drphilmarshall/Combining_Four_Lenses.ipynb

## Combining_Four_Lenses.ipynb
{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Combining Four Lenses: What \"Tensions\" Should We Expect?\n",
    "\n",
    "_Phil Marshall, August 2018_\n",
    "\n",
    "In this notebook we use a simple univariate Gaussian model to explore plausible tensions between 4 independent measurements of the same parameter (in our case, $H_0$).\n",
    "\n",
    "### Requirements\n",
    "\n",
    "For the animations in this notebook to run, you will need the `ffmpeg` package to be installed."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import pandas as pd\n",
    "import matplotlib\n",
    "import matplotlib.pyplot as plt\n",
    "import matplotlib.animation as animation\n",
    "matplotlib.rc('animation', html='html5')\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Context\n",
    "\n",
    "Simon and Vivien made the following plots after unblinding J1206. At first glance, all 4 lens systems seem consistent with each other, in the lefthand panel. But is RXJ1131 the odd one out, in tension with the other three?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<table><tr>\n",
    "    <td><img src=\"august2018/h0licow_august2018_4lensesH0.png\"></img></td>\n",
    "    <td><img src=\"august2018/h0licow_august2018_3lenses-no-rxj1131-H0.png\"></img></td>\n",
    "</tr></table>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Simple Model\n",
    "\n",
    "Let's pretend that each lens acts as a direct measuring instrument for $H_0$, in that we observe a data point for each lens, $d_i$, and that this is a noisy estimator of $H_0$, with Gaussian sampling distribution:\n",
    "\n",
    "${\\rm Pr}(d_i|H_0) = \\frac{1}{\\sqrt{2\\pi\\sigma_i^2}} \\exp\\left[ - \\frac{(d_i - H_0)^2}{2\\sigma_i^2} \\right]$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To simulate a 4-lens dataset using this model, we just need the uncertainty $\\sigma_i$ for each lens."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "sigma = {'B1608':3.45, 'RXJ1131':5.1, 'HE0435':5.75, 'J1206':5.25}\n",
    "colors = {'B1608':'crimson', 'RXJ1131':'royalblue', 'HE0435':'lightslategrey', 'J1206':'seagreen'}\n",
    "actuals = {'B1608':69.8, 'RXJ1131':80.9, 'HE0435':74.5, 'J1206':68.8}"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now, we can draw a random $H_0$ and 4 random observed datapoints, and combine them as in the lefthand figure. For this, we'll need the prior PDF for $H_0$ and remember that\n",
    "\n",
    "${\\rm Pr}(H_0|\\boldsymbol{d}) = \\frac{1}{Z} {\\rm Pr}(H_0) \\prod_i {\\rm Pr}(d_i|H_0)$\n",
    "\n",
    "where $Z$ is the evidence for the model. Let's implement the above system in a simple class, and have it emulate the above plots as well as investigate the \"tension\" with any \"outliers\"."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "class SimpleModel(object):\n",
    "    \n",
    "    def __init__(self, sigma, colors):\n",
    "        self.sigma = sigma\n",
    "        self.lenses = list(sigma.keys())\n",
    "        self.data = {key: None for key in self.lenses}\n",
    "        self.colors = colors\n",
