06_pmf_soln.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Bite Size Bayes\n",
    "\n",
    "Copyright 2020 Allen B. Downey\n",
    "\n",
    "License: [Attribution-NonCommercial-ShareAlike 4.0 International (CC BY-NC-SA 4.0)](https://creativecommons.org/licenses/by-nc-sa/4.0/)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Review\n",
    "\n",
    "[In the previous notebook](https://colab.research.google.com/github/AllenDowney/BiteSizeBayes/blob/master/05_test.ipynb) ...\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import pandas as pd\n",
    "\n",
    "def make_bayes_table(index, prior, likelihood):\n",
    "    \"\"\"Make a Bayes table.\n",
    "    \n",
    "    index: sequence of hypotheses\n",
    "    prior: prior probabilities\n",
    "    likelihood: sequence of likelihoods\n",
    "    \n",
    "    returns: DataFrame\n",
    "    \"\"\"\n",
    "    table = pd.DataFrame(index=index)\n",
    "    table['prior'] = prior\n",
    "    table['likelihood'] = likelihood\n",
    "    table['unnorm'] = table['prior'] * table['likelihood']\n",
    "    prob_data = table['unnorm'].sum()\n",
    "    table['posterior'] = table['unnorm'] / prob_data\n",
    "    return table"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## 101 Bowls\n",
    "\n",
    "In a previous notebook, we saw that the Bayes table works with more than two hypotheses.  As an example, we solved a cookie problem with five bowls.\n",
    "\n",
    "Now, just to prove a point, I'll solve a cookie problem with 101 bowls:\n",
    "\n",
    "* Bowl 0 contains no vanilla cookies,\n",
    "\n",
    "* Bowl 1 contains 1% vanilla cookies,\n",
    "\n",
    "* Bowl 2 contains 2% vanilla cookies,\n",
    "\n",
    "and so on, up to\n",
    "\n",
    "* Bowl 99 contains 99% vanilla cookies, and\n",
    "\n",
    "* Bowl 100 contains all vanilla cookies.\n",
    "\n",
    "Suppose we choose a bowl at random, choose a cookie at random, and it turns out to be vanilla.  What is the probability that the cookie came from Bowl $x$, for each value of $x$?\n",
    "\n",
    "To solve this problem, I'll use `np.arange` to represent 101 hypotheses, numbered from 0 to 100. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "xs = np.arange(101)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The prior probability for each bowl is $1/101$.  I could create a sequence with 101 identical values, but if all of the priors are equal, we only have to probide one value:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "prior = 1/101"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Because of the way I numbered the bowls, the probability of a vanilla cookie from Bowl $x$ is $x/100$.  