{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Bayesian Statistics Seminar\n",
    "\n",
    "Copyright 2017 Allen Downey\n",
    "\n",
    "MIT License: https://opensource.org/licenses/MIT"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from __future__ import print_function, division\n",
    "\n",
    "%matplotlib inline\n",
    "\n",
    "import warnings\n",
    "warnings.filterwarnings('ignore')\n",
    "\n",
    "import numpy as np\n",
    "import pandas as pd\n",
    "\n",
    "import thinkbayes2\n",
    "import thinkplot"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Survival analysis\n",
    "\n",
    "Suppose that you are an auto insurance company interested in the time between collisions for a particular driver.  If the probability of a collision is roughly constant over time, the time between collisions will follow an exponential distribution.\n",
    "\n",
    "Here's an example with parameter $\\lambda = 0.5$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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oklEjhjBq5BBGjRjKyGGDGTa0iuFDBjFsSFXL9NAqXnrhORbdfFNqx0964/sr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\nmVlRHBhmZlaU/w8YTXqofMvfSAAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f7c939a9410>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from thinkbayes2 import MakeExponentialPmf\n",
    "\n",
    "pmf = MakeExponentialPmf(lam=0.5, high=30)\n",
    "thinkplot.Pdf(pmf)\n",
    "thinkplot.Config(xlabel='Lifetime', ylabel='PMF')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For the exponential distribution, the mean and standard deviation are $1/\\lambda$.\n",
    "\n",
    "In this case they are only approximate because we truncated the distribution."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(1.9255614181751446, 1.9994621154028254)"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "pmf.Mean(), pmf.Std()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "From the PMF, we can compute the CDF."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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cA7wSeClwcZIXjxxzIfC8qnoBsBe4tsuaZtXKysq0S+iU19df83xtMP/XtxVdtxjOBe6p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\nYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f7c9148e1d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "cdf = pmf.MakeCdf()\n",
    "thinkplot.Cdf(cdf)\n",
    "thinkplot.Config(xlabel='Lifetime', ylabel='CDF')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And from the CDF, we can compute the survival function, which is the complement of the CDF.\n",
    "\n",
    "$SF(x) = Prob\\{X > x\\} = 1 - Prob\\{X \\le x\\} = 1 - CDF(x)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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1ZsxgF5VEIpEKi+T3ZIgkzDiTG4lEcsHMMJLbzdK+B8da86eXaPraA6ntqfd0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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f7cc47955d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from survival import MakeSurvivalFromCdf\n",
    "\n",
    "sf = MakeSurvivalFromCdf(cdf)\n",
    "thinkplot.Plot(sf)\n",
    "thinkplot.Config(xlabel='Lifetime', ylabel='Survival function')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "From the survival function we can get the hazard function, which is the probability of a collision at $x$, given no collision prior to $x$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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RWg5j38m1uSvJzMzGqK3v7iSfzwMwcXwrrePKfxYDODCYmdW9ak5uAwcGM7O6t7mKdySB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\nluLAYGZmKQ4MZmaW8v8BymyxnSV6z5EAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f7c910bb650>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "hf = sf.MakeHazardFunction()\n",
    "thinkplot.Plot(hf)\n",
    "thinkplot.Config(xlabel='Lifetime', ylabel='Hazard function')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If the distribution is truly exponential, the hazard function is constant for all $x$.\n",
    "\n",
    "In this case it goes to 1 at the end, again because we truncated the distribution."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise:** Go back and increase the value of `high`, and confirm that the hazard function is a constant until we approach the point where we cut off the distribution."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Given the survival function, we can compute the distribution of remaining lifetime, conditioned on current age.  The following function computes the mean remaining lifetime for a range of ages."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def RemainingLifetime(sf):\n",
    "    \"\"\"Computes remaining lifetime as a function of age.\n",
    "    \n",
    "    sf: survival function\n",
    "    returns: Series that maps from age to remaining lifetime\n",
    "    \"\"\"\n",
    "    pmf = sf.MakePmf()\n",
    "    d = {}\n",
    "    for t in sorted(pmf.Values()):\n",
    "        pmf[t] = 0\n",
    "        if pmf.Total():\n",
    "            pmf.Normalize()\n",
    "            d[t] = pmf.Mean() - t\n",
    "\n",