    "        self.started_plot = False\n",
    "        return\n",
    "        \n",
    "    def choose_H0(self, to_be=None):\n",
    "        if to_be == None:\n",
    "            self.H0 = 60.0 + (80.0 - 60.0) * np.random.random()\n",
    "        else:\n",
    "            self.H0 = to_be\n",
    "        return\n",
    "    \n",
    "    def make_data(self):\n",
    "        for lens in self.lenses:\n",
    "            self.data[lens] = self.H0 + self.sigma[lens] * np.random.randn()\n",
    "        return\n",
    "    \n",
    "    def start_new_plot(self):\n",
    "        self.npts = 100\n",
    "        self.x = np.linspace(35.1, 109.9, self.npts)\n",
    "        self.y = {key: np.zeros(self.npts) for key in self.lenses}\n",
    "        if self.started_plot:\n",
    "            self.fig.clear()\n",
    "            self.ax.clear()\n",
    "        else:    \n",
    "            self.fig = plt.figure()\n",
    "            self.started_plot = True\n",
    "        self.ax = self.fig.add_subplot(111)\n",
    "        #self.ax.set_xlim(1.01*self.x[0], 0.99*self.x[-1])\n",
    "        self.ax.set_xlim(48, 99)  # same than J1206 plot       \n",
    "        self.ax.set_ylim(0.0015,0.20)  # slightly above 0 to hide the 0-wings of the Gaussian\n",
    "        self.ax.set_xlabel(r\"$H_{\\rm{0}}\\rm{\\ [km\\,s^{-1}\\,Mpc^{-1}]}$\", fontsize=18)\n",
    "        self.ax.set_ylabel('probability density', fontsize=18)\n",
    "        self.ax.tick_params(axis='both', which='major', labelsize=15)\n",
    "        self.ax.set_yticks([])\n",
    "        \n",
    "        \n",
    "        return\n",
    "        \n",
    "    def plot(self):\n",
    "        if not self.started_plot:\n",
    "            self.start_new_plot()\n",
    "        artists = []\n",
    "             \n",
    "        printval = r\"${{{0}}}_{{-{1}}}^{{+{2}}}$\"\n",
    "        fmt = \"{{0:{0}}}\".format(\".1f\").format\n",
    "        for lens in self.lenses:\n",
    "            self.y[lens] = self.gaussian(self.x, self.data[lens], self.sigma[lens])\n",
    "            artists.append(self.ax.fill(self.x, self.y[lens], color=self.colors[lens], alpha=0.5, linewidth=0.0, label=lens)[0])   \n",
    "            txt = r'$H_{0}: $' + printval.format(fmt(self.data[lens]), fmt(self.sigma[lens]), fmt(self.sigma[lens]))\n",
    "            # can't figure out how to add annotation to a list of artists...??\n",
    "            #artists.append(self.ax.annotate(txt, xy=(0.0, 0.0), xytext=(0.02, 0.9 - 0.1 * self.lenses.index(lens)), xycoords='axes fraction', fontsize=14, color=self.colors[lens]))\n",
    "            \n",
    "\n",
    "        yA = self.gaussian(self.x, self.measured_H0_value, self.measured_H0_error)\n",
    "        artists.append(self.ax.plot(self.x, yA, color='black', label='All')[0])\n",
    "        x0 = self.H0 * np.ones(self.npts)\n",
    "        y0 = np.linspace(0.0, 1.0, self.npts)\n",
    "        artists.append(self.ax.plot(x0, y0, color='black', linestyle=':', label='Truth')[0])\n",
    "        self.ax.legend(artists, self.lenses+['All', 'Truth'])\n",
    "        \n",
    "        txt = r'$H_{0}: $' + printval.format(fmt(self.measured_H0_value), fmt(self.measured_H0_error), fmt(self.measured_H0_error))\n",
    "        #self.ax.annotate(txt, xy=(0.0, 0.0), xytext=(0.02, 0.9 - 0.1 * len(self.lenses)), xycoords='axes fraction', fontsize=14, color='black')       \n",
    "        \n",
    "        \n",
    "        return artists\n",
    "\n",
    "    def gaussian(self, x, x0, s):\n",
    "        return np.exp(-(x-x0)*(x-x0)/(s*s))/np.sqrt(2*np.pi*s*s)\n",
    "    \n",
    "    def infer_H0(self):\n",
    "        d = pd.Series(self.data, name='d')\n",
    "        s = pd.Series(self.sigma, name='sigma')\n",
    "        self.df = pd.concat([d, s], axis=1)\n",
    "        self.measured_H0_value, self.measured_H0_error = weighted_mean(self.df.d, self.df.sigma)\n",
    "        return\n",
    "    \n",
    "    def get_outlier_distance(self):\n",
    "        self.infer_H0()\n",
    "        self.df.sort_values(by='d')\n",
    "        H0low, errlow = weighted_mean(self.df.d[1:], self.df.sigma[1:])\n",
    "        distlow = distance(H0low, errlow, self.df.d[0], self.df.sigma[0])\n",
    "        H0high, errhigh = weighted_mean(self.df.d[:3], self.df.sigma[:3])\n",
    "        disthigh = distance(H0high, errhigh, self.df.d[3], self.df.sigma[3])\n",
    "        return np.max([distlow, disthigh])\n",
    "    \n",
    "# Utility functions:\n",
    "def weighted_mean(x, s):\n",
    "    w = 1.0 / (s * s)\n",
    "    mean = np.sum(w * x) / np.sum(w)\n",
    "    err = np.sqrt(1.0 / np.sum(w))\n",
    "    return mean, err\n",
    "    \n",
    "def distance(x1, s1, x2, s2):\n",
    "    dx = np.abs(x1 - x2)\n",
    "    ds = np.sqrt(s1*s1 + s2*s2)\n",
    "    return dx/ds"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here's a quick demo: we'll set the true value of $H_0$, draw some mock observations given the obersvational uncertainties in each lens system, then infer $H_0$ from those data, and make a plot like the left-hand above."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "True H0 =  72.0\n"
     ]
    }
   ],
   "source": [
    "lets = SimpleModel(sigma, colors)\n",
    "lets.choose_H0(72.0)\n",
    "print(\"True H0 = \",lets.H0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Mock observations:  {'B1608': 74.22056036485584, 'RXJ1131': 76.73590064670726, 'HE0435': 72.41937341520436, 'J1206': 70.11123622090422}\nInferred H0 =  73.65511821977293  +/-  2.300281576073681\n"
     ]
    }
   ],
   "source": [
    "lets.make_data()\n",
    "print(\"Mock observations: \",lets.data)\n",
    "\n",
    "lets.infer_H0()\n",
    "print(\"Inferred H0 = \",lets.measured_H0_value, \" +/- \", lets.measured_H0_error)\n",
    "\n",
    "frame = lets.plot()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Developing Intuition\n",
    "\n",
    "Let's make 10 mock datasets, and make an animated plot of them to show us what we kinds of tensions to expect."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "scrolled": false
   },
   "outputs": [],
   "source": [
    "frames = []\n",
    "lets.start_new_plot()\n",
    "for x in np.arange(10):\n",
    "    lets.make_data()\n",
    "    lets.infer_H0()\n",