So we can compute the likelihoods like this:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "likelihood = xs/100"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And that's all we need; the Bayes table does the rest:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "table = make_bayes_table(xs, prior, likelihood)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here's a feature we have not seen before: we can give the index of the Bayes table a name, which will appear when we display the table."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [],
   "source": [
    "table.index.name = 'Bowl'"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here are the first few rows:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
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       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>prior</th>\n",
       "      <th>likelihood</th>\n",
       "      <th>unnorm</th>\n",
       "      <th>posterior</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Bowl</th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>0.009901</td>\n",
       "      <td>0.00</td>\n",
       "      <td>0.000000</td>\n",
       "      <td>0.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>0.009901</td>\n",
       "      <td>0.01</td>\n",
       "      <td>0.000099</td>\n",
       "      <td>0.000198</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>0.009901</td>\n",
       "      <td>0.02</td>\n",
       "      <td>0.000198</td>\n",
       "      <td>0.000396</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>0.009901</td>\n",
       "      <td>0.03</td>\n",
       "      <td>0.000297</td>\n",
       "      <td>0.000594</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>0.009901</td>\n",
       "      <td>0.04</td>\n",
       "      <td>0.000396</td>\n",
       "      <td>0.000792</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "         prior  likelihood    unnorm  posterior\n",
       "Bowl                                           \n",
       "0     0.009901        0.00  0.000000   0.000000\n",
       "1     0.009901        0.01  0.000099   0.000198\n",
       "2     0.009901        0.02  0.000198   0.000396\n",
       "3     0.009901        0.03  0.000297   0.000594\n",
