MyMC3.ipynb

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      "name": "MyMC3.ipynb",
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      "authorship_tag": "ABX9TyMJ9qj/Qxn55GR4AbJ95ohk",
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      "cell_type": "markdown",
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        "id": "view-in-github",
        "colab_type": "text"
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      "source": [
        "<a href=\"https://colab.research.google.com/github/mattbarrett98/Hamiltonian-Monte-Carlo/blob/main/MyMC3.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
      ]
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    {
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      "source": [
        "# for reading RData file into a pandas dataframe\n",
        "!pip install pyreadr\n",
        "import pyreadr\n",
        "\n",
        "# for fast maths\n",
        "import numpy as np\n",
        "\n",
        "# for visualisations\n",
        "import seaborn as sns\n",
        "from matplotlib import pyplot as plt\n",
        "\n",
        "# for autocorrelation plots to monitor efficiency of Markov chains\n",
        "import statsmodels.graphics.tsaplots\n",
        "\n",
        "# for normal pdf\n",
        "import scipy.stats\n",
        "\n",
        "# for finding maximum likelihood estimate\n",
        "from scipy.optimize import minimize\n",
        "\n",
        "# MCMC library to compare results with\n",
        "import pymc3 as pm\n",
        "import arviz as az\n",
        "\n",
        "# for hiding warnings\n",
        "import warnings\n",
        "warnings.filterwarnings(\"ignore\")"
      ],
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "data = pyreadr.read_r('contracep.RData')\n",
        "data = data['contracep']\n",
        "data"
      ],
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        "colab": {
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          "height": 1000
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      "execution_count": 46,
      "outputs": [
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              "      <th></th>\n",
              "      <th>age</th>\n",
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              "      <th>use</th>\n",
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            "text/plain": [
              "     age  total   use\n",
              "0   18.0   35.0  16.0\n",
              "1   19.0   27.0  17.0\n",
              "2   20.0   29.0  12.0\n",
              "3   21.0   27.0  13.0\n",
              "4   22.0   35.0  26.0\n",
              "5   23.0   34.0  23.0\n",
              "6   24.0   33.0  28.0\n",
              "7   25.0   26.0  23.0\n",
              "8   26.0   28.0  26.0\n",
              "9   27.0   29.0  23.0\n",
              "10  28.0   22.0  22.0\n",
              "11  29.0   22.0  18.0\n",
              "12  30.0   30.0  28.0\n",
              "13  31.0   28.0  28.0\n",
              "14  32.0   31.0  29.0\n",
              "15  33.0   32.0  32.0\n",
              "16  34.0   26.0  24.0\n",
              "17  35.0   35.0  32.0\n",
              "18  36.0   30.0  27.0\n",
              "19  37.0   26.0  22.0\n",
              "20  38.0   29.0  27.0\n",
              "21  39.0   24.0  23.0\n",
              "22  40.0   29.0  25.0\n",
              "23  41.0   28.0  13.0\n",
              "24  42.0   27.0  13.0\n",
              "25  43.0   29.0  16.0\n",
              "26  44.0   31.0  13.0\n",
              "27  45.0   27.0  12.0\n",
              "28  46.0   43.0  27.0\n",
              "29  47.0   22.0   3.0\n",
              "30  48.0   27.0   8.0\n",
              "31  49.0   31.0   2.0\n",
              "32  50.0   29.0   1.0"
            ]
          },
          "metadata": {},
          "execution_count": 46
        }
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      "cell_type": "markdown",
      "metadata": {
        "id": "aV9Oxiotix8J"
      },
      "source": [
        "We first visualise the data: since we have varying numbers of observations depending on the age, we plot the proportion of women which use contraception, rather than raw figures."
      ]
    },
    {
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      "metadata": {
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          "height": 305
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        "id": "2GgGnwPrke0G",
        "outputId": "243e970c-0ffe-4a5a-d9f8-3ca38cb1e260"
      },
      "source": [
        "age = np.array(data.age)\n",
        "use = np.array(data.use)\n",
        "total = np.array(data.total)\n",
        "proportion = use / total\n",
        "data['proportion'] = proportion\n",
        "\n",
        "sns.set_theme(style='darkgrid')\n",
        "sns.scatterplot(x=age, y=proportion)"
      ],
      "execution_count": 47,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "<matplotlib.axes._subplots.AxesSubplot at 0x7fd8f906cc90>"
            ]
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          "execution_count": 47
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          "data": {
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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "First we will use the Hamiltonian Monte Carlo algorithm with the standard normal as our target distribution to show the algorithm does provide us samples of the correct distribution. To implement HMC we need the negative log of the target density and its gradient, denoted U and grad_U in the function below. The density of the standard normal is $$\\pi(x)=\\frac{1}{\\sqrt{2\\pi}}e^{-\\frac{x^2}{2}}$$and the taking the negative of the natural log of this gives $$-\\log\\pi(x)=-\\log\\frac{1}{\\sqrt{2\\pi}} - \\log e^{-\\frac{x^2}{2}} = \\frac{1}{2}\\log 2\\pi + \\frac{x^2}{2}.$$ The gradient of this is simply $$\\frac{d}{dx}-\\log\\pi(x)=x.$$"
      ],
      "metadata": {
        "id": "Wf7q_h88Acbm"
      }
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 572
        },
        "id": "LhdqugaGFxvs",
        "outputId": "fe388850-af5c-4cf2-e4b2-802d50f84407"
      },
      "source": [
        "def HMC(U, grad_U, epsilon, L, current_q, n_sim):\n",
        "    \"\"\"The Hamiltonian Monte Carlo algorithm. It simulates a Markov chain \n",
        "    which has a target distribution of our choosing as its invariant \n",
        "    distribution. Hence after the Markov chain has converged to its invariant \n",
        "    distribution all subsequent values in the chain will be samples from our \n",
        "    target distribution, albeit correlated samples.\n",
        "\n",
        "    Parameters\n",
        "    ----------\n",
        "    U : function which calculates the value of the negative log of our target\n",
        "    distribution. \n",
        "\n",
