS9omyuSTiuJiEgpCgcRESklWsNhrN8F+EDbHPmibXtB21xh\nonLMQUREflq0HjmIiMhPiKpwMLNBZrbGzDLNbLTf9ZwKM2thZvPMLMPM0s3sDq+9gZnNMbN13r/x\nXruZ2Yvetq8ws55BrzXC67/OzEb4tU1lYWYxZrbUzKZ7663MbKG3XRO8S75jZtW89Uzv8eSg13jA\na19jZgP92ZKyM7P6ZjbZzFab2Soz6xfJ+9nM7vJ+plea2Tgzqx6J+9nM3jKzHWa2Mqit3ParmfUy\nszTvOS+amZ1Qgc65qPgicCnw9UBrIA5YDnT0u65T2J5mQE9vuQ6wFugIPAOM9tpHA097y4OBWYAB\nfYGFXnsDYIP3b7y3HO/39v1WSarMAAADdUlEQVTEdt8N/AOY7q1PBIZ5y68CN3vLtwCvesvDgAne\nckdv31cDWnk/EzF+b9dxtvld4P95y3FA/UjdzwRuGbwRqBG0f2+IxP0MnA30BFYGtZXbfgW+9fqa\n99yLTqg+v79Blbgj+gGzg9YfAB7wu65y3L6PgAHAGqCZ19YMWOMtvwYMD+q/xnt8OPBaUPv3+oXS\nF4E7BH4GnAdM937odwGxx+5jAvcG6ectx3r97Nj9HtwvFL+Aet4vSzumPSL3M/+9n3wDb79NBwZG\n6n4Gko8Jh3LZr95jq4Pav9evLF/RdFrp6A/dUVleW9jzDqV7AAuBJs65HO+hbUATb/nHtj+cvi/P\nA/cBJd76acBe51yRtx5c+3+2y3s8z+sfTtsLgb96dwJve6fT3jCzWkTofnbOZQPPAVuAHAL7bQmR\nv5+PKq/9muAtH9teZtEUDhHJzGoDU4A7nXP7gh9zgT8ZImI6mpn9AtjhnFvidy2VLJbAqYdXnHM9\ngIMETjf8R4Tt53hgCIFQbA7UAgb5WpRP/N6v0RQO2UCLoPVEry1smVlVAsHwd+fcB17zdjNr5j3e\nDNjhtf/Y9ofL96U/cKmZbQLGEzi19AJQ38yO3tEwuPb/bJf3eD1gN+GzvUdlAVnOuYXe+mQCYRGp\n+/kCYKNzbqdzrhD4gMC+j/T9fFR57ddsb/nY9jKLpnBYBKR4sx7iCAxeTfO5ppPmzTx4E1jlnPtL\n0EPTgKMzFkYQGIs42n69N+uhL5DnHb7OBi40s3jvr7YLvbaQ4px7wDmX6JxLJrDv5jrnrgXmAUO9\nbsdu79Hvw1Cvv/Pah3mzXFoBKQQG7kKSc24bsNXMTveazgcyiND9TOB0Ul8zq+n9jB/d3ojez0HK\nZb96j+0zs77e9/H6oNcqG78HZCp58GcwgVk964GH/K7nFLflZwQOOVcAy7yvwQTOt34GrAM+BRp4\n/Q14ydv2NCA16LV+BWR6Xzf6vW1l2PZz+e9spdYE/tNnApOAal57dW8903u8ddDzH/K+D2s4wRkc\nPm1vd2Cxt6+nEpiVErH7GXgMWA2sBN4nMOMo4vYzMI7AuEohgSPEkeW5X4FU73u4Hvgbx0xqON6X\nPiEtIiKlRNNpJRERKSOFg4iIlKJwEBGRUhQOIiJSisJBRERKUTiIiEgpCgcRESlF4SAiIqX8f6xw\nTcXpaEkeAAAAAElFTkSuQmCC\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {
            "tags": []
          }
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "jhuuMhFpVA4x",
        "colab_type": "text"
      },
      "source": [
        "Looking at that multinomial factor."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "id": "DPI6Oet6VAqK",
        "colab_type": "code",
        "outputId": "af8e1d2c-729c-4a1e-c001-7bc9e45f1da9",
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 215
        }
      },
      "source": [
        "z_k = 32\n",
        "n = 100\n",
        "\n",
        "num = np.zeros(n-z_k)\n",
        "denom = np.zeros(n-z_k)\n",
        "for i in range(1, n-z_k+1):\n",
        "  num[i-1] = z_k + i\n",
        "  denom[i-1] = i\n",
        "\n",
        "print(num)\n",
        "print(denom)\n",
        "\n",
        "new_num = [x for x in num if x not in denom]\n",
        "print(new_num)\n",
        "new_denom = [x for x in denom if x not in num]\n",
        "print(new_denom)"
      ],
      "execution_count": 0,
      "outputs": [
        {
          "output_type": "stream",
          "text": [
            "[ 33.  34.  35.  36.  37.  38.  39.  40.  41.  42.  43.  44.  45.  46.\n",
            "  47.  48.  49.  50.  51.  52.  53.  54.  55.  56.  57.  58.  59.  60.\n",
            "  61.  62.  63.  64.  65.  66.  67.  68.  69.  70.  71.  72.  73.  74.\n",
            "  75.  76.  77.  78.  79.  80.  81.  82.  83.  84.  85.  86.  87.  88.\n",
            "  89.  90.  91.  92.  93.  94.  95.  96.  97.  98.  99. 100.]\n",
            "[ 1.  2.  3.  4.  5.  6.  7.  8.  9. 10. 11. 12. 13. 14. 15. 16. 17. 18.\n",
            " 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36.\n",
            " 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51. 52. 53. 54.\n",
            " 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68.]\n",
            "[69.0, 70.0, 71.0, 72.0, 73.0, 74.0, 75.0, 76.0, 77.0, 78.0, 79.0, 80.0, 81.0, 82.0, 83.0, 84.0, 85.0, 86.0, 87.0, 88.0, 89.0, 90.0, 91.0, 92.0, 93.0, 94.0, 95.0, 96.0, 97.0, 98.0, 99.0, 100.0]\n",
            "[1.0, 2.0, 3.0, 4.0, 5.0, 6.0, 7.0, 8.0, 9.0, 10.0, 11.0, 12.0, 13.0, 14.0, 15.0, 16.0, 17.0, 18.0, 19.0, 20.0, 21.0, 22.0, 23.0, 24.0, 25.0, 26.0, 27.0, 28.0, 29.0, 30.0, 31.0, 32.0]\n"
          ],
          "name": "stdout"
        }
      ]
    }
  ]
}