       "4     0.009901        0.04  0.000396   0.000792"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "table.head()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Because Bowl 0 contains no vanilla cookies, its likelihood is 0, and its posterior probability is 0.  That is, the cookie cannot have come from Bowl 0.\n",
    "\n",
    "Here are the last few rows of the table."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
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       "\n",
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       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>prior</th>\n",
       "      <th>likelihood</th>\n",
       "      <th>unnorm</th>\n",
       "      <th>posterior</th>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>Bowl</th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "      <th></th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>96</th>\n",
       "      <td>0.009901</td>\n",
       "      <td>0.96</td>\n",
       "      <td>0.009505</td>\n",
       "      <td>0.019010</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>97</th>\n",
       "      <td>0.009901</td>\n",
       "      <td>0.97</td>\n",
       "      <td>0.009604</td>\n",
       "      <td>0.019208</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>98</th>\n",
       "      <td>0.009901</td>\n",
       "      <td>0.98</td>\n",
       "      <td>0.009703</td>\n",
       "      <td>0.019406</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>99</th>\n",
       "      <td>0.009901</td>\n",
       "      <td>0.99</td>\n",
       "      <td>0.009802</td>\n",
       "      <td>0.019604</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>100</th>\n",
       "      <td>0.009901</td>\n",
       "      <td>1.00</td>\n",
       "      <td>0.009901</td>\n",
       "      <td>0.019802</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "         prior  likelihood    unnorm  posterior\n",
       "Bowl                                           \n",
       "96    0.009901        0.96  0.009505   0.019010\n",
       "97    0.009901        0.97  0.009604   0.019208\n",
       "98    0.009901        0.98  0.009703   0.019406\n",
       "99    0.009901        0.99  0.009802   0.019604\n",
       "100   0.009901        1.00  0.009901   0.019802"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "table.tail()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The posterior probabilities are substantially higher for the high-numbered bowls.\n",
    "\n",
    "There is a pattern here that might be clearer if we plot the results."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "\n",
    "def plot_table(table):\n",