        "    grad_U : function which calculates the gradient of the function U.\n",
        "\n",
        "    epsilon : float. The step size used to discretise the differential \n",
        "    equations.\n",
        "\n",
        "    L : int. The number of steps to take in the simulation of the differential\n",
        "    equations. \n",
        "\n",
        "    current_q : The inital value of the Markov chain. \n",
        "\n",
        "    n_sim : int. The number of samples to generate.\n",
        "\n",
        "    Returns\n",
        "    -------\n",
        "    samples : an array containing the values from our markov chain. \n",
        "\n",
        "    n_accepted : The number of proposals that were accepted. Useful for tuning \n",
        "    epsilon and L. \n",
        "    \"\"\"\n",
        "    samples = []\n",
        "    n_accepted = 0\n",
        "    for j in range(n_sim):\n",
        "        q = current_q\n",
        "        p = np.random.normal(size=1) \n",
        "        current_p = p\n",
        "        # perform L leapfrog steps\n",
        "        p = p - epsilon*grad_U(q)/2\n",
        "        for i in range(L):\n",
        "            q = q + epsilon*p\n",
        "            if i != L-1:\n",
        "                p = p - epsilon*grad_U(q)\n",
        "        p = p - epsilon*grad_U(q)/2\n",
        "        p = -p\n",
        "        # calculate inital and proposed potential and kinetic energies\n",
        "        current_U = U(current_q)\n",
        "        proposed_U = U(q)\n",
        "        current_K = sum(0.5 * current_p**2)\n",
        "        proposed_K = sum(0.5 * p**2)\n",
        "        # Metropolis acceptance rule\n",
        "        prob = np.exp(current_U + current_K - proposed_U - proposed_K)\n",
        "        if np.random.uniform(size=1) < prob:\n",
        "            current_q = q\n",
        "            n_accepted = n_accepted + 1\n",
        "        samples.append(current_q)   \n",
        "    return np.array(samples), n_accepted   \n",
        "\n",
        "def U(x):\n",
        "    \"\"\"define potential energy function (minus log of target distribution).\"\"\"\n",
        "    return 0.5*x**2 + 0.5*np.log(2 * np.pi)\n",
        " \n",
        "def grad_U(x):\n",
        "    \"\"\"gradient of potential energy function.\"\"\"\n",
        "    return x\n",
        "\n",
        "# Tune epsilon and L appropriately to obtain low autocorrelation and optimal\n",
        "# acceptance rate of roughly 0.65 (https://arxiv.org/pdf/1206.1901.pdf)\n",
        "n_sim, epsilon, L, current_q = 20000, 1.7, 10, 0\n",
        "samples, n_accepted = HMC(U, grad_U, epsilon, L, current_q, n_sim) \n",
        "samples = samples.reshape(n_sim,)\n",
        "\n",
        "# show acceptance rate, trace plot and autocorrelation of Markov chain\n",
        "acc_rate = n_accepted / n_sim\n",
        "print(acc_rate)\n",
        "sns.lineplot(x = range(n_sim), y = samples)\n",
        "statsmodels.graphics.tsaplots.plot_acf(samples);"
      ],
      "execution_count": 13,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "0.6134\n"
          ]
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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/bucFyWWTyswSJFmvN4frtqSi3n6Tv8TS1VdITX0zPl55mLNpSQpb5wchL3++06m5bP4Xu/Acy2/CbLHiUok2CdB8jS6aa5QaOeb447zI0k7Mxjaq8dBZ8S9LxcQMx2U27S9wauf+bsMp3nVtk2o8N+Va1u/da02eHcAdR5yH0lfUNOH7jS41JJdd7BVZu9l+uAgfrVCnm2C5QNPZD63/BtfjJSdlQU19s6q5hFxfLAPC19n/uo4adjhHVoYRnSjPk+YyG850GBLtPFqMAekJWLLuOH7fX4A5UwZ6vM2/vyv8zsOX6aIm7xZQHKg3dNh778s/c+mmt2bbOdnb2neiBA8sXC/Y5dATYoLgeyLHEGzxQl9pLlz/DDmh+ol3NuHJ9zZ7VB4liA2m63aIf/qokpm2WCq+ZiFb1kjbiHTbS+Ythzy/fuSmZbbTOCurLoK8LB4eeE9WF7Pusx9uwXcbTqGuQf4FxpZQavcREQmsHAroGtDktlc6Djrx1iAQIXz/lpKKenszjdq/UW8nMxNDrQnRxXLNLFtZ0yiqF5EY3gy5u4+VsPYe8yZFmmvKy8sxe/ZsnD9/HiaTCampqZg3bx7attU+v3JZVSNvX3JftmbbOfymwVt52+M3wzDYsJc9FwpbN0kbtqBom0uzoqYRT4vsYaIlx6YMtxiscEzeuK8AYwYkK7tRSL95KJkp8cu1xzxa33Ucw5J1wqPRfVFxRT2KOX5D3qJITd5gMOCBBx7A2rVrsXLlSnTq1AlvvPGGEptWxC629KssP4DzxTUoq2rAxZIafLXOs4tUKaJzY7R673tpL8GWrDsG16hlOzSnLlVhvcsFWlPfjKZmi0CKZfdgYfuRSg3w6vbFZt/2xyu9myqAs8eMH+OqHHD5bQ9/ZUbqMVJ74hrXmeh8mSJBPjY2FkOGDLH/3b9/fxQUyEvApaXy6kbM+mAL3v7Pft6aKuC7WRW50i9w2XGkGGYLe7BjezH5wr+3qfYiVEnr91yUPUG0reuszf6Tyvel90VSKxRsxDbFWVxm81FyXgZvNH8tEOiZ5UsUb5O3Wq1YtmwZsrKylN60okp4MtG5po1ls2Gf/93EAKCuUXwbP9uPpb7RIvlGooWv1h0XTF0gNhbk/qhO6lu1adHUL7byw5a2VylC3Z8rWL6vrG3Cf7efE7xB/CfvpEdl04LiXSjnz5+P8PBwTJkyRdJ68fGRSheFlxJ9e6VISIiy/7/JpF3P1Z2H3YfTR0e3cfssISEK0Ze4B/GEhYVwfmfk+KU7HgOx+I4V5/aCg4WXgfwEWcEh7HUj274aJW7WFBIk69gI+WHzGTwwqbesdcPCQuxlknq98v1bbN+xzWC1SKHfZHx8JIytF2FoqHvZXccEBIeG4KnW5GIj+qfwbvvnHefx2N0DBMsQEREqtriqnHtHikabBQsW4Ny5c1i8eDGMRmkPCaWlNbL6Eat9gJRSUnI1l3YT2zRkGioqqXb7rKSkGlVV3E1WDTy9frhOYwnLfoQ08czcxbW9f362XXAZT5zjyFZp21dZmbSBRU3NFlXKufVgIbbKzJ3S0NBsL1OjlKc/8B9zvu8OSJg8m8/WfRft41zWiuhuvGbT1XEnpSLOnZhzVVsrfm4KuefeaDSIqhwr1lzz1ltvIT8/H7m5uTCZ3PNjkKt8rcccVxImuRObiJ1SUQw5h8qxffdKRb2o5jcpuN5h2LrKrdkqfxyDv/t6rfdnPHMlJYsrAFwWeP/mavaHWwSX8aVusYrU5E+cOIGPPvoIXbp0wT333AMASElJQW5urhKbJxo5LGEUrxiuMwOJceiMuGnjuHhzLti8PRcxbXxP6ZNdMAwYhpGVWsDX7D9RglsGdxJeUEVS35dJ7Ql0RcTMUqd4mjq9TZEg36NHDxw75htdDv3ByUu+ldFPDjnzZEqZGUgMhmGwlm1kpkZdn+RW3g6dLceMBb6ZVpnIIzetsxp0kbvGH5xyCOxKNmdohXXsgZc9uHADa/IwOU8MRDl8uWw27S/AwIwEL5ZGmvlf7NK6CIqjIO8lr34lPPsRkUZu9kn1+Fp5tMGWvtfms/8exf5T+h93oGZuKKkoyBNCeEm5dVVUNwo+SSmRcpuIF7gJygivBh/r5ukPfO7BwgMb9xXgf97cCIZhcExgDmRHxeX1aGSbo8CRrw4X1ym/D/Kb9mmb/EevfuBJ30wCgy3NgRLpDoh2/DrIV9c1YeFX+ntR4gtsWSMJUdpJH5svVu/8Osgv+ta7qQkCCz1TS6Wj1hqiI34d5IUm8CXy1emgm6e3Hb9QgZ82nxFekBAv8usgT9TjC/3g/U1xeb3ugjw9nfg/vw7yGk+dSAghPs+vgzwhRF3LfjmhdRGIhyjIE0I4CU3LR3wfBXlCCNExCvKEEKJjFOQJIUTHKMgTQoiOUZAnhBAdoyBPCCE6RkGeEEJ0zK+DvC/NvkIIIb7Ir4M8IYQQfhTkCSFExyjIE0KIjlGQJ4QQHaMgTwghOkZBnhBCdIyCPCGE6BgFeUII0TEK8oQQomMU5AkhRMcoyBNCiI5RkCeEEB2jIE8IITpGQZ4QQjRUVduk6vYpyBNCiIZqG5pV3T4FeUII0TEK8oQQoiGDwaDq9inIE0KIhtQN8QoG+TNnzuDuu+/GuHHjcPfdd+Ps2bNKbZoQQohMigX5F198Effddx/Wrl2L++67D3PnzlVq04QQol8qV+UVCfKlpaU4fPgwJk6cCACYOHEiDh8+jLKyMiU2TwghuuUXzTWFhYVITExEUFAQACAoKAjt27dHYWGhEpsnhBAiU7DWBbCJj4/UugiEEOJ18fGRSIiPUG37igT5pKQkFBUVwWKxICgoCBaLBcXFxUhKShK9jdLSGlitjBLFIYQQv1FaVosgq1XyekajQVTlWJHmmvj4eGRmZmLVqlUAgFWrViEzMxNt27ZVYvOEEKJbarfJK9Zc89JLLyE7OxsffPABoqOjsWDBAqU2TQghuuU3QT4tLQ3ffvutUpsjhBCiABrxSgghWvKHfvKEEELkMagc5SnIE0KIjlGQJ4QQDamchJKCPCGE6BkFeUII0TEK8oQQoiGaNIQQQohsFOQJIUTHKMgTQoiGqHcNIYQQ2SjIE0KIhvxiZihCCCEyUe8aQgghclGQJ4QQDVFzDSGE6Bn1riGEECIXBXlCCNEQNdcQQgiRjYI8IYRoiBKUEUIIkY2CPCGE6JhfB/leXeK0LgIhhHiEEpTxCAkO0roIhBDiEepdw2Pa+Ayti0AIIT7Nr4N8TGSo1kUghBAPUe8aQgjRLWqTJ4QQIhsFeUII0TEK8oQQomMU5AkhREPUJk8IITpmoN41hBBC5KIgTwhxcu8NPbQuQmCh5hp5jGo3dBFCiAIorYEIw3p14Pxu7LXJsrfbv3s72esS3xATYcI/HxqqdTEI0YzfB/mVb96OB2+7xu1zW0W+W1K07G0/Orm37HWVNKpfktZF8Ko5UwYqtq2YSBM6tA0Xvfw9Wd0V2zcJHMN6JTr9LaVySb1rPMQw8tf1nRYfnymId3jhn/v3P/dVfyd+KixU3eyuwUHeu55jIk1e2c/o/s5BvVP7SNHr0sxQRFEZqZSDHwB6pMSwf+HBD07KE4OawkyeBekRvdV9cuzQNkLV7Tsa0CPBK/sJ8uKNSyqPg/zLL7+M8ePHY9KkSbjnnntw8OBBJcrlEzx5CvBVQ1X+ASshMa6Nx9uY+7dBvN+H8gTCZ+7pj9H9O0rep688+cVFeZad1WgU9w/p2TmW87t+afEelSE+Osyj9W3CQrSZc6JTgpSavIoFgQJBftSoUVi5ciVWrFiBhx9+GE899ZQS5fLYpBFdkJ4Sg2vTne/kfD9uX6XkRZAs4eLztnYxYfg0OwtR4Z4/YnsywKRXl7aYNr4nMgP4qUdMpwM1K0Hjh3RWZDsj+mpTqUlL5nhSZOHzg6HGjh2LkJAQAED//v1x+fJlWK1WjwvmqXaxbZA9ZSDCw4KdPn/ST9ti5824TpHt+Eptk02b0GDhhVjw1SiV1iY0GPOms58LKS/5g0TWlgPVDQNTMKrf1aep1x6W10PKL46zykWU96visHTpUowZMwZGo/R7R3y8/BpmQkKU22fRUWGsn3dOFh8Q2rOs701fvjQOn/x0CDNu74UgGcfUF3RPicHJi5Wilg0ONrKeMyGv/300bnvmJ6fPYuPCW7cZxLrNhHbs+4mKDLUvb2J56msTGoQBvdhrh09PGYQ/sleJKvNT916LN5buFrWsVMHBnl0rCQlRMJncQ8N9N2fg63XHru6HpynExHPDDnV4sZs1qBPydl1gLUObNiH2v9vGyWvHb+ul9v+4WOf3MQkJURg7MAXrd18UXLd9QpSqL18Fg/zkyZNRUFDA+t2WLVsQFNRywlavXo2VK1di6dKlsgpSWloDq1X6819CQhRKSqrdPq+qbmD9vKy8TnCb7/59JC6V1KCsrNbp8+gIE6pqmySVb2TfJGw6UChpHRtzQzOmjUuHuaEZFfXNsrahtWaz+Kc6s9nKes6EsK1TXl7buk0L6/clV9j3U1PTaF++udni9r3FwnCWsbKiDm8+NgL/WX8S2w8X8ZbZajbjpkGd8EtrgAsJNmLOlIF4+fOdvOuJYZZwzNmUlFSjqcns9vmN1yY7BflmlmVsmhq5vzM3Xy1fj47RyOMoQ3391d9aqctvUaxygfWiwkNQXef5b6u8wjmulJRUY+pN6aKC/JUrNbL2aTQaRFWOBYP8jz/+KLiRX375BW+//TY+//xztGunzQCiSSO6YMUfZ+1/t2GpiYgV2SYEGZ3d22PHDkjG7dd3xfQctsuSnZbNI8ntInDpirwfh68b3LM9dh4tFlxO7fZOV3FRoWgXI+6l4b039rAH+c7tI5HaQdsnR0eMiAZ3tiWMhpbuhOXVjZzrjerfEV+tbblZiP19GAAsfGQYThZUorisHss3nxFc58W/Deb9/o5R3TC0VyJmf7hVXCH8lMdtAOvXr8drr72GTz75BCkpKUqUSRZb+11kmxBMvzUT/bp79nafDdeFP6CH+jc2OU2L8x8Yosi+X7qf/8fiC954dLj9//89eyznS0Guw/i3W3oCAPo6XDfdJbw8c2Tbd2eevtKuN5/wsBCOJaVTomlPzDM12zLvzhqLqeMyeNcbO0D6KHQGLe/Zhl7TAROHdxG1TmqHKNYTHh1hwuSRXTFhWCpiFZonOrKNcudPaR5fDc899xyam5vxxBNP4Pbbb8ftt9+O8vJyJcomS0iwEdf3TVJ9gIGauiZF40/Xd3X6jC8I9PWwu5ojtpw/nRPF1TB7dW0ra5+DerYHwB2AXYPGvTe6J9Bq69DlzqkLoMjLYFS/jvg0OwuJcVfbVieN6Oq2nOuxnjEhk3Obtn8XH9sxGzNAepdNLlxPEn8a2dWjNB+uEmPdu7qmdmh5+WyrEM2YkIkHJ7qPSLfJTI1DdHgI6zJK9N5hO/0JsWG4bURXRWOE43Xjazx+8bpt2zYlyuHXlL6h/GMaex/vyaO6oXtyDF5fttfp8yfv6ufWhPTCX9m3ITQYymCAYDUuOMgIs0W5HlQThqZiF0/Ti2PMXvjIMLSLbSP6Za4Yz/+VPY0CW39x1xsO2zJSLgdb7w8lE+oltm2DiLBg1DY4t4tfl5mIuMhQrN9zSZH9/Hlsd/yRf5l3mYiwEPTv0Q7/WnWY9fuocBMWPTESADiXAcQ1HwkZ2TcJMZEmjOnPfqN7eFIvfLTikMf78TX+2WWDhxbZJ/kvQOXKc9vwLqL6bndoG45uHdm788VFcbcXi+0XPvSaROGFbDgOzZuPjRC/idZtTBiWinYstUdPpXUU3ywTHCT+J8MbllwuC6W6nD99dz/cOTqN83ux40SEYmpwkMGenkDKMdHS/bdm4o5RaU5PfTYRYcFIkHltmUJ8+9/v26WTIC4qFLcM7Yyn7+7HvyDD4IW/DlJsRJ2ve/Fvg0X3MRY8dgBSEiIl37fuZkn6JWVUJgPg0+wsp+Blu5dPuTmdfZ3WIOVWVF9txVMoyvfuGq9I0GUkFMjkYZdN7jJc5dh2LqkeJ7BwcJAR997YA8//dRDromzvZRJir8aOx+/og/kzhN99De/NnSlXbboJ8gaDAXeN6Y6keN/et2gAABF3SURBVOF+sd06Ris2uMjXpXaIEt1eKOaFndAPLLS17/QIh4t63HWdMZ/leN8ytDOevXeA4D6Ded46Rwi8sPTVVzMGl/96c5+uEtly7nDE+Osyhd8zKO1vt/R0GihnMBgwYViq23KP39EHD0x0fkci5vjeNKgTZ96hAenunSpsL81TE6NwbXqC2xMAW0oHLS9D3QR5qcSOrnyZY3SjWLFeyoLnaNx1nWSv62nTpy3rX+9uwi+D7xrTXbCJ6Pbru2J4H+m1ICk1UT6PTO6jyHaEKFFex/cF90iY3Wk8y/XCVZoZE66+IDUFt9zQR7Hk+dGi48O16QkYrmBupqnjMlh7zdjOFVfvq7/fJfxE7E0BEeS5coR/mp0luG6n9pGYNKKL7H1rkStHzNOMGNdLzfvBd4fg+dFHR7TcGPqw1IBuv76rh10CPQs2oweKu2E6prS1VSDEZIPkCoZSu+S1iwlz6po4ok8SwkVWZBzTBwixnYrM1LYICTbio1ljcNcY93cAU8dlYOyAZPTuJq/HFQD0bu15lMrTu0uoR5eUsx+tQM4kXxQQQf7m6zqjYzvvpTd1JHUwzmtenMVo6rgMTOVo0waA+2/piX/NHuP0mRpJqeKiQvHFi+MweWQ35TbK2U9e2vlwbCl64k72vEe5T43CgoeH2f++eXAn3JPV3Snouk4qIVSKIVJebgO4Y7T7sevK8fLdlcFgwFN/ca593jGqG9o7ZAO1jUMIMhrx6oND7BPqhAQbWW9UcVGhmDouQ9L7Addr7brMROQ+NYp3kFh6J+XyFsXHhGHBI8OcJukJZUndIGuAXesqWsyfGxBBXmsPT+olelnW9lEebC8whbqbpSREID46FGMHJGPstc4D2NI7tTyCTr05HQaDwWs5c9pGh4lOcSuFLf689vBQPHmX9OR04WEhuP/Wnnjj0eHozzLorUdKDNqEBsPkEAyCg4y4+brOTgFuxsRrcBvLIB5bBwC3WrfEm+nQa8Q3af179lj7/+c80nJz6tMtHneO7oY5U1u6k3ZNikaOw43LsUdKUnwEa/DzFNu1JjdpnVwJsW2u9tBjGAzq2R69uiiXjVTtCVnYePcI6kTnxEicLxKfb0JsJrzH75DW/vvxs2NkvVicde8AWCzsUSSoNTDxdSdrx9EziW2LtvZLrd9/JsaFO72AlvJEMrIvd3NGuxhx3e6MBgMmj+qGExcrcPR8hf3u85esNKR3jmVNo6G41pPgeDNt73CeJwzron4ZvCAm0mQfyNZWZIoJJw4/KqPBgIcm9cLf392MpPhwFJbWYUSfDrhQXIPbXJpxn713ACLCBEKqBnNUUE1eBNfHUaEeHa5iXWrbGZ1i3WpCMREmt9z3QoKDjJJq2lHhLeWODjd5NLHEhOGpnO2kE4a29HromxaPsdcm43/+pO48uVwvLLl+S77W2yYkOAiDHUbG3n9rT9Y2bldfvjiOt6kN0P7GyoZvhLBS3n78entTmRLjZqLCTfg0Owu9urS0/4eZgvHo5D72d0k2malxnKPDvZ1DyRHV5L0grWM0hl6TiG2tmQkfu6MPDp0pwyerD8PMUaNW2kv3D1YkT4fB0PJYfW1GAs4VuWdjHNmvI0a2vsibejN/DhNVcfWT15jQE4TtqWHpuuOcy8x/YAjiosMw9toUfMWznC9Z+D/DEBEW4vXmFznSU2KwYe8lpEiYp9WX+f4RV1l8dBhKqxokreNaORD64RoMBvRMjcO2w0W4LrM9ItuEYMg1iRhyTaKkjJZisU0iLDb/jBC5L17jY8IQGhKERpb0vf7o/SdHwiIjNbaN0M2Hr0tlsshOBL40e6XYZi01zJiQiaMS0mAM7dUBPVPjFEte5sjxnDx5V1+cLVY/Syw110j4KYgNcE//pR/6ckyfZuJ4YaVUmtlPs7M8ymh4fZ+WngVSeyOldojCXWPYu6oCLb0UPnxmNICWpik1xEeH2meJYlSuyoeHhciaplBu4L1hoHYZXv3diD5JyP6rtEyqigd4luuwb1o7zJikbnMmQDV5UVzPD1/cCA4yone3eJTWNuPAySuIaON8iLnWfeR28T1w1GR7wpBKKHe3zdszr1dtGPzrj7rnw+FqC9V6knapTcVpHaPxm4cTSfla01UgeGxyb1gZ4ODpUs3KEPBBXs5vvUN8BA6ddU+nPGZAMm5tnYB48ug0GBkrRvQRN6AozINJTvyJWrV4sXztxatYSndZHJjO/k6FKGtgRstLdccgP2lEF2zczz7bnhoCI7KIMDAjAbuPlYha9i9ju6N/93Z48//2OX3ep2tbe5bEoCCjU9c7JVKl2rSPbYPiinq3z9WcuOCuMWn4dsMp+9/XdInDj7+rtjuPaV1T5ySxYH8ek4bQkCDWPvqeeExid12lvXT/YJ+eaENptw3vgsIrtRiYkYCIsBD8ScmBfwICpk0+xNZE4FKVs/3FlimRb1tyJ8hQoib54v2DnWZCAlqaS5SaCYrNtRkt3TttGfjSOsaISguhOT+tudtuBaEhQbhhYIq8XDAuNxRfuu91ToxiTfkrx41+8L4iIbYNnv/rIMndr5UQMDX5xyf3waYDBegY7zyiVMqFz7XshOGpOH6hAj1kDLGefmum0/BxMdqEBrt1RVN7ftDEuHDMvKMPeorMOe+plIRIZHTmPp7jruuMfSevIDNVbm4UbUKeJ3t949Hh9sFq5Kr7bkrHfTfxjxkIZAET5ONjwngfkXgHKwhUotI6xuD9p0bJKpfkJGAaGiBxsJYnhFJBd02KxuJnxghuh2XeJvhCnVZOzdzTmq+fPtQQDwVMkNfa4J6J2HGkGLcNd583lChPzDuQByZmonN7dZ+AiH4M690Bv+6+qOicyt4Q8EHeWy/owsOCRU2QoYXkdhHox9GvX8jzUweihOUlsK/gq70qmXtcrKHXJOLExUrBqeZsOWU8ST9BlNU1Kdo/3kO5CPggb2MwAK8+OATVdc1aF8XrPHlhm5YcgzSOyRN8go/1mRwzIBkj+3UUTMF70+BOSE6IsOdLkUOJ+sstQzsjRqd51gMFBXkHSfERSGJ5Ehvcsz2WbzqDIRpMfUbksY3Y5ep5oVUXS4Ph6gTYfIwGA3p31b5ZgG8UM/EPFORFSIqP8MvHtEBmyxzoyscq9t7lB//4CcNSBaeEBIBhvTpg9dZzThk8CTsK8oT4sf7d29lTSOvBnaOF0ywDLU9qVPESh4I8IX7siT9Ln+2KBBYaWUECkrea5NOSxc2zqoYbWqd2NIXQzzyQ0dknAcXbrdLP3jMAb8+83st7bTEgPQGfZmchmnrHBLSAb65RMnEYIa5MIUGccwgQ4g1Uk28lKwEUIX6ErvDAFPBBnurxgcU2NaKRIh4JEAHfXEMCy5N/6YeLxTUICaYmFBIYAr4mTwJLRFgIMjp7J10yIb6Agjy11xCdG9Q6BV14GD24ByI6663ovSvRqz+PScOtw1I1mZWIaI9q8oTonNFoCKj5VImzgK/JP3N3f2zcX4CYCBowQgjRn4AP8intI/H/aH5IQjwSGhKEkGBqGPBFAR/kCSGee/+pkVoXgXBQ7Na7fft2ZGZmYsmSJUptkhDiJ4KMRgQZqSbvixQ5KzU1NXjjjTcwatQoJTZHCCFEIYoE+ZycHMyYMQNxcTTIhBBCfInHQX7jxo2orq7G+PHjlSiPX4mPDtW6CIQQwsvACOTanTx5MgoKCli/+/nnnzFt2jR89tlniI+PR3Z2Nnr37o0pU6aoUlhf02y2gmEYSiVLCPFZgkGez65duzBz5ky0adMGAFBeXg6TyYSpU6fi8ccfl7St0tIaWK3Si5KQEIWSkmrJ66mNyiUNlUsaKpc0eiyX0WhAfHyk4HIedaEcNGgQtm7dav870GryhBDi66jPEyGE6Jiig6FycnKU3BwhhBAPUU2eEEJ0jII8IYToGAV5QgjRMZ9JUGb0YGZlT9ZVE5VLGiqXNFQuafRWLrHredRPnhBCiG+j5hpCCNExCvKEEKJjFOQJIUTHKMgTQoiOUZAnhBAdoyBPCCE6RkGeEEJ0jII8IYToGAV5QgjRMZ9JayDVmTNnkJ2djYqKCsTGxmLBggXo0qWLKvsqLy/H7Nmzcf78eZhMJqSmpmLevHlo27YtMjIykJ6eDqOx5X65cOFCZGRkAADy8vKwcOFCWCwW9OrVC6+99pp9Fi2+76TIysqCyWRCaGjLfLOzZs3CyJEjsW/fPsydOxeNjY1ITk7G66+/jvj4eACQ/Z1YFy9exGOPPWb/u7q6GjU1NdixYwdnedUq14IFC7B27VpcunQJK1euRHp6OgD+60eN78SUi+86A+CVa43reKlx3qScU7Zy8V1napXZFd85U+O4yPodMH5q6tSpzPLlyxmGYZjly5czU6dOVW1f5eXlzLZt2+x/5+TkMM899xzDMAyTnp7O1NTUuK1TU1PDDB8+nDlz5gzDMAwzZ84c5r333hP8TqqxY8cyx44dc/rMYrEwN954I7Nz506GYRgmNzeXyc7O9ug7T7zyyivMyy+/zFleNcu1c+dOpqCgwG2/fNePGt+JKRffdcYw3rnWuI6X0udN6jnlKpcjx+tMjTKz4TpnahwXub8DvwzyV65cYQYOHMiYzWaGYRjGbDYzAwcOZEpLS72y/59//pmZNm0awzDcP7w1a9YwDz30kP3vAwcOMLfeeqvgd1KxXcj79+9nJkyYYP+7tLSU6d+/v0ffydXY2MgMGTKEyc/P5yyvN8rluF++60eN78SWy5XjdcYw3r3WxAZ5b19rXOVwvc7UKLMYtnOmxnGRWza/bK4pLCxEYmIigoKCAABBQUFo3749CgsL7Y+2arFarVi2bBmysrLsn02dOhUWiwWjRo3CzJkzYTKZUFhYiI4dO9qX6dixIwoLC+3l5/pOjlmzZoFhGAwcOBBPP/202/bbtm0Lq9WKiooK2d/FxsbKKlteXh4SExPRq1cvzvJGR0d7tVx81w/DMIp/J+eaZLvOAG2vNSXPm9LnlO06U7rMQuVyPGdqHBe5ZaMXrxLNnz8f4eHh9snKN2zYgB9++AFLly7FyZMnkZub69XyLF26FCtWrMD3338PhmEwb948r+5fyPfff48777zT/revl9dXuF5ngLbXmq+fN9frDPB+mdnOmS/wyyCflJSEoqIiWCwWAIDFYkFxcTGSkpJU3e+CBQtw7tw5LFq0yP7yy7bPyMhI3HXXXdizZ4/984KCAvu6BQUF9mX5vpPKtp7JZMJ9992HPXv2uG2/rKwMRqMRsbGxsr+To6ioCDt37sRtt93GW17b594qF9/1o8Z3UrFdZ7ZyA9pca0qfNyXPKdt1pkaZ+bieMzWOi9yy+WWQj4+PR2ZmJlatWgUAWLVqFTIzM1VtqnnrrbeQn5+P3NxcmEwmAEBlZSUaGhoAAGazGWvXrkVmZiYAYOTIkTh48CDOnj0LAPjmm29wyy23CH4nRV1dHaqrqwEADMNgzZo1yMzMRO/evdHQ0IBdu3bZtz9+/HgAkP2dHD/++CNGjx6NuLg43vJ6u1x8148a30nBdp0B2l5rapw3Jc+p63WmVpm5sJ0zNY6L3GPmt5OGnDp1CtnZ2aiqqkJ0dDQWLFiAbt26qbKvEydOYOLEiejSpQvCwsIAACkpKXjggQcwd+5cGAwGmM1mDBgwAHPmzEFERAQA4Ndff8Xrr78Oq9WKzMxM5OTkIDw8XPA7sS5cuICZM2fCYrHAarUiLS0NL7zwAtq3b489e/bgxRdfdOpq1a5dOwCQ/Z1U48aNw/PPP49Ro0YJlletcr3yyitYt24drly5gri4OMTGxmL16tW8148a34kp16JFi1ivs9zcXOzdu9cr1xpbuRYvXqzKeZNyTrnOI9t1BnjvWuOKDbm5uaocFzm/A78N8oQQQoT5ZXMNIYQQcSjIE0KIjlGQJ4QQHaMgTwghOkZBnhBCdIyCPCGE6BgFeUII0TEK8oQQomP/H6eW6hDrXwitAAAAAElFTkSuQmCC\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {}
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "This is what we want to see from our Markov chain. The trace plot moving consistently around the same set of values implies the Markov chain has converged to its invariant distribution. Also, the autocorrelations drop off quickly. This is good since lower autocorrelations means a lower variance when we come to estimate the mean of our samples. \n",
        "\n",
        "Now we compare the density of our samples with the density of the actual standard normal, to prove the algorithm does indeed provide samples from the standard normal."
      ],
      "metadata": {
        "id": "aFUzAdEpL69j"
      }
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 305
        },
        "id": "kUImeSzkgTf3",
        "outputId": "2f040031-acb5-4c54-b17f-1ad724003be5"
      },
      "source": [
        "sns.kdeplot(samples)\n",
        "xx = np.arange(-4, 4, 0.01)\n",
        "yy = scipy.stats.norm.pdf(xx)\n",
        "sns.lineplot(x=xx, y=yy)"
      ],
      "execution_count": 14,
      "outputs": [
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "<matplotlib.axes._subplots.AxesSubplot at 0x7f845cde6210>"
            ]
          },
          "metadata": {},
          "execution_count": 14
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "wUH7PvLPrcru"
      },
      "source": [
        "Now we will use HMC with the contraceptive data from before. First we must specify a suitable model for this data. Note that $n_i$ women of age $z_i$ were asked if they use contraceptives ($n_i$'s in column 'total', $z_i$'s in column 'age'). We let $y_i$ be the number of women of age $z_i$ using contraceptives and assume that $y_i$ is binomially distributed i.e. $$y_i \\sim Bin(n_i, p_i)$$ \n",
        "where $p_i = \\Phi(t_i)$ with $t_i = \\beta_0 + \\beta_1z_i$ . Here $\\Phi$ denotes the cumulative distribution function of the standard normal. This model is referred to as a probit model.\n",
        "\n",
        "With the model specified we seek to find the values of the parameters $\\beta_0$ and $\\beta_1$, which we will achieve by using the Hamiltonian Monte Carlo algorithm to obtain samples from our posterior distribution. We will use uninformative normal priors with mean 0 and standard deviation 10 for both parameters. Hence the posterior density satisfies \\begin{align} \\pi(\\beta_0, \\beta_1 | y) &\\propto \\frac{1}{10\\sqrt{2\\pi}} e^{-\\frac{\\beta_0^2}{200}}\\frac{1}{10\\sqrt{2\\pi}} e^{-\\frac{\\beta_1^2}{200}}\\prod_{i=1}^n \\mathbb{P}(y_i|\\beta_0, \\beta_1) \\\\ &\\propto e^{-\\frac{\\beta_0^2}{200}}e^{-\\frac{\\beta_1^2}{200}}\\prod_{i=1}^n\\mathbb{P}(y_i|\\beta_0, \\beta_1). \\end{align}\n",
        "Our function U, given by the negative log of this target density is $$ \\frac{\\beta_0^2}{200} + \\frac{\\beta_1^2}{200} - \\sum_{i=1}^n \\log\\mathbb{P}(y_i|\\beta_0, \\beta_1).$$ Now we will find the gradient of this, first with respect to $\\beta_0$. First we replace the $\\mathbb{P}$ term with the pmf of the binomial distribution: \\begin{align} &\\frac{\\beta_0^2}{200} + \\frac{\\beta_1^2}{200} - \\sum_{i=1}^n \\log \\left[{n_i \\choose y_i}\\Phi(\\beta_0 + \\beta_1z_i)^{y_i}[1-\\Phi(\\beta_0 + \\beta_1z_i)^{n_i-y_i}\\right]\\\\ = &\\frac{\\beta_0^2}{200} + \\frac{\\beta_1^2}{200} - \\sum_{i=1}^n \\log {n_i \\choose y_i}+y_i\\log\\Phi(\\beta_0 + \\beta_1z_i)+(n_i-y_i)\\log[1-\\Phi(\\beta_0 + \\beta_1z_i)].\\end{align} Now differentiating with repect to $\\beta_0$ we have\n",
        "\\begin{align}&\\frac{\\partial}{\\partial \\beta_0}\\left[\\frac{\\beta_0^2}{200} + \\frac{\\beta_1^2}{200} - \\sum_{i=1}^n \\log {n_i \\choose y_i}+y_i\\log\\Phi(\\beta_0 + \\beta_1z_i)+(n_i-y_i)\\log[1-\\Phi(\\beta_0 + \\beta_1z_i)]\\right] \\\\= &\\frac{\\beta_0}{100} - \\sum_{i=1}^n y_i\\frac{\\partial}{\\partial \\beta_0}\\log\\Phi(\\beta_0 + \\beta_1z_i) + (n_i-y_i)\\frac{\\partial}{\\partial \\beta_0}\\log[1-\\Phi(\\beta_0 + \\beta_1z_i)]\\\\ = &\\frac{\\beta_0}{100} - \\sum_{i=1}^n y_i\\frac{\\phi(\\beta_0+\\beta_1z_i)}{\\Phi(\\beta_0 + \\beta_1z_i)} + (y_i-n_i)\\frac{\\phi(\\beta_0+\\beta_1z_i)}{1-\\Phi(\\beta_0 + \\beta_1z_i)}\\end{align}\n",
        "where $\\phi()$ is the pdf of the standard normal. Similar calculations yields for the derivative with respect to $\\beta_1$: $$\\frac{\\beta_1}{100} -\\sum_{i=1}^n y_i\\frac{z_i\\phi(\\beta_0+\\beta_1z_i)}{\\Phi(\\beta_0 + \\beta_1z_i)}+(y_i-n_i)\\frac{z_i\\phi(\\beta_0+\\beta_1z_i)}{1-\\Phi(\\beta_0 + \\beta_1z_i)}.$$"
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 839
        },
        "id": "ed0RPdmMl5X4",
        "outputId": "fc8c18eb-d4c8-42c9-d076-20695f1d133f"
      },
      "source": [
        "def U(beta):\n",
        "    \"\"\"Define potential energy function (minus log of posterior density).\"\"\"\n",
        "    p_i = scipy.stats.norm.cdf(beta[0] + beta[1]*age)\n",
        "    prob = scipy.stats.binom.pmf(use, total, p_i)\n",
        "    neg_log_posterior = -sum(np.log(prob)) + beta[0]**2/200 + beta[1]**2/200\n",
        "    return neg_log_posterior\n",
        "  \n",
        "def grad_U(beta):\n",
        "    \"\"\"Gradient of U as we found above.\"\"\"\n",
        "    t_i = beta[0]+beta[1]*age\n",
        "    phi = scipy.stats.norm.pdf(t_i)\n",
        "    Phi = scipy.stats.norm.cdf(t_i)\n",
        "    a = use * phi / Phi\n",
        "    b = (use - total) * phi / (1 - Phi)\n",