    "    return pd.Series(d)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And here's what it looks like for the exponential survival function."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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y9DjvbN3X5n0qGEREspiZpfzuJAWDiEiWmzTmZL9JqzbsaPP+FAwiIlluwuhB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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f7c90ff4590>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "mean_rem_life = RemainingLifetime(sf)\n",
    "thinkplot.Plot(mean_rem_life)\n",
    "thinkplot.Config(xlabel='Lifetime', ylabel='Survival function')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The mean time until a collision is pretty much constant, until we approach the point where we truncate the distribution."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The Weibull distribution\n",
    "\n",
    "The Weibull distribution is a generalization of the exponential distribution that takes an additional \"shape\" parameter, `k`.\n",
    "\n",
    "When `k=1`, the Weibull is an exponential distribution.  Other values of `k` yield survival curves with different shapes, and hazard functions that increase, decrease, or both.  So the Weibull family can capture a wide range of survival patterns."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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Yd390lPu/Adju7rsBzOxxYAWwNVBmBfBY4nhrzWySmTW6e6e7dyfK1CTqOmQXWVVId0gl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\nUWCIiEhW/j+9nm9e+kHU7wAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f7c90fbd0d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from thinkbayes2 import MakeWeibullPmf\n",
    "\n",
    "pmf = MakeWeibullPmf(lam=2.0, k=1.5, high=30)\n",
    "thinkplot.Pdf(pmf)\n",
    "thinkplot.Config(xlabel='Lifetime', ylabel='PMF')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise**: In the previous section, replace the exponential distribution with a Weibull distribituion and run the analysis again.  What can you infer about the values of the parameters and the behavior of the hazard function and remaining lifetime?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Bayesian survival analysis"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Suppose you are the manager of a large building with many light fixtures.  To figure out how often you will need to replace lightbulbs, you install 10 bulbs and measure the time until they fail.\n",
    "\n",
    "To generate some fake data, I'll choose a Weibull distribution and generate a random sample (let's suppose it's in years):"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([ 3.93306269,  0.72048   ,  0.78116042,  1.22621775,  1.45191502,\n",
       "        1.78698079,  3.18159171,  0.97084327,  3.00128723,  1.48932615])"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def SampleWeibull(lam, k, n=1):\n",
    "    return np.random.weibull(k, size=n) * lam\n",
    "\n",
    "data = SampleWeibull(lam=2, k=1.5, n=10)\n",
    "data"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "**Exercise:** Write a class called `LightBulb` that inherits from `Suite` and provides a `Likelihood` function that takes an observed lifespan as data and a tuple, `(lam, k)`, as a hypothesis.  It should return a likelihood proportional to the probability of the observed lifespan in a Weibull distribution with the given parameters.\n",
    "\n",
    "Test your method by creating a `LightBulb` object with an appropriate prior and update it with the data above.\n",
    "\n",
    "Plot the posterior distributions of `lam` and `k`.  As the sample size increases, does the posterior distribution converge on the values of `lam` and `k` used to generate the sample?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Hint\n",
    "\n",
    "from thinkbayes2 import Suite, Joint, EvalWeibullPdf\n",
    "\n",
    "class LightBulb(Suite, Joint):\n",
    "    \n",
    "    def Likelihood(self, data, hypo):\n",
    "        lam, k = hypo\n",
    "        x = data\n",
    "        like = 1\n",
    "        return like"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# Solution goes here"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# Solution goes here"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# Solution goes here"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# Solution goes here"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# Solution goes here"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# Solution goes here"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise:**  Go back and run this analysis again with `n=20` and see if the posterior distributions seem to be converging on the actual parameters. "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Censored data"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "**Exercise:** Now suppose that instead of observing a complete lifespan, you observe a lightbulb that has operated for 1 year and is still working.  Write another version of `LightBulb` that takes data in this form and performs an update. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Hint\n",
    "\n",
    "from thinkbayes2 import EvalWeibullCdf\n",