    "    table['prior'].plot()\n",
    "    table['posterior'].plot()\n",
    "\n",
    "    plt.xlabel('Bowl #')\n",
    "    plt.ylabel('Probability')\n",
    "    plt.legend()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
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49yFonv9v69JH0mwzS4u3LZlnKvFScP4MVpQ2BekJvBKzPBZINbOWwCT+dwa0Y+dmQ8wszczS6tSpU8S3cs65Yvr+U3i6A3z4MBzTE/rOKBMJpTDJHKjPAhrELNcHlhfQJktSReAAYHVhHUs6BqhoZrPz1pnZqpgmz/K/8ZlSx0vfO1eKbdkAkwcGd3cd0AAufQMad4w6qr0mmUllFtBEUiOCO7J6Ahfna5MO9AamA92BKVa063EXseNZCpLqmlnefbJdgQW7G7iZFfnuK7ejZF1Oda5UyJwEY2+EdcvghKuDBxmr7Bd1VHtV0pKKmeVI6gdMJLileJiZzZM0EMgws3RgKDBCUibBGUrPvP0lLQGqA5UlnQecaWZ5xawuAM7J95bXS+oK5IR99dmduKtWrcqqVauoVauWJ5ZiMjNWrVpF1apVow7Fub3rl9Uw8Xb4/BWo3RQunwgNT4g6qkgkbaC+NEhLS7OMjIwd1m3bto2srCw2by7wxjO3C1WrVqV+/fpUqlQ6S0w4V2zz3gpqdv26Gtr3h9//FSqV7T+sdjVQ72Va8qlUqRKNGjWKOgznXEm34UcYfzMsGAt1jwmeOanbMuqoIudJxTnnisMM5r4cXO7athlOvxvaXQcp/usUPKk451zRrVkKY2+AxVOh4YlBAcjajaOOqkTxpOKcc4XJ3R7cIjx5YPAU/DkPQdoVUMFr8ubnScU553YleyGM6QdZM6HxGdD5UajRoPD9yilPKs45F8/2bfDxY/DBA1B5P/jDEGh5QZms15VInlSccy6/5XOCs5OfvoKjzoezH4D9vKxTUXhScc65PNt+hffvg0/+A/vWgZ7/hWbnRh1VqeJJxTnnAJZ8HEyetXoRHNsLzvgn7FMj6qhKHU8qzrnybfN6mHQ3ZAyFGodBrzHwu1MiDqr08qTinCu/vnkXxt0I63+Atn3htL9BZZ8qe094UnHOlT+bVsHE2+CLUVCnGVzxHjRoE3VUZYInFedc+WEG894MCkBuXgsnD4CTboKKVaKOrMzwpOKcKx/Wr4C3/wIL34ZDW0PXMXDI0VFHVeZ4UnHOlW1m8NmL8O4dsH0LnHkPnHCNF4BMEv+pOufKrtXfwdjr4btpcFgH6PoE1Do86qjKtKRWQ5PUSdJCSZmSBsTZXkXSqHD7DEmp4fpakqZK2ihpUL593g/7nBt+HbSrvpxz5VDudpj+JAw+EX6YA50fg95jPaHsBUk7U5GUAjwJnAFkAbMkpcdMCQxwBbDGzBpL6gncD1wIbAbuAI4Ov/K7xMwy8q0rqC/nXHmyckFQYuWHDGhyVlAA8oB6UUdVbiTzTOV4INPMFpvZVmAk0C1fm27A8PD1aKCjJJnZJjP7iCC5FFXcvnY/fOdcqZKzFd6/H54+CdZ8B38cCheP8oSylyVzTKUesCxmOQs4oaA2ZpYjaR1QC/i5kL6fl7QdeB24x8ysqH1Jugq4CqBhw4a7cVjOuRLnh9kw5jpYOQ+O/mNQAHLf2lFHVS4l80wl3lmC7Uab/C4xsxbASeHXZcXpy8yGmFmamaXVqeNVR50r1bb+Au/+HZ47HX5dDT1fge7DPKFEKJlJJQuIncmmPrC8oDaSKgIHAKt31amZ/RB+3wD8l+Ay22715Zwrxb77EJ5uH1QUPrYX9J0Bzc6JOqpyL5lJZRbQRFIjSZWBnkB6vjbpQO/wdXdgSngpKy5JFSXVDl9XAjoDX+1OX865UmrzumCe+OGdwXKhVzp0eRyqHhB1ZI4kjqmE4xr9gIlACjDMzOZJGghkmFk6MBQYISmT4KyiZ97+kpYA1YHKks4DzgSWAhPDhJICTAKeDXcpsC/nXBmx8J2gAOTGH6FdPzj1b1C5WtRRuRgqz3/Mp6WlWUZG/juTnXMlzqafYcKt8NVoOKg5dB0E9Y+LOqpyS9JsM0uLt82fqHfOlVxm8NXrMOGWYN6TU26DDjdBxcpRR+YK4EnFOVcyrV8O426CbyZAveOCs5ODm0cdlSuEJxXnXMliBp8NDwtAboMz/wVtr4EKKVFH5orAk4pzruRYtSi4s2vJh5B6UlAAsubvoo7KFYMnFedc9HK3w6dPwZR/QUql4BbhY3uDV1oqdTypOOei9dN8GNMXln8GR5wD5z4M1Q+NOiq3mzypOOeikbMFPnwEPnw4eHCx+/Nw1B/87KSU86TinNv7sjKC8vTZC6DFBdDpPti3VtRRuQTwpOKc23u2bgrGTT59KrjEdfFr0PTMqKNyCeRJxTm3dyz+IJjad80SSLsCTr8bqlaPOCiXaJ5UnHPJ9etaeO8O+OxFqHk49HkbUjtEHZVLEk8qzrnk+frt4Kn4TSuh/Q1BmZVK+0QdlUsiTyrOucTbmB3U65r3Bhx8NFz0CtQ7Nuqo3F7gScU5lzhm8MWr8M6twaD8qX+HDv2DBxpdueBJxTmXGOuygrlOvn0X6rcJCkAe1CzqqNxe5knFObdncnNh9jB4726w7cEzJ8df5QUgy6lkTieMpE6SFkrKlDQgzvYqkkaF22dISg3X15I0VdJGSYNi2leT9LakryXNk3RfzLY+krIlzQ2/rkzmsTnngJ8zg2l93/5LMGZy7XSvKFzOJe1MRVIK8CRwBpAFzJKUbmbzY5pdAawxs8aSegL3AxcCm4E7gKPDr1gPmdnUcN77yZLONrMJ4bZRZtYvWcfknAttz4Hpg+D9e6FiFej2JLS6xEusuKRe/joeyDSzxQCSRgLdgNik0g24O3w9GhgkSWa2CfhIUuPYDs3sF2Bq+HqrpM+A+kk8Budcfj9+GZRYWTEXmnWGcx6C6nWjjsqVEMm8/FUPWBaznBWui9vGzHKAdUCRCgBJqgF0ASbHrP6jpC8kjZbUoID9rpKUISkjOzu7aEfinAsKQE65B4acAut/gB7D4cKXPKG4HSQzqcQ7D7bdaLNzx1JF4BXgibwzIWAskGpmLYFJwPB4+5rZEDNLM7O0OnXqFPZWzjmAZTPh6Q4w7cGgAGTfmXDUeX65y+2kSElF0uuSzpVUnCSUBcSeLdQHlhfUJkwUBwCri9D3EOBbM3ssb4WZrTKzLeHis8BxxYjVORfPlo0w4VYYeiZs+xUueR3+MBiq1Yw6MldCFTVJDAYuBr6VdJ+kotx8PgtoIqlROKjeE0jP1yYd6B2+7g5MMbNdnqlIuocg+fTPtz72HLwrsKAIMTrnCrJoCgxuBzOehuP/L7izq8npUUflSrgiDdSb2SRgkqQDgIuA9yQtIzgjeMnMtsXZJ0dSP2AikAIMM7N5kgYCGWaWDgwFRkjKJDhD6Zm3v6QlQHWgsqTzgDOB9cDfgK+BzxSceg8ys+eA6yV1BXLCvvoU94fhnAN+XQMT/w5zX4JajeFP78Bh7aKOypUSKuTE4H8NpVrApcBlBJexXgY6AC3M7JRkBZhMaWlplpGREXUYzpUcC8YGz5xs+jkoAHnyrVCpatRRuRJG0mwzS4u3rUhnKpLeAJoBI4AuZrYi3DRKkv9Wdq602/ATTPgrzB8Dh7SAS16DusdEHZUrhYr6nMpzZjY+doWkKma2paBs5ZwrBczg85HwzoBgIL7jnXDi9V4A0u22oiaVe4Dx+dZNB7yWtXOl1drvYWx/WDQZGrSFrv+BOk2jjsqVcrtMKpIOIXhAcR9JrfnfcyXVgWpJjs05lwy5uTDrOZh0d7B89oPQ5kqokNRSgK6cKOxM5SyCu6jqA4/ErN8A3J6kmJxzyZL9DaRfB8s+hcNPgy6PQ42GUUflypBdJhUzGw4Ml/RHM3t9L8XknEu07dvgkyfg/fuD6XzPexqO6elPxLuEK+zy16Vm9hKQKumm/NvN7JE4uznnSpIVn8OYvkEhyObdggKQ+x0UdVSujCrs8te+4ff9kh2Icy7Btm2GD+6Hjx+HfWvDBSOgedeoo3JlXGGXv54Jv/9j74TjnEuIpdMhvR+syoRWl8JZ98A+B0YdlSsHCrv89cSutpvZ9YkNxzm3R7ZsgMkDYeazUKMBXPYWHH5q1FG5cqSwy1+z90oUzrk99+0kGNcf1mXBCVfDaXdAFb9y7fauotz95ZwryX5ZDRNvh89fgdpHwBXvQoPjo47KlVOFXf56zMz6SxpLnMmzzMxH/ZyLillQq2v8zUFl4ZNuhpNvCeaMdy4ihV3+GhF+fyjZgTjnimHDj0E14a/HQd1WcNmbQSFI5yJW2OWv2eH3D8KJtpoRnLEsNLOteyE+51wsM5jzErz7t2DO+NPvhnbXQUpRy/g5l1xFLX1/LvA0sIig/lcjSVeb2YRkBueci7FmCYy9ARa/Dw1PDApA1m4cdVTO7aCoFeQeBk41s1PM7GTgVODRwnaS1EnSQkmZkgbE2V5F0qhw+wxJqeH6WpKmStooaVC+fY6T9GW4zxMKp3+UVFPSe5K+Db/7TfmubMjdDp8OhqfaQVYGnPsw9HnbE4orkYqaVFaaWWbM8mJg5a52kJQCPAmcDTQHLpLUPF+zK4A1ZtaYIEndH67fDNwB3Byn68HAVUCT8KtTuH4AMNnMmgCTw2XnSreVX8OwTsF8J4e1h74zvKKwK9EKu/vr/PDlPEnjgVcJxlR6ALMK6ft4INPMFod9jQS6AfNj2nQD7g5fjwYGSZKZbQI+krTDn2KS6gLVzWx6uPwicB4wIezrlLDpcOB94NZCYnSuZNq+DT56DKY9AJX3g/OfhRY9vACkK/EKG1PpEvP6J+Dk8HU2UNjlpXrAspjlLOCEgtqYWY6kdUAt4Odd9JmVr8964euD86Y5NrMVkuJWzJN0FcGZDg0b7l7J73+Mncf85et3a1/nCtNo27dcs/YRDsv5jk+qnszz+1/D+uk1YPqnUYfmypDmh1bnri5