        "    d_beta0 = beta[0]/100 - sum(a + b)\n",
        "    d_beta1 = beta[1]/100 - sum(age * (a + b))\n",
        "    return d_beta0, d_beta1\n",
        "\n",
        "# tune epsilon and L appropriately\n",
        "n_sim, L = 2000, 48\n",
        "epsilon = np.array([0.017, 0.0013])\n",
        "current_q = np.array([1.5, -0.03])\n",
        "samples, n_accepted = HMC(U, grad_U, epsilon, L, current_q, n_sim) \n",
        "\n",
        "# show acceptance rate, trace plots and autocorrelations\n",
        "acc_rate = n_accepted / n_sim\n",
        "print(acc_rate)\n",
        "fig, (ax1, ax2) = plt.subplots(2, 1, sharey=False, sharex=True)\n",
        "ax1.plot(range(n_sim), samples[:,0])\n",
        "ax2.plot(range(n_sim), samples[:,1])\n",
        "statsmodels.graphics.tsaplots.plot_acf(samples[:,0]);  \n",
        "statsmodels.graphics.tsaplots.plot_acf(samples[:,1]);  "
      ],
      "execution_count": 17,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "0.6765\n"
          ]
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 2 Axes>"
            ]
          },
          "metadata": {}
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {}
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "hcuP83yv5noe"
      },
      "source": [
        "Now we have an efficient Markov chain providing us samples from our posterior, we can estimate the posterior means of $\\beta_0$ and $\\beta_1$."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 321
        },
        "id": "G3dRz0fjE1fs",
        "outputId": "f00ee5d7-c67f-44d7-c2a0-6f9629d7d764"
      },
      "source": [
        "beta_0_hat = round(np.mean(samples[:,0]), 4)\n",
        "beta_1_hat = round(np.mean(samples[:,1]), 4)\n",
        "print(beta_0_hat, beta_1_hat)\n",
        "\n",
        "#Having estimated the parameters, let us visualise the actual data compared\n",
        "#with the predicted values of the probabilites implied by these estimates\n",
        "t_i = beta_0_hat + beta_1_hat*age\n",
        "p_i = scipy.stats.norm.cdf(t_i)\n",
        "sns.scatterplot(x = age, y = proportion)\n",
        "plt.plot(age, p_i, color='red')"
      ],
      "execution_count": 21,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "1.59 -0.0327\n"
          ]
        },
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "[<matplotlib.lines.Line2D at 0x7f845adfcd10>]"
            ]
          },
          "metadata": {},
          "execution_count": 21
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "8RMao-T3_vgh"
      },
      "source": [
        "Clearly the fitted probabilities don't follow the observed probabilities very well. Originally we modelled the probabilities as $p_i = \\Phi(\\beta_0 + \\beta_1z_i)$. The linear relationship inside $\\Phi()$ doesn't allow the probabilities to first increase and then decrease with age, as seen in the data. To fix this we slightly amend the model so that $p_i = \\Phi(\\beta_0 + \\beta_1z_i + \\beta_2z_i^2)$. "
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 1000
        },
        "id": "Yn_ylgkWBXwR",
        "outputId": "b49530c6-c105-4b57-90b6-6ef244bbe9ae"
      },
      "source": [
        "def U(beta):\n",
        "    \"\"\"Define new U function (minus log of posterior density).\"\"\"\n",
        "    p_i = scipy.stats.norm.cdf(beta[0] + beta[1]*age + beta[2]*age**2)\n",
        "    prob = scipy.stats.binom.pmf(use, total, p_i)\n",
        "    neg_log_posterior = -sum(np.log(prob)) + beta[0]**2/200 + beta[1]**2/200 + beta[2]**2/200 \n",
        "    return neg_log_posterior\n",
        "  \n",
        "def grad_U(beta):\n",
        "    \"\"\"Gradient of new U.\"\"\"\n",
        "    t_i = beta[0] + beta[1]*age + beta[2]*age**2\n",
        "    phi = scipy.stats.norm.pdf(t_i)\n",
        "    Phi = scipy.stats.norm.cdf(t_i)\n",
        "    a = use * phi / Phi\n",
        "    b = (use - total) * phi / (1 - Phi)\n",
        "    d_beta0 = beta[0]/100 - sum(a + b)\n",
        "    d_beta1 = beta[1]/100 - sum(age * (a + b))\n",
        "    d_beta2 = beta[2]/100 - sum(age**2 * (a + b))\n",
        "    return d_beta0, d_beta1, d_beta2\n",
        "\n",
        "# tune epsilon and L appropriately \n",
        "n_sim, L, current_q = 1000, 580, np.array([-7.6, 0.57, -0.009])\n",
        "epsilon = np.array([0.002, 0.0001, 0.00003])\n",
        "samples, n_accepted = HMC(U, grad_U, epsilon, L, current_q, n_sim) \n",
        "\n",
        "# show acceptance rate, trace plots and autocorrelations\n",
        "acc_rate = n_accepted / n_sim\n",
        "print(acc_rate)\n",
        "fig, (ax1, ax2, ax3) = plt.subplots(3, 1, sharey=False, sharex=True)\n",
        "ax1.plot(range(n_sim), samples[:,0])\n",
        "ax2.plot(range(n_sim), samples[:,1])\n",
        "ax3.plot(range(n_sim), samples[:,2])\n",
        "statsmodels.graphics.tsaplots.plot_acf(samples[:,0]);  \n",
        "statsmodels.graphics.tsaplots.plot_acf(samples[:,1]);\n",
        "statsmodels.graphics.tsaplots.plot_acf(samples[:,2]);    "
      ],
      "execution_count": 25,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "0.577\n"
          ]
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 3 Axes>"
            ]
          },
          "metadata": {}
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {}
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {}
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "We estimate the posterior expectations of $\\beta_0$, $\\beta_1$, $\\beta_2$ and plot the probabilities implied by these estimates against the observed probabilities. "
      ],
      "metadata": {
        "id": "wH8-7qMSxDmH"
      }
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 321
        },
        "id": "pcosMF1hCSmK",
        "outputId": "223b816c-deaf-442b-9678-d8d94729304b"
      },
      "source": [
        "beta_0_hat = round(np.mean(samples[:,0]), 5)\n",
        "beta_1_hat = round(np.mean(samples[:,1]), 5)\n",
        "beta_2_hat = round(np.mean(samples[:,2]), 5)\n",
        "print(beta_0_hat, beta_1_hat, beta_2_hat)\n",
        "t_i = beta_0_hat + beta_1_hat*age + beta_2_hat*age**2\n",
        "p_i = scipy.stats.norm.cdf(t_i)\n",
        "sns.scatterplot(x = age, y = proportion)\n",
        "plt.plot(age, p_i, color='red')"
      ],
      "execution_count": 26,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "-7.63624 0.57 -0.00896\n"
          ]
        },
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "[<matplotlib.lines.Line2D at 0x7f845ab75f10>]"
            ]
          },
          "metadata": {},
          "execution_count": 26
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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cq/Vr30W6yp9yEnywCSZNwrqgCGorSXn0Ud12GNXVa9+GQOTvsMCnpaWxZ88e3G43RqMRt9vN3r17SUtLO+S4kpIS5s6di8FgIDY2lmHDhrFlyxafCrysg4fIt98i/oZr8PRIpWblq3h69gUf8nQ1/+599jYvjO6utGFSAvveaP3ad5Vu8xctwpJ4DNaHHsTZ6Mb2wEO6K/K6fe3306wffHJyMunp6ZSWlgJQWlpKenr6IdMzAL179+a9994DoKmpiQ8++IATT5S2tL4wr15F/LVZuI/vT/WaDXiO69vtGeTCaBhSFBwzZ8FddxH9/NPEzLxdllGGCK/+mc7Pz6ekpISMjAxKSkooKCgAICcnh61btwJw99138+mnnzJy5EhGjx5N3759ufLKKwOXPMREPf0EsTfegGvIGdSsfh1VowvUcmE0TCkKzJmDY8o0op95kpi77pAiHwKkVcFhuv2rnqpiWTgfa9EcGi8eQd3jz0J0dKdPF3Tb5X0Qrl+zg0VKSiwVe+uwFszCsnQxjgk3Yp8zXxetpkPitZdWBSFGVbHm3YNl2SM0jBlL/cPFEBkEI2V/bpcX+qIo2PPuA48Hy7JHwGDAft88XRR5cSQp8BqKLl6MZdkjLSOl2UW6u7AlQpSiYC+YA6oHy6NLQTFgL5wrRV6HpMBrxLRmNTGFs2gYfbkUdxF8FAV74f0tI/lHi8FobBnZS5HXFSnwgdBBi9yITz4ibvJEXGcMpX7xMinuIjgpCvbZRShuN5ali1uma2YVSJHXESnw/tbBTlDDrp3EXzcWT880ap99oWUzkxDBSlGw3f9gy0j+kYdRTZEtSyqFLsjQ0c+O1iJXqa4iPvsKaG6m9h8voR5zjMZphfCComCbtwDn1ddhXfgA5lde0jqR8JIUeD9rr0VubXU9cddfg7HsJ+qefQH3CbIJTOiIwYCtaCGuoX8k9rbJGP+zTetEwgtS4P2szZ2gEQZOLLwT0+aN1D9cjOuPZ2uUToguMJmofeI5PPEJxI/LRqmu6vgxQlNS4P2srZ2gC2reIfblFdhn3EPjFVkaJxSi89TUVOqeeh7Dr/8l7qbx4HZ3/CChGSnw/nZQi9z8CUMpTi3juMceomHs1Tim36l1OiG6rPn0P2CbtwDTO//COm+21nHEUcgqmkDYvxM0+bMPiZ9xG03n/h/1Dy6S5WUiZDRcO46ILz7HsmgBrlNOpWnkaK0jiTbICD5AjD98R9y4q3Ef34+6p54Hk3RjFKHFNnc+riFnEDflZozffK11HNEGKfCB0NRE7MQbwGigdvlK1PgErRMJ4X9mM3VPl+CJiSHur1eh1NZonUgcRgp8AFiL5hC59UvqHyrWpKe7EN3F0zONuiefx/hzGbE3TwCPR+tI4iBS4P0sctP7RD/yMM5rx9F0yaVaxxEi4JqHnoltdhHmtzZgeeB+reOIg0iB9yOlpprYW27EfXw/bIXyQRfho+H6CTivugbrgiJMb7yudRyxnxR4f1FVYmZMx7C7nPq/PwFWq9aJhOg+itKy03XwacROnojxx++1TiSQAu835pdeJOqVVTj+dhfNpw3x7kEK1DldlFXYqWtoBllFKfQsKoq6p5dDhJHYqZNkE1QQkHXwfmAo+4mYmXfg+sOZOG693bsHddB1Ugg98hzbG9t984ibchPRTz2GM+dmrSOFNRnBd5XbTewtN4KqUlf8GBiNHT+Go3edFELPGq+8isbhF2OdU4Bh549axwlrUuC7yLLkIUwfbsY270GflkS213Wyxt7k54RCdDNFwfbgIlRjBLHTp8jSSQ1Jge+CiC8+wzJ/Lg2jL6dxzFifHttm18lIIwlW2fEq9M/T61jshXMxbXqfqGef0jpO2JIC31l2O7E3T8DTIxXb/Id87jPTVtfJqVcOJs4SGYi0QnS7huxrafq/C7AW5mL4uUzrOGFJLrJ2Ukzu3Rh//IHaVWtQExJ9P8FBXSdr7E0kWE0txV0usIpQoSjUL1xC4nlnEjt9CrUrVkvDvW4mI/hOMK1bS/TzT+OcNBXXOed1/kT7u072OcZKXLQUdxF6PL/pgz23ENO/3yFq+XNaxwk7UuB9pNTWEDt9Cq6TT8E+816t4wgR9Br+egNNZ5+LNe8eDL/+V+s4YUUKvI+s82ajVFViW1QMZrPWcYQIfgYD9Q89guJuJub2qaDKV9XuEh4F3k87Ro1bvyLq6SdoGDee5kGn+jejECHM0/d47PfkYf7Xm5hf/Ef3BwjTXeOhf5HVXztGPR5iZ96OmpQkUzNCdIJz/I2YX1tNzKy7cJ0/DE/PtO554jDeNR7yI3h/7Rg1r3iByI+3YJtV2LlVM0KEO4OB+kXFKI0NxNw5rdumasJ513jIF3h/7BhVamuIKZyFa8gZNGZl+zuiEGHD3e8E7DNnYV63FvPLK7vlOcN513jIF3h/7Bi1FM1BqarCVrQADCH/kgkRUM4bJ+EacgYxs2ai1NUG/PnCede4V9Vq586dZGVlkZGRQVZWFrt27WrzuLVr1zJy5EgyMzMZOXIk+/bt82fWTunqjlHjtq1EP/U4DX+9geZTBgcyqhDhwWjENu9BlMpKLA89GPCnC+dd44qqdjwRdt111/GXv/yFUaNG8eqrr7Jq1Sqee+7QTQtbt25lxowZPPvss6SkpFBfX4/JZMLsw1LCykobHk8A5uWUlnk4b3aMpqTEUlFR3/IbVSVhZAbGH76j6oPPdDH3fkh+ndFzdpD8voqdejPml1dS9f5HeI7v16VzdZjdhxqghc6+9gaDQnJyTPt/39EJKisr2b59O5mZmQBkZmayfft2qqqqDjnumWee4YYbbiAlJQWA2NhYn4p7QHVyx6h5xQtEfvQh9nsLdFHchdAT+925EBFJTGFu4J8sTHeNd1jgy8vLSU1Nxbi/z7nRaKRHjx6Ul5cfctwPP/zAzz//zNVXX81ll13G0qVL8eLLQdBS6mqJKczFNeR0Gq66Rus4QoQcT880HFOnYX79NSI3b9Q6Tkjy2zp4t9vNjh07ePrpp2lqamLChAn06tWL0aNHe32Oo33V6E4pKbEwZxbsq8DwxlpSUuO1juSTlJRYrSN0mp6zg+T3We7dsPxZEgrugY8/9vqGOW2R1/5IHRb4tLQ09uzZg9vtxmg04na72bt3L2lph25S6NWrFyNGjMBkMmEymbjwwgv56quvfCrwAZuD90FKSixV735A4iOP0HDdDdj6DAAdzavqeR5Yz9lB8neW+Z584m4aT13xYzR28ttyuL72XZ6DT05OJj09ndLSUgBKS0tJT08nKSnpkOMyMzPZuHEjqqricrn48MMP+e1vf+tzYM2pasuO1fh47HfP0jqNECGv8bIrcA05A+ucAhSbfot0MPJqmWR+fj4lJSVkZGRQUlJCQUEBADk5OWzduhWASy+9lOTkZP70pz8xevRoTjjhBK644orAJQ+U5cuJ3PJBy4XVxKSOjxdCdI2iYJs9D+PePUQveUjrNCHFq2WS3UXrKRqlrpZjzjkDV69jqVn7L11uatLzV1U9ZwfJ31WxN0/AXPoqVZs/xfObPj49VuvsXaXZFE04sSxaCHv2YJsnO1aF6G72e/PBYMA6O0/rKCFDqth+hvJfiX7873D11TQP/r3WcYQIO55je+OYNJWoV1YR8dEWreOEBCnw+1kWzAe3G/ZfXxBCBMhRerM7brkNd880YnJngsejXcYQIQUeMP74PVHLn6XhuuuhX9e2TAshjmJ/b/YZSzeT/+QWZhRv4uuy2v8VeasV+z15RH72KeZVKzSNGgqkwNPSLRKzGfu0O7WOIkRI86Y3e+OYsbgGn4Z1dj7Y7doEDRFhX+Ajtn5J1CurcNw4CbVHD63jCBHSvOrNbjBgK5yHsfxXLEsXd3PC0BL2Bd46pwBPYiLOybdqHUWIkOdtb/bmM/9Iw58vw1K8CMOv/+3OiCElrAt85OaNmN5+C8fU21Hj9NVvRgg98qU3u31WAbhcWBY+0N0xQ0bo33S7PaqKdXY+7rReOG/I0TqNEOFBhfQ+8RRNOqvD3uye4/rScM1fiXr+GRxTbsNzXN9uj6t3YTuCN61/g8hPPsJxx0yIjtY6jhDhw4fe7I5pf4OICKwLirovXwgJzwLvdmOdW0Bz/xOk17sQQczTMw3nX8djXvECxh++0zqO7oRlgTe/9CIR33yN/a5ZEBG+s1RC6IFj6nSIisLywP1aR9Gd8CvwjY1YH7gf1ymDacocpXUaIUQH1JQUnBNuwvzKKoxfb9c6jq6EXYGPev5pjGU/Yb8nTxqKCaETjklTUK0xWGUU75OwqnCKrR7rwvk0nXMervOHaR1HCOElNSkZ542TMJe+SsTWL7WOoxthVeCjH12KYd++lru5K0rHDxBCBA3nTZPxJCS0tBYRXgmbAq9UVhJdvJjGSzJpPv0PXThR+53whAh7Afz5UOMTcE6ainnDOiI+/dh/Jw5hYbOExLJ4IYrD3rJyprP2d8I70CzpwC689D7xR13LK0RY6IafD8eEm4h+tBjrvNnUrnzVPycNYWExgjfsLif6qcdoHDMW92/TO30ebzrhCRGuuuXnIyYGx5TpmP79DpEfbPLfeUNUWBT46EcehuZm7LfP6NJ5vOqEJ0Qo8mLqpbt+PpzjxuPukYpl3mwInltKB6WQn6Ix7NlN9HNP0zhmLJ6+x3fpXAc64R38IW6rE54QIcXLqZdu+/mwWHBMu4PYu/5G5Hvv4vq/C/x7/hCi7xG8F6OK6EcWgcuF/bY7uvx0vnTCEyJUeDv10p0/Hw3XjMN9bG+sMoo/Kv2O4L0YVSh79xL93FM0/uVKPP36d/05feiEJ0SoONrUS1z0QcW7O38+zGYc0/5G7B23YnprPWSPCcCT6J9uR/DejCosxYugsRHHtK6P3lv50AlPiFDg7U06gG79+Wi46hrcffsSdf8ctn63V5Ytt0G3Bb6jCzpKRQXRzz5J42VX4O5/ohYRhQgJQTs1aYrk+3FTMW/7ktduf+jIG3gL/U7RdHRBx/L3JeB04pguN9IWokuCdGqyzuFilr0fCxN7cfXmf7DlhD+weMUXFE0669CpozCm2xH80UYVSmUl0U89TuNlf8F94gCNkwoRAoJwarLG1oTTDf88M4u+lWWc+f0WWbZ8GN2O4I82qmgZvTtwTO/aunchRPA68C3+/YHncNUHL5L14Qo+Sz9bli0fRLcjeKDNUYVSVUnUk4/ROOoy3AMGap1QCBEgB77FR5pNrBh6Bf0rdpKftk/7awNBRN8Fvg3Ry4pRHHYZvQuhdx3tcznoW/zwh2fg6tOXgc8uAU8QzB8FCf1O0bRBqa4i+olHaRw52veeM0rLRZvd31dgMUcQFx0RFPOMQoQlbxuX7f8W379PEvXT7iB22i2Y3n6Tpgsv1ix6MAmpEXz0o8UYbPW+r5zZ/2GasXQzd/99syy3EkJjnWlc1jBmLO7ev8HyYJHsbt3PqwK/c+dOsrKyyMjIICsri127drV77I8//sipp55KUVGRvzJ6RampJvrxR2nMHIX7dyf59FjpEilEcOlU4zKTCcfU6UR++jGR770b2IA64VWBz8vLIzs7m/Xr15OdnU1ubm6bx7ndbvLy8hg+fLhfQ3oj+tGlGOrrOtUxUrpEChFcfNo9e5CGq67BndYLy4LuHWAGqw4LfGVlJdu3byczMxOAzMxMtm/fTlVV1RHHPvbYY5x//vn07dvX70GPRqmtIfrxZTT+aSTuk072+fGd/TAJIQKj07tnzWYcU6dh+nAzkZs3dkPS4NZhgS8vLyc1NRWjseWFNhqN9OjRg/Ly8kOO++abb9i4cSPjxo0LSNCjiX58GYa62k73ew/ardhChKuDVsjkTxhK0aSzvL4zVEP2dS394mUU759VNC6Xi1mzZnH//fe3/kPQGcnJMb4/qLYWHlsKo0aRNOzszj93Ugz9eydQVe8kKTaatGOsGAz6vMqakhKrdYRO03N2kPz+luLLsa3ZY2HmDIzTp5Oy40s455xARPO7QLz2HRb4tLQ09uzZg9vtxmg04na72bt3L2lpaa3HVFRUUFZWxsSJEwGoq6tDVVVsNhv33Xef12EqK214fFzDan7lZeJqaqiecjvNFfU+PfZwJnMw0uEAABBzSURBVAUG9U+hoqKeykpbl86llZSUWCq6+DpoRc/ZQfJr6Yjsl2eTPPd+mmflUbtitXbBvNTZ195gUI46MO6wwCcnJ5Oenk5paSmjRo2itLSU9PR0kpKSWo/p1asXW7Zsaf39kiVLcDgczJgR+M1GjRdfQvX6d2g+ZXDAn0sIoRMWC45JU4kpnEXEpx/TPOQMrRNpwqtVNPn5+ZSUlJCRkUFJSQkFBQUA5OTksHXr1oAG7JDVSvNpQ7TNIIQIOs5x4/EkJWFZOF/rKJpRVDV4dgR0ZorG3/T8NRX0nV/P2UHya6m97JaHH8Q6t5DqN/9N86mnaZDMO4GaogmpnaxCCHEw5/iJeOITsCwIz1G8FHghRMhSY+Nw3jgJ87rXMW7TeDpZA1LghRAhzZlzE57YOKwPPaB1lG4nBV4IEdLU+AScOTdiXrMa49fbtY7TraTACyFCnnPiJDwxsWG3okYKvBAi5KlJyS2j+NdeCatRvBR4IURYcN50C2pMLNYH52kdpdtIgRdChAU1MQlnzk0tc/H/2aZ1nG4hBV4IETacN03GExeP9YH7tY7SLaTACyHChpqQiHPizZjXrsG49Sut4wScFHghRFhx3jgpbEbxUuCFEGFFjU/AedNkzOteJ+KrL7SOE1BS4IUQYcc58eaWHjUhPoqXAi+ECDtqXDzOm2/BvP4NIr74zLcHK1DndFFWYaeuoRmC+MZvUuCFEGHJmXMTnsRE30bxCnxdVsuMpZvJf3ILM4o38XVZbdAWeSnwQoiwpMbG4bx5CuY31xPx2SdePabO4WLxii9odLkBaHS5WbziC+ocrkBG7TQp8EKIsOWccGPLXZ+8HMXX2Jpai/sBjS43NfamQMTrMinwQoiwpcbE4ph0K+Z/vUnEJx91eHxCrBlzpPGQPzNHGkmwmgIVsUukwAshwprzhhw8yclerYuPi45g6pWDW4u8OdLI1CsHE2eJDHTMTonQOoAQQmgqJgbH5NuIKZxFxMdbaD5jaPvHqpDeJ56iSWdRY28iwWpqKe5Bc2frQ8kIXggR9pzXT8BzTArW+XM7PliFuOhI+hxjJS46eIs7SIEXQgiwWnHcchumf79DxJYPtU7jN1LghRACcI4bjyelB9ai2aAG8bDcB1LghRACwGLBPu0OTBvfw/SvDVqn8Qsp8EIIsV/DX8fT3P8ErHn3gCs4Ny/5Qgq8EEIcEBmJPW82Ed99S9Tzz2idpsukwAshxEGaMi6h6ZzzsD4wF6W2Rus4XSIFXgghDqYo2AvmoFRVYXl4gdZpukQKvBBCHKZ50Kk0ZmUT/fjfMfy0S+s4nSYFXggh2mC/axZERGCdna91lE6TAi+EEG3wpPXCMflWol59mYiPtmgdp1OkwAshRDsck6bi7plGTO5M8Hi0juMzrwr8zp07ycrKIiMjg6ysLHbt2nXEMcXFxVx66aWMHDmSyy+/nPfff9/fWYUQontZrdjvziXys08xr16ldRqfeVXg8/LyyM7OZv369WRnZ5Obm3vEMaeccgovvfQSa9asYe7cuUybNo2Ghga/BxZCiO7UeOVVuAad2jIX73RqHccnHRb4yspKtm/fTmZmJgCZmZls376dqqqqQ44799xziY6OBmDgwIGoqkpNjb7XkAohBAYD9oI5GH/5mejH/651Gp90WODLy8tJTU3FaGxpcG80GunRowfl5eXtPmb16tX06dOHnj17+i+pEEJoxHXOeTSOuBTLwwtQ9u7VOo7X/H7Dj48++ohFixbx1FNP+fzY5OQYf8fplJSUWK0jdIme8+s5O0h+LQU8+6KFcNJJHLPkAVi2zO+nD0T+Dgt8Wloae/bswe12YzQacbvd7N27l7S0tCOO/fzzz/nb3/7G0qVL6devn89hKitteDzatulMSYmloqJe0wxdoef8es4Okl9L3ZI9MQ3r9ROIfvwxqrOvx53+O7+durP5DQblqAPjDqdokpOTSU9Pp7S0FIDS0lLS09NJSko65LivvvqKadOmsXjxYk466SSfgwohRLBz3D4DNTaOmPx7tI7iFa9W0eTn51NSUkJGRgYlJSUUFBQAkJOTw9atWwEoKCigoaGB3NxcRo0axahRo9ixY0fgkgshRDdTk5Jx3H4npnf+hWnDG1rH6ZCiqsFz6xKZouk6PefXc3aQ/Frq1uxNTSRedB5KZSXV732ImpTc5VNqNkUjhBDiICYTdY88hqG6ipg7pwf17f2kwAshhI/cg07BfufdRL32CuaXV2odp11S4IUQohOct9yG64yhxMy8A8Ov/9U6TpukwAshRGcYjdQtWYbichF766SgbEYmBV4IITrJ068/toI5mP79DlFPP651nCNIgRdCCAAF6pwuyirs1DU0g+Ldwxquu57GCy8ipjAX4/ffBTajj/zeqkAIIXRHga/Lalm84gsaXW7MkUamXjmY9D7x0NEiGUXB9nAxiecNJfaWidSUvgkRwVFaZQQvhAh7dQ5Xa3EHaHS5WbziC+ocLq8e70ntiW3+Q0R+9imWRcFzo24p8EKIsFdja2ot7gc0utzU2Ju8PkfjqMtpuHwMlgVFRHz5ub8jdooUeCFE2EuINWOONB7yZ+ZIIwlWk0/nsc17EE9KD2InTwyKm4NIgRdChL246AimXjm4tcgfmIOPs0T6dB41IZH6RUuJ+HYH1rkFgYjqk+C4EiCEEFpSIb1PPEWTzqLG3kSC1dRS3DvRhcB1/jCcN+RgeXQpTRl/wnXOef7P6yUZwQshBIAKcdGR9DnGSlx054r7Abbc+2jufwKxk3IwlP3kv4w+kgIvhBD+ZrFQ9+TzKA1OEv4yEsPu9m9xGkhS4IUQIgDcvzuJ2hdWoezbR/yYUSiVld2eQQq8EEIESPOQM6greRHjT7uIH3s5Sn1dtz6/FHghhAgg19nnUvfkc0T8ZytxV18JDke3PbcUeCGECLCmi0ZQ//cniPzoQ+KvvxoaG7vleaXACyFEN2gcdTm2hUswvfMv4m6eAM3NAX9OKfBCCOFv7XSmbMi+Ftt992MufZXY2yYHvIe8bHQSQgh/6qAzpfPGySg2G9aiOagxMdjufzBgUWQEL4QQfuRNZ0rH9DtxTJpK9FOPY51bGLAsMoIXQgg/Olpnyrjo/b1tFAV73n0o9fUt7YVHXQonn+73LFLghRDCjw50pjy4yLfZmVJRsM1fiOucc4k7/XRo8H8WmaIRQgg/8qkzpdFI42VXQGxsQLLICF4IIfzJl86USsuc/e7vK7CYI4iLjuhSk7PDSYEXQgh/29+ZsnXOvZ3i3un7wHpJpmiEEEIDXb0PrDekwAshhAb8cR/YjkiBF0IIDfjrPrBHIwVeCCE04K/7wB6NXGQVQggtHLTaxuFyY4k0dvo+sO3xagS/c+dOsrKyyMjIICsri127dh1xjNvtpqCggOHDh3PRRRexcuVK/6UUQohQtH+1zaD+KV2+D2xbvCrweXl5ZGdns379erKzs8nNzT3imDVr1lBWVsaGDRt48cUXWbJkCb/88ot/0wohhPBahwW+srKS7du3k5mZCUBmZibbt2+nqqrqkOPWrl3LmDFjMBgMJCUlMXz4cNatWxeY1EIIITrU4Rx8eXk5qampGI0tFwKMRiM9evSgvLycpKSkQ47r1atX6+/T0tLYvXu3T2GSk2N8Oj5QUlICs224u+g5v56zg+TXkp6zQ2DyB9VF1spKGx6PnyehfJSSEktFRb2mGbpCz/n1nB0kv5b0nB06n99gUI46MO6wwKelpbFnzx7cbjdGoxG3283evXtJS0s74rhff/2VU045BThyRO9t2GAQLDk6S8/59ZwdJL+W9JwdOpe/o8d0WOCTk5NJT0+ntLSUUaNGUVpaSnp6+iHTMwAjRoxg5cqVXHzxxdTU1PDWW2+xfPlyn8ImJlp9Oj5QgmWqqLP0nF/P2UHya0nP2SEw+RVVVTucE/nhhx+YOXMmdXV1xMXFUVRURL9+/cjJyWHq1KkMGjQIt9tNYWEhmzZtAiAnJ4esrCy/BxZCCOEdrwq8EEII/ZFWBUIIEaKkwAshRIiSAi+EECFKCrwQQoQoKfBCCBGipMALIUSIkgIvhBAhKqh60XSn6upq7rzzTsrKyjCZTBx33HEUFhaSlJTEF198QW5uLo2NjRx77LE88MADJCcnax35EEfLP3DgQAYMGIDB0PLv9/z58xk4cKDGiQ81adIkfvnlFwwGAxaLhVmzZpGens7OnTuZOXMmNTU1JCQkUFRURN++fbWOe4T28g8bNgyTyYTZbAbgjjvu4Nxzz9U4bdseeeQRlixZwpo1axgwYIAuPvcHOzy/Hj73QLufkYC8/mqYqq6uVj/88MPW38+bN0+96667VLfbrQ4fPlz9+OOPVVVV1eLiYnXmzJlaxWxXe/lVVVUHDBig2mw2raJ5pa6urvW/33zzTXX06NGqqqrqtddeq65evVpVVVVdvXq1eu2112qSryPt5b/gggvUHTt2aBXLa9u2bVPHjx/fmlcvn/sDDs+vqvr43Ktq25+RQL3+YTtFk5CQwNChQ1t/P3jwYH799Ve2bduG2Wzm9NNPB2Ds2LFB2de+vfx6ERv7v9aoNpsNRVG8vvdAMGgrv140NTVRWFhIfn5+65/p5XMPbefXu0C9/mE7RXMwj8fDCy+8wLBhw47ogpmUlITH42mdMghGB+c/4Nprr8XtdnPeeecxZcoUTCb/3andX+655x42bdqEqqo88cQTXt97IFgcnv+AO+64A1VVGTJkCNOnTycuLk7DlEdatGgRf/7zn+ndu3frn+npc99W/gP08LmHIz8jgXr9w3YEf7D77rsPi8XCNddco3WUTjk8/7vvvsvLL7/M8uXL+f777ykuLtY4YdvmzJnDu+++y7Rp05g/f77WcXzWVv7ly5fz2muvsWrVKlRVpbCwUOOUh/r888/Ztm0b2dnZWkfplKPl18vnvjs/I2Ff4IuKivjpp594+OGHMRgMrX3tD6iqqsJgMATdKOaAw/MDrb36Y2JiGDNmDJ999pmWETs0evRotmzZQs+ePVvvPQC0e++BYHMgf3V1dWtWk8lEdnZ20L32H3/8MT/88AMXXnghw4YNY/fu3YwfP56ffvpJF5/79vJv3LhRN5/7tj4jgao7YV3gFy5cyLZt2yguLm79KnfyySfT0NDAJ598AsA///lPRowYoWXMdrWVv7a2loaGBgCam5tZv3496enpWsY8gt1up7y8vPX3b7/9NvHx8YfcewBo994DWmsvv9lspr6+5a48qqqydu3aoHvtJ06cyMaNG3n77bd5++236dmzJ08++SQTJkzQxee+vfyDBg0K+s89gMPhaPMzEqi6E7Zz8N999x2PPvooffv2ZezYsQD07t2b4uJi5s+fT15e3iHLlYJNe/knTJhAbm4uiqLQ3NzMaaedxq233qpx2kM5nU5uvfVWnE4nBoOB+Ph4li1bhqIo5OfnM3PmTJYuXdp674Fg017+yspKpkyZgtvtxuPx0L9/f/Ly8rSO6xWDwaCLz317fvzxx6D/3APtfkYC9fpLP3ghhAhRYT1FI4QQoUwKvBBChCgp8EIIEaKkwAshRIiSAi+EECFKCrwQQoQoKfBCCBGipMALIUSI+n81AXycWSVVfQAAAABJRU5ErkJggg==\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "It is very clear looking at this plot that this model is much better than the original. However for the sake of being rigorous we will use a more concrete method to show the difference in quality of the 2 models. In particular, we will calculate the Bayesian information criterion (BIC) for the 2 models. It is defined as $$\\text{BIC} = k\\log n - 2\\log L(\\hat{\\beta})$$ where k is the number of parameters in the model, n the number of data points and $L(\\hat{\\beta})$ the value of the likelihood function evaluated at the maximum likelihood estimate. "
      ],
      "metadata": {
        "id": "mHy3KqtGQW8X"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "# to calculate BIC we first need to find the MLEs for both of the models\n",
        "def neg_log_lik_1(beta):\n",
        "    p_i = scipy.stats.norm.cdf(beta[0] + beta[1]*age)\n",
        "    prob = scipy.stats.binom.pmf(use, total, p_i)\n",
        "    negative_log_likelihood = -sum(np.log(prob))\n",
        "    return negative_log_likelihood\n",
        "\n",
        "x0 = [0, 0]  \n",
        "opt1 = scipy.optimize.minimize(neg_log_lik_1, x0, method='BFGS')\n",
        "\n",
        "def neg_log_lik_2(beta):\n",
        "    p_i = scipy.stats.norm.cdf(beta[0] + beta[1]*age + beta[2]*age**2)\n",
        "    prob = scipy.stats.binom.pmf(use, total, p_i)\n",
        "    negative_log_likelihood = -sum(np.log(prob))\n",
        "    return negative_log_likelihood\n",
        "\n",
        "x0 = [0, 0, 0]\n",
        "opt2 = scipy.optimize.minimize(neg_log_lik_2, x0, method='BFGS')\n",
        "\n",
        "BIC1 = 2*np.log(33) + 2*opt1.fun\n",
        "BIC2 = 3*np.log(33) + 2*opt2.fun\n",
        "print(f'BIC1 = {BIC1:.1f} \\nBIC2 = {BIC2:.1f}')"
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/"
        },
        "id": "vSkNwG-OLsx0",
        "outputId": "7e82ce7a-402e-478d-baac-48e7f12f2430"
      },
      "execution_count": 33,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "BIC1 = 420.6 \n",
            "BIC2 = 180.6\n"
          ]
        }
      ]
    },
    {
      "cell_type": "markdown",
      "metadata": {
        "id": "Kw4_QM2vlH5M"
      },
      "source": [
        "The BIC for the second model is much smaller hence confirming it is the more appropriate model for the data.\n",