    "\n",
    "class LightBulb2(Suite, Joint):\n",
    "    \n",
    "    def Likelihood(self, data, hypo):\n",
    "        lam, k = hypo\n",
    "        x = data\n",
    "        like = 1\n",
    "        return like"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# Solution goes here"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# Solution goes here"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# Solution goes here"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# Solution goes here"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# Solution goes here"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# Solution goes here"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note: based on this data alone, we can rule out some small values of `lam` and `k`, but we can't rule out large values.  Without more data or a more informative prior, the results are not useful.\n",
    "\n",
    "To see why, try increasing the upper bounds in the prior distribition."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "**Exercise:** Suppose you install a light bulb and then you don't check on it for a year, but when you come back, you find that it has burned out.  Extend `LightBulb` to handle this kind of data, too."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# Hint\n",
    "\n",
    "class LightBulb3(Suite, Joint):\n",
    "    \n",
    "    def Likelihood(self, data, hypo):\n",
    "        lam, k = hypo\n",
    "        x = data\n",
    "        like = 1\n",
    "        return like"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# Solution goes here"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# Solution goes here"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# Solution goes here"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# Solution goes here"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# Solution goes here"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# Solution goes here"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This example has some of the same problems as the previous one.  Based on this data alone, we can't pin down the parameters much."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Pulling it together"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise:** Suppose you have 15 lightbulbs installed at different times over a 10 year period.  When you observe them, some have died and some are still working.  Write a version of `LightBulb` that takes data in the form of a `(flag, x)` tuple, where:\n",
    "\n",
    "1. If `flag` is `eq`, it means that `x` is the actual lifespan of a bulb that has died.\n",
    "2. If `flag` is `gt`, it means that `x` is the current age of a bulb that is still working, so it is a lower bound on the lifespan.\n",
    "3. If `flag` is `lt`, it means that `x` is the elapsed time between installation and the first time the bulb is seen broken, so it is an upper bound on the lifespan."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To help you test, I will generate some fake data.\n",
    "\n",
    "First, I'll generate a Pandas DataFrame with random start times and lifespans.  The columns are:\n",
    "\n",
    "* `start`: time when the bulb was installed\n",
    "\n",
    "* `lifespan`: lifespan of the bulb in years\n",
    "\n",
    "* `end`: time when bulb died or will die\n",
    "\n",
    "* `age_t`: age of the bulb at t=10"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>lifespan</th>\n",
       "      <th>start</th>\n",
       "      <th>end</th>\n",
       "      <th>age_t</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>0.812265</td>\n",
       "      <td>1.451079</td>\n",
       "      <td>2.263344</td>\n",
       "      <td>8.548921</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>2.010584</td>\n",
       "      <td>3.822779</td>\n",
       "      <td>5.833363</td>\n",
       "      <td>6.177221</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>0.404421</td>\n",
       "      <td>1.048181</td>\n",
       "      <td>1.452601</td>\n",
       "      <td>8.951819</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>0.506743</td>\n",
       "      <td>3.503315</td>\n",
       "      <td>4.010057</td>\n",
       "      <td>6.496685</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>1.014877</td>\n",
       "      <td>4.508463</td>\n",
       "      <td>5.523340</td>\n",
       "      <td>5.491537</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "   lifespan     start       end     age_t\n",
       "0  0.812265  1.451079  2.263344  8.548921\n",
       "1  2.010584  3.822779  5.833363  6.177221\n",
       "2  0.404421  1.048181  1.452601  8.951819\n",
       "3  0.506743  3.503315  4.010057  6.496685\n",
       "4  1.014877  4.508463  5.523340  5.491537"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import pandas as pd\n",
    "\n",
    "lam = 2\n",
    "k = 1.5\n",
    "n = 15\n",
    "t_end = 10\n",
    "starts = np.random.uniform(0, t_end, n)\n",
    "lifespans = SampleWeibull(lam, k, n)\n",
    "\n",
    "df = pd.DataFrame({'start': starts, 'lifespan': lifespans})\n",
    "df['end'] = df.start + df.lifespan\n",