    "    frames.append(lets.plot());"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
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       "\">\n",
       "  Your browser does not support the video tag.\n",
       "</video>"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# @Phil, any idea how to make the animation larger? The xlabel is cut out of the gif.\n",
    "play = animation.ArtistAnimation(lets.fig, frames, interval=500, blit=True, repeat_delay=1000)\n",
    "play"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As you can see 1-2 sigma \"outliers\" happen most of the time."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For some other good resources for how to use the `matplotlib.animation` module, see this nice [blog post on \"Embedding Matplotlib Animations in Jupyter Notebooks\" by Louis Tiao](http://louistiao.me/posts/notebooks/embedding-matplotlib-animations-in-jupyter-notebooks/), and this [example](https://matplotlib.org/gallery/animation/dynamic_image.html#sphx-glr-gallery-animation-dynamic-image-py) of using the [`ArtistAnimation` class](https://matplotlib.org/api/_as_gen/matplotlib.animation.ArtistAnimation.html#matplotlib.animation.ArtistAnimation) from the matplotlib docs. Note that for the above inline animation to run, the `ffmpeg` package has to be installed - I did this with `brew install ffmpeg`."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "> Warning: Once you have made an animation, the `fig` attribute of the SimpleModel instance seems to be altered, such that it can no longer be used to `show` static plots:\n",
    ">\n",
    "> ```\n",
    "> lets.fig.show()\n",
    ">     UserWarning: matplotlib is currently using a non-GUI backend, so cannot show the figure\n",
    "> ```\n",
    "> If you need a static plot, start a new SimpleModel instance..."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "play.save('fourlenses.gif', dpi=80, writer='imagemagick')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Investigating Outliers\n",
    "\n",
    "Now let's look at how often we get an \"outlier\" like RXJ1131. In any given realization of the 4 measurements, there will be one (with either the highest, or lowest, $d$ value) that is furthest away from the cluster of the other three. Let's look at how many \"sigma\" that \"outlier\" is from the other three, adding the outlier's uncertainty to that of the cluster in quadrature."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Actual inferred H0 =  72.61831728377038  +/-  2.300281576073681\n"
     ]
    }
   ],
   "source": [
    "lets.data = actuals\n",
    "lets.infer_H0()\n",
    "print(\"Actual inferred H0 = \",lets.measured_H0_value, \" +/- \", lets.measured_H0_error)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "RXJ1131 lies  1.8194381134950584  sigma from the other three.\n"
     ]
    }
   ],
   "source": [
    "RXJ1131 = lets.get_outlier_distance()\n",
    "print(\"RXJ1131 lies \",RXJ1131,\" sigma from the other three.\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [],
   "source": [
    "N = 3000\n",
    "nsigma = np.zeros(N)\n",
    "for k in np.arange(N):\n",
    "    lets.make_data()\n",