        "\n",
        "We are interested in finding the age at which the use of contraceptives is maximal. To do this we must maximise $\\Phi(\\beta_0 + \\beta_1z + \\beta_2z^2)$. Since the CDF of the standard normal is strictly increasing this is equivalent to maximising the quadratic function. Differentiating wrt $z$ gives $2\\beta_2z + \\beta_1$. Setting to 0 and solving for $z$ gives $z = \\frac{-\\beta_1}{2\\beta_2}$."
      ]
    },
    {
      "cell_type": "code",
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 321
        },
        "id": "SE4AuZyTmPyj",
        "outputId": "ac77f8aa-c47f-4564-f5e1-03a6db772f26"
      },
      "source": [
        "print(f'z_max = {-beta_1_hat/(2*beta_2_hat):.1f}')\n",
        "# we can also find the posterior distribution of the age of maximal use using \n",
        "# our posterior samples. We plot its distribution and also show the upper and \n",
        "# lower bounds of the 95% credible interval\n",
        "ss = -samples[:,1]/(2*samples[:,2])\n",
        "sns.kdeplot(ss, fill=True) \n",
        "low = np.quantile(ss, 0.025)\n",
        "upp = np.quantile(ss, 0.975)\n",
        "plt.axvline(low, color='green')\n",
        "plt.axvline(upp, color='green')"
      ],
      "execution_count": 36,
      "outputs": [
        {
          "output_type": "stream",
          "name": "stdout",
          "text": [
            "z_max = 31.8\n"
          ]
        },
        {
          "output_type": "execute_result",
          "data": {
            "text/plain": [
              "<matplotlib.lines.Line2D at 0x7f845afd8f50>"
            ]
          },
          "metadata": {},
          "execution_count": 36
        },
        {
          "output_type": "display_data",
          "data": {
            "image/png": 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\n",
            "text/plain": [
              "<Figure size 432x288 with 1 Axes>"
            ]
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "Finally we use the same model with PyMC3 to check we obtain similar results."
      ],
      "metadata": {
        "id": "9Dq9GIAjWBCb"
      }
    },
    {
      "cell_type": "code",
      "source": [
        "model = pm.Model()\n",
        "with model:\n",
        "    beta0 = pm.Normal(\"beta0\", mu=0, sd=10)\n",
        "    beta1 = pm.Normal(\"beta1\", mu=0, sd=10)\n",
        "    beta2 = pm.Normal(\"beta2\", mu=0, sd=10)\n",
        "  \n",
        "    def phi(x):\n",
        "        \"\"\"Probit transformation.\"\"\"\n",
        "        import theano.tensor as t\n",
        "        return 0.5 * (1 + t.erf(x / t.sqrt(2)))\n",
        "\n",
        "    t = beta0 + beta1*age + beta2*age**2\n",
        "    Y = pm.Binomial(\"Y\", n=total, p=phi(t), observed=use)\n",
        "    trace = pm.sample(1000,start={'beta0':-7.6, 'beta1':0.57, 'beta2':-0.009})"
      ],
      "metadata": {
        "id": "Op74D3GsWAZL"
      },
      "execution_count": null,
      "outputs": []
    },
    {
      "cell_type": "code",
      "source": [
        "with model:\n",
        "    display(az.summary(trace)) "
      ],
      "metadata": {
        "colab": {
          "base_uri": "https://localhost:8080/",
          "height": 160
        },
        "id": "Hu-O7fM0W-hO",
        "outputId": "07bf09f0-933d-437c-8ea2-107169fc7d36"
      },
      "execution_count": 53,
      "outputs": [
        {
          "output_type": "display_data",
          "data": {
            "text/html": [
              "\n",
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              "  <thead>\n",
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              "      <th></th>\n",
              "      <th>mean</th>\n",
              "      <th>sd</th>\n",
              "      <th>hdi_3%</th>\n",
              "      <th>hdi_97%</th>\n",
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              "  <tbody>\n",
              "    <tr>\n",
              "      <th>beta0</th>\n",
              "      <td>-7.662</td>\n",
              "      <td>0.616</td>\n",
              "      <td>-8.826</td>\n",
              "      <td>-6.539</td>\n",
              "      <td>0.035</td>\n",
              "      <td>0.025</td>\n",
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              "      <td>0.572</td>\n",
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              "      <td>319.0</td>\n",
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              "          document.querySelector('#df-b42e700c-727a-42ea-ab68-88f34a80a800 button.colab-df-convert');\n",
              "        buttonEl.style.display =\n",
              "          google.colab.kernel.accessAllowed ? 'block' : 'none';\n",
              "\n",
              "        async function convertToInteractive(key) {\n",
              "          const element = document.querySelector('#df-b42e700c-727a-42ea-ab68-88f34a80a800');\n",
              "          const dataTable =\n",
              "            await google.colab.kernel.invokeFunction('convertToInteractive',\n",
              "                                                     [key], {});\n",
              "          if (!dataTable) return;\n",
              "\n",
              "          const docLinkHtml = 'Like what you see? Visit the ' +\n",
              "            '<a target=\"_blank\" href=https://colab.research.google.com/notebooks/data_table.ipynb>data table notebook</a>'\n",
              "            + ' to learn more about interactive tables.';\n",
              "          element.innerHTML = '';\n",
              "          dataTable['output_type'] = 'display_data';\n",
              "          await google.colab.output.renderOutput(dataTable, element);\n",
              "          const docLink = document.createElement('div');\n",
              "          docLink.innerHTML = docLinkHtml;\n",
              "          element.appendChild(docLink);\n",
              "        }\n",
              "      </script>\n",
              "    </div>\n",
              "  </div>\n",
              "  "
            ],
            "text/plain": [
              "        mean     sd  hdi_3%  hdi_97%  ...  mcse_sd  ess_bulk  ess_tail  r_hat\n",
              "beta0 -7.662  0.616  -8.826   -6.539  ...    0.025     320.0     315.0   1.01\n",
              "beta1  0.572  0.040   0.496    0.644  ...    0.002     315.0     319.0   1.01\n",
              "beta2 -0.009  0.001  -0.010   -0.008  ...    0.000     321.0     339.0   1.01\n",
              "\n",
              "[3 rows x 9 columns]"
            ]
          },
          "metadata": {}
        }
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "Recall our estimates of the means: $\\beta_0=-7.636$, $\\beta_1=0.57$, $\\beta_2=-0.009$. We can see that our algorithm yields very similar results to PyMC3. "
      ],
      "metadata": {
        "id": "v7DEFVWDJyeB"
      }
    }
  ]
}