HJbzfwu7++tMe9B3vT6r8z7oUpc2etN+5sdkQYAhAWlpasfZ1Lpkq2RZ6bHiJLpteZ22FGjx44F1kVG0XdVjOFUtR7/6qSjD+cRRQNW+9mV2+i92ygAYxy/WB5QW0yZJUETgAWF1In/UL6PMnSXXDs5S6FDLmsyeSkd1dObfkY0i/CTYtgmN7UfOMf/LXfWpEHZVzxVbU0b4RwCEEM0F+QPDLfEMh+8wCmkhqFD7j0hNIz9cmHegdvu4OTDGzAs8ewstbGyS1De/66gWMidNX75j1zpVcm9cHDzG+cA7k5kCvMcGtwp5QXClV1CemGptZD0ndzGy4pP8CE3e1QzhG0i9slwIMM7N5kgYCGWaWDgwFRkjKJDhD6Zm3v6QlQHWgsqTzgDPNbD5wDfACsA/BAH3eszL3Aa9KugL4nuBmAudKrm/fg7H9Yf0P0LYvnPY3qLxv4fs5V4IVNalsC7+vlXQ08COQWthOZjYeGJ9v3Z0xrzdTwC9/M4vbv5llAEfHWb8K6FhYTM5FbtMqmHgbfDEK6jSDK96DBm2ijsq5hChqUhkSPkx4B8Flpv3C1865ojKDeW/A+Ftg81o4+VY46S9eANKVKUVKKmb2XPjyA+B3yQvHuTJq/Qp4+yZYOB4ObQ1dx8AhO51wO1fqFfXur1oEDym2J7iF90Pgn+ElJ+dcQczgsxfh3Ttg+xY445/Q9lovAOnKrKL+yx4JTAP+GC5fAowCTk9GUM6VCau/g7HXw3fTIPUk6PI41Do86qicS6qiJpWaZvbPmOV7wjuynHP55W6HGU/D5H9CSiXo/Bgc29vrdblyoahJZaqkngS1vyB4puTt5ITkXCm2cgGM6Qc/ZEDTTnDuI3BAvcL3c66MKKyg5AaCMRQBNwEvhZsqABuBu5IanXOlRc5W+OhRmPYgVK0OfxwKR//RC0C6cqew2l/7761AnCu1fpgdnJ2snA9Hd4ez74d9a0cdlXORKPItKJK6Ar8PF983s3HJCcm5UmLrLzD1X/DpU7DfIXDRSDji7Kijci5SRb2l+D6gDfByuOoGSR3MzCfCcuXTdx9C+nWw5js47k9wxj+g6gFRR+Vc5Ip6pnIO0MrMcgEkDQfm4LMruvJm8zp47y6Y/Twc2Ah6j4VGvy98P+fKieI8gVWD/5Wl9z/JXPmzcAKMuwk2/gjt+sGpf4PK1aKOyrkSpahJ5V5gjqSpBHeC/R64LWlROVeSbPoZJtwKX42Gg46Cni9BveOijsq5EqnQpBLOW/IR0KqbO/8AABWmSURBVJZgXEXArWb2Y5Jjcy5aZvDlaJhwC2zZAKfcDh1uhIqVo47MuRKr0KRiZibpLTM7jp0n2XKubFr3Q1AA8pt3grOSbk/CQUdGHZVzJV5RL399KqmNmc1KajTORS03Fz57Ad69M5iJ8ax/wwl/hgopUUfmXKlQ1GJEpxIklkWSvpD0paQvCttJUidJCyVlStrpTjFJVSSNCrfPkJQas+22cP1CSWeF646QNDfma72k/uG2uyX9ELPtnCIem3OBVYtgeBcYdyPUaw3XTod2fT2hOFcMRT1TKfYTXZJSgCeBM4AsYJak9HBK4DxXAGvMrHFYW+x+4EJJzQmmFj4KOBSYJKmpmS0EWsX0/wPwZkx/j5rZQ8WN1ZVz23OCBxin/gtSqkCXJ+DYXl5ixbndUFjtr6rAn4HGwJfAUDPLKWLfxwOZZrY47Gsk0A2ITSrdCOZpARgNDApvDOgGjDSzLcB34Rz2xwPTY/btCCwys6VFjMe5nf00LyixsvwzOOJcOPdhqF436qicK7UKu/w1HEgjSChnAw8Xo+96wLKY5axwXdw2YbJaB9Qq4r49gVfyresXXp4bFk5/vBNJV0nKkJSRnZ1djMNxZUrOFpj6b3jm97D2e+j+PPR82ROKc3uosKTS3MwuNbNnCMrdn1SMvuNdO7AittnlvpIqA12B12K2DwYOJ7g8toICEqCZDTGzNDNLq1OnTsHRu7Jr2awgmXxwf1BJuN8sOPp8v9zlXAIUNqayLe+FmeWoeP/psoAGMcv1geUFtMmSVJHgSf3VRdj3bOAzM/spJr7fXkt6FvCCl25HWzfBlLAAZPV6cPFr0PTMqKNyrkwpLKkcI2l9+FrAPuGyCB5hqb6LfWcBTSQ1IhhQ7wlcnK9NOtCbYKykOzAlfC4mHfivpEcIBuqbADNj9ruIfJe+JNU1sxXh4h+Arwo5NleeLH4f0q+HtUuhzZXQ8a5g3hPnXEIVNp/Kbt9LGZ7Z9AMmAinAMDObJ2kgkGFm6cBQYEQ4EL+aIPEQtnuVYFA/B+hrZtsBJFUjuKPs6nxv+YCkVgSXyZbE2e7Ko1/Xwrt/hzkjoObh0Gc8pLaPOirnyiyZ5R/mKD/S0tIsIyMj6jBcsiwYB2//BTZlw4nXwSkDoNI+UUflXKknabaZpcXbVpwqxc6VDhtXwvi/wvy34OAWcPFIOLR11FE5Vy54UnFlhxl8MQreGRAMyp/2d2jfH1IqRR2Zc+WGJxVXNqxdFpRXyXwP6h8P3QZBnSOijsq5cseTiivdcnMhYyhMuhssