Notebooks/Chap_5_Bayesian_exercise.ipynb

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        "<a href=\"https://colab.research.google.com/github/blhuillier/2024A_MathematicalAstronomy/blob/main/Notebooks/Chap_5_Bayesian_exercise.ipynb\" target=\"_parent\"><img src=\"https://colab.research.google.com/assets/colab-badge.svg\" alt=\"Open In Colab\"/></a>"
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      "source": [
        "# Monte-Carlo Integration"
      ],
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      "source": [
        "import pandas as pd\n",
        "import ssl\n",
        "import numpy as np\n",
        "import matplotlib.pyplot as plt\n",
        "from scipy import stats\n",
        "print (pd.__version__)\n",
        "\n",
        "# print (plt.rcParams)\n",
        "mycols = ['#1f77b4', '#ff7f0e', '#2ca02c', '#d62728', '#9467bd', '#8c564b', '#e377c2', '#7f7f7f', '#bcbd22', '#17becf']"
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          "text": [
            "2.0.3\n"
          ]
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      "source": [
        "## 1. We are interested in calculating\n",
        "$$I=\\int_0^1  {\\sqrt{1-u^2}}\\mathrm{d}u$$\n",
        "\n",
        "1. Draw the integrand. What do you expect the answer will be? (do not calculate analytically!)\n",
        "\n",
        "2. Numerically calculate $I$ with the Monte-Carlo method\n",
        "\n",
        "3. Conclude"
      ]
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    {
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    {
      "cell_type": "markdown",
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      "source": [
        "## 2. Let us consider  $$I=\\int_{-1}^{1} \\cos 2\\pi t \\mathrm d t$$\n",
        "\n",
        "\n",
        "\n"
      ]
    },
    {
      "cell_type": "markdown",
      "source": [
        "### 2.1 Plot the function & guess the area"
      ],
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    {
      "cell_type": "code",
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    {
      "cell_type": "markdown",
      "source": [
        "### 2.2 Calculate it using the Monte-Carlo Method"
      ],
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    {
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      "cell_type": "markdown",
      "source": [
        "# 3. Model Selection"
      ],
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      "cell_type": "markdown",
      "source": [
        "In Assignment 3, we performed MCMC on the Pantheon dataset.\n",
        "\n",
        "The [Data](\"https://raw.githubusercontent.com/dscolnic/Pantheon/master/lcparam_full_long.txt\")\n",
        "and the systematics part of the covariance matrix are [here](https://raw.githubusercontent.com/dscolnic/Pantheon/master/sys_full_long.txt)\n",
        "\n",
        "Among the following models, which model fits the Pantheon supernovae compilation better? why?\n",
        "$\\Lambda$CDM, Flat $\\Lambda$CDM, $w$CDM, Flat $w$CDM"
      ],
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      "cell_type": "markdown",
      "source": [
        "### 3.1 Which tools do you know can compare these models?"
      ],
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      "source": [
        "## 3.2 Calculate them and rank the models."
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