    "df['age_t'] = t_end - df.start\n",
    "\n",
    "df.head()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now I'll process the DataFrame to generate data in the form we want for the update."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "('eq', 0.81226508619456328)\n",
      "('eq', 2.0105842471709634)\n",
      "('eq', 0.40442059107469491)\n",
      "('eq', 0.50674258157079088)\n",
      "('eq', 1.0148765503730666)\n",
      "('eq', 2.265141425374412)\n",
      "('eq', 1.3281182217357757)\n",
      "('gt', 2.25256715166429)\n",
      "('gt', 0.46707158609037691)\n",
      "('eq', 2.7495410426490592)\n",
      "('eq', 0.25270950048067764)\n",
      "('eq', 2.0144323296919548)\n",
      "('eq', 1.0093772276491204)\n",
      "('eq', 1.5122724223847952)\n",
      "('eq', 1.4181328310784498)\n"
     ]
    }
   ],
   "source": [
    "data = []\n",
    "for i, row in df.iterrows():\n",
    "    if row.end < t_end:\n",
    "        data.append(('eq', row.lifespan))\n",
    "    else:\n",
    "        data.append(('gt', row.age_t))\n",
    "        \n",
    "for pair in data:\n",
    "    print(pair)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "# Solution goes here"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# Solution goes here"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# Solution goes here"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 37,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# Solution goes here"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 38,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# Solution goes here"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": true
   },
   "source": [
    "## Prediction\n",
    "\n",
    "Suppose we know that, for a particular kind of lightbulb in a particular location, the distribution of lifespans is well modeled by a Weibull distribution with `lam=2` and `k=1.5`.  If we install `n=100` lightbulbs and come back one year later, what is the distribution of `c`, the number of lightbulbs that have burned out?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The probability that any given bulb has burned out comes from the CDF of the distribution."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 39,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.29781149867344037"
      ]
     },
     "execution_count": 39,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "lam = 2\n",
    "k = 1.5\n",
    "p = EvalWeibullCdf(1, lam, k)\n",
    "p"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The number of bulbs that have burned out is distributed Binom(n, p).\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 40,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
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5UxjS5wpoXwGRKFESKDJh1ATq68ZRWVEOxNYt6rlyPS/vKyJjpyRQZILbSuYrCZhZypLS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\n9Vv7ZX4P1OAAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f7c90ecc410>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "from thinkbayes2 import MakeBinomialPmf\n",
    "\n",
    "n = 100\n",
    "pmf_c = MakeBinomialPmf(n, p)\n",
    "thinkplot.Pdf(pmf_c)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Or we can approximate the distribution with a random sample."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 41,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(29.827999999999999, 4.4346833032359818)"
      ]
     },
     "execution_count": 41,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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NmPqypB+j7twt5cDRs07bSi/Et/sraoH/DuydB23RdNMwlvRj1M4DJ/H5KgDo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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f7c9124a490>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "n = 100\n",
    "sample = np.random.binomial(n, p, 1000)\n",
    "pdf_c = thinkbayes2.EstimatedPdf(sample)\n",
    "thinkplot.Pdf(pdf_c)\n",
    "np.mean(sample), np.std(sample)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise:** Now suppose that `lam` and `k` are not known precisely, but we have a `LightBulb` object that represents the joint posterior distribution of the parameters after seeing some data.  Compute the posterior predictive distribution for `c`, the number of bulbs burned out after one year."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 42,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# Solution goes here"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 43,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# Solution goes here"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 44,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# Solution goes here"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 45,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "# Solution goes here"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 2",
   "language": "python",
   "name": "python2"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 2
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython2",
   "version": "2.7.11"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 0
}