    "    nsigma[k] = lets.get_outlier_distance()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Median of Monte Carlo distribution of outlier distances =   1.0161519615489754\n"
     ]
    }
   ],
   "source": [
    "median = np.median(nsigma)\n",
    "print(\"Median of Monte Carlo distribution of outlier distances =  \",median)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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qIm0XySwrK6vY17l7d7q/7sK7SMvMzCz27RrjUpA7uKeFEcSUHE2aNHEdwal2\n7axzKJN6ghxZGHOEq666ynUEp666aovrCMaELsh9FsYYY1Jc4GIhIiclMogpOQYPHpzS/ULdcUc6\nd9yR7jqGMaGKWSxEpL2IfAKs9d+3EJHHE57MJK2WLVvSsmVL1zGMMSEK0mbxD+ASvDGyUdWPRaRD\nQlOZpHbFFVe4jmCMCVmg01Cqmr9F71ACshhjjElSQYrFFhFpD6iInCgid+GfkiqKiFQQkWUi8rGI\nrBGRYf70BiLyvohsEJFZInKiP728/36j/3n94/heJoEGDhyYsuNvG5OqgpyGugUYC5wB5ACvA7cF\nWO4HoJOq7hWRcsDbIvIqXl9T/1DV50VkItAXeMJ//kZVG4nI1cBDQMKu0UzEzVqpol27dq4jOJWZ\nud11BGNCF+SmvJ3AdfGuWFUV2Ou/Lec/FOgERDoXmgY8gFcsuvmvAeYA40VE/PWYJNK9e3fXEZzq\n3n2r6wjGhK7QYiEij3G4S/KjBOkbSkROALKBRsAEYBOwW1UP+rPk4B2x4D9v8dd9UERygeqA9XZr\nksr+/d7Z2woVbOBIkzqKOrJYfrwrV9VDQLo/jveLwDkFzeY/SxGf5RGRm4CbAOrVq3e8Ec0xuPPO\nOwF45JFHHCdxY9Cg5gCMGVN431DGlDaFFov8fUKJSBVvsn4b70ZUdbeIZAFtgaoiUtY/uqgDRI7p\nc4C6QI6IlAXSgK8LWNckYBJARkaGnaJyoGPHjq4jGGNCFuSmvAwRWQWsBFb7Vze1CrBcTf+IAhGp\nCPwS7yqqJUDkQv1ewEv+6/n+e/zP37T2iuTUtWtXunbt6jqGMSZEQa6GmgLcqqr/ByAiFwBPA81j\nLFcbmOa3W5QBZqvqy/7d4M+LyIPAR8BT/vxP4Y33vRHviOLquL+NMcaYhAhSLL6NFAoAVX1bRGKe\nilLVlcBRfUKo6qdA6wKm7wd+HyCPceyOO+4AYMyYMY6TGGPCEqRYLBORfwIz8RqcrwKyROQ8AFX9\nMIH5TBLq3Lmz6whOde78pesIxoQuSLGIdK85NN/09hy+b8KkECsWVixM6glyU55d+mKOcPCgd5tM\n2bKpOXZWbm45ANLSDjhOYkx4Yv7a/SuaegL1o+cPclOeKZ3uuusuIHXbLIYO9YaVtfssTCoJ8qfh\nK8B7wCrAblk1XHbZZa4jGGNCFqRYVFDV/glPYkqMiy++2HUEY0zIgnRR/oyI/EFEaotItcgj4clM\n0tq/fz/79+93HcMYE6IgRxY/Ag8DgzncV5MCDRMVyiS3QYMGAanbZmFMKgpSLPoDjfyuyo3h8ssv\ndx3Bqcsv/5/rCMaELkixWAM8ga6aAAATd0lEQVR8n+ggpuTo1Cm1b63p1GmH6wjGhC5IsTgErBCR\nJXij3wF26Wwq27vXG9Pq5JNPdpzEje3bywNQq9YPMeY0pvQIUizm+Q9jALjvvvuA1G2zGDnSG5bF\n7rMwqSTIHdzTYs1jUstvf/tb1xGMMSEraljV2ap6pT+WRfS4EoI3CFKsLspNKdWhQwfXEYwxISvq\nyOJ2/9lGuTFHyM3NBSAtLc1xEmNMWIoaVnWb/3InsE9VfxKRnwONgVfDCGeS09ChXgfEqdpmYUwq\nCtLAvRT4fyJyCrAYWI43psV1iQxmkteVV17pOoJTV165xXUEY0IXpFiIqn4vIn2Bx1T17yLyUaKD\nmeTVvn171xGcat9+l+sIxoQuSN9QIiLt8I4kFvjTUnMgAwPA119/zddff+06hjNffFGRL76o6DqG\nMaEKUixuB+4BXlTVNSLSEFgSayERqSsiS0RkrYisEZHb/enVRGSRiGzwn0/xp4uIjBORjSKyMjJs\nq0k+w4cPZ/jw4a5jOPPoo2fz6KNnu45hTKiC3GexFK/dIvL+UyDI3dsHgTtV9UMRqQxki8gioDew\nWFVHicggYBAwEOgCnOU/2gBP+M8myVx77bWuIyS9rKwsZ9vOzMx0tm1TeiXsdJJ/NdU2//W3IrIW\nOAPoBmT6s00DsvCKRTdguqoq8J6IVBWR2lFXZZkk0bp1a9cRjDEhC3Ia6riJSH2gJfA+cGqkAPjP\ntfzZzgCiLzPJ8aeZJLN9+3a2b9/uOoYxJkSFFgsRech//v3xbEBETgZeAO5Q1T1FzVrAND1qJpGb\nRGS5iCzfscN6/3Rh5MiRjBw50nUMY0yIijoNdamI3IfXuP2vY1m5iJTDKxTPqepcf/JXkdNLIlIb\niPyJmgPUjVq8DrA1/zpVdRIwCSAjI+OoYmISr0ePHq4jONWjx+euIxgTuqKKxUK8u7dPEpE9+H1C\ncbhvqCpFrVhEBHgKWKuqj0Z9NB/oBYzyn1+Kmt5PRJ7Ha9jOtfaK5NSqVSvXEZxq1eob1xGMCV2h\np6FU9W5VTQMWqGoVVa0c/Rxg3ecDPYBOIrLCf1yKVyQuFpENwMX+e4BXgE+BjcCTwK3H8b1MAm3d\nupWtW4866EsZGzeezMaNqTmWh0ldQS6d7SYipwK/8Ce9r6oxGwtU9W0KbocAuKiA+RW4LdZ6jXt/\n//vfgdTtG2r8+EaAjWdhUkvMYuE3cI/Gu8RVgMdE5G5VnZPgbCZJ9e7d23UEY0zIgtxncR/wC1Xd\nDiAiNYE3ACsWKSo9Pd11BGNMyILcZ1EmUih8uwIuZ0qpL774gi+++MJ1DGNMiIIcWSwUkdeAmf77\nq/Aao02KevRR7+K2VG2zMCYVBWngvltEfgtcgNdmMUlVX0x4MpO0brzxRtcRnLrxxk9dRzAmdIH6\nhvJvqJsbc0aTEpo2beo6glNNmxbVEYExpZO1PZi4ffbZZ3z22WeuYzizenUVVq8OcquRMaWHFQsT\nt7FjxzJ27FjXMZyZPLkhkyc3dB3DmFDZiHcmbrfccovrCMaYkB3TkYWIPFDMOUwJ0rhxYxo3buw6\nhjEmRMd6Giq7WFOYEmXjxo1s3LjRdQxjTIiOqVio6r+LO4gpOcaPH8/48eNdxzDGhChI31B1gMfw\n7rP4CXgbuF1VcxKczSSpfv36uY7gVL9+dlRlUk+QBu6ngRlAZMS86/1pFycqlElujRo1ch3BqUaN\n9rqOYEzogpyGqqmqT6vqQf8xFaiZ4Fwmia1bt45169a5juFMdvYpZGef4jqGMaEKcmSxU0Su53Df\nUNfgdSZoUtTEiROB1O0b6plnfgbYiHkmtQQpFn2A8cA/8IZVfcefZlLU7bff7jqCMSZkQToS/AK4\nPIQspoRo0KCB6wjGmJAVWixEZEgRy6mq/jUBeUwJsHr1asA6FDQmlRTVwP1dAQ+AvsDABOcySWzy\n5MlMnjzZdQxjTIgKPbJQ1Ucir0WkMnA7cAPwPPBIYctFLTMF6ApsV9Wm/rRqwCygPrAZuFJVvxER\nAcYClwLfA71V9cNj+0om0fr37+86glP9+693HcGY0BV56ayIVBORB4GVeIXlPFUdmG+Y1cJMBTrn\nmzYIWKyqZwGL/fcAXYCz/MdNwBOBv4EJXb169ahXr57rGM7Uq7ePevX2uY5hTKgKLRYi8jDwAfAt\n0ExVH1DVwNcKqupS4Ot8k7sB0/zX04DuUdOnq+c9oKqI1A66LROuFStWsGLFCtcxnHnnneq88051\n1zGMCVVRRxZ3AqcD9wFbRWSP//hWRI51qLBTVXUbgP9cy59+BrAlar4cf9pRROQmEVkuIst37Nhx\njDHM8Zg6dSpTp051HcOZ2bPrMnt2XdcxjAlVUW0WYQ6MJAVFKGhGVZ0ETALIyMgocB6TWAMGDHAd\nwRgTsrAHP/pKRGqr6jb/NFOk7SMHiP5TrQ6wNeRsJqDTTz/ddQRjTMjCHlZ1PtDLf90LeClqek/x\ntAVyI6erTPLJzs4mO9uGNDEmlSTsyEJEZgKZQA0RyQGGAqOA2SLSF/iCwz3ZvoJ32exGvEtnb0hU\nLnP8nnnmGQBatWrlOIkxJiwJKxaqek0hH11UwLwK3JaoLKZ43Xvvva4jOHXvvWtdRzAmdGG3WZhS\noFatWrFnKsVq1frBdQRjQmfFwsRt2bJlALRu3dpxEjfefNMbzqVTp+S8dDsrK8vJdjMzM51s14TD\nioWJ24wZM4DULRbz53u3ACVrsTAmEaxYmLgNGVJUh8TGmNLIioWJW7Vq1VxHMMaELOz7LEwp8M47\n7/DOO++4jmGMCZEdWZi4zZ49G4D27ds7TmKMCYsVCxO3YcOGuY7g1LBha1xHMCZ0VixM3NLS0lxH\ncCot7YDrCMaEztosTNyWLl3K0qVLXcdwZuHC01i48DTXMYwJlR1ZmLjNnTsXgA4dOjhO4kakUHTu\n/KXjJMaEx4qFiduDDz7oOoIxJmRWLEzcTj75ZNcRjDEhs2Jh4vbmm28C0KlTJ8dJTDJx1ScVWL9U\nYbBiYeI2f/58wIqFManEioWJ26hRo1xHcGrUqJWuIxgTOisWJm4VKlRwHcGpChV+ch3BmNDZfRYm\nbosWLWLRokWuYzgzb97pzJt3uusYxoTKioWJ24IFC1iwYIHrGM5kZdUiKyu1Rws0qSepTkOJSGdg\nLHACMFlVU/vkeJIaPXq06wjGHMFGB0y8pDmyEJETgAlAF+Bc4BoROddtKlOQsmXLUrZsUv2dYYxJ\nsKQpFkBrYKOqfqqqPwLPA90cZzIFWLhwIQsXLnQdwxgTomT68/AMYEvU+xygjaMspgiRQtG5c2fH\nSYxxK5VuREymYiEFTNOjZhK5CbjJf7tXRNYfw7ZqADuPYbmwlIh8HTt2dJ2jMKHsv+P4+iXiv6/r\nEIVI5mxQcvL9LN4Fk6lY5AB1o97XAbbmn0lVJwGTjmdDIrJcVTOOZx2JZPmOj+U7PsmcL5mzQenO\nl0xtFh8AZ4lIAxE5EbgamO8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      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f68c07626a0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.hist(nsigma, color='gray', alpha=0.5, label=\"Monte Carlo\")\n",
    "plt.xlabel('Outlier distance (sigma)')\n",
    "plt.ylabel('No. of simple model realizations')\n",
    "plt.axvline(RXJ1131, linestyle='dashed', color='blue', label=\"RXJ1131\")\n",
    "plt.axvline(median, linestyle='dotted', color='black', label=\"MC median\")\n",
    "plt.legend();"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "According to our simple model, having an \"outlier\" like RXJ1131 is completely expected: its distance of 1.1 sigma is close to the median of the distribution of expected \"outlier\" distances."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.6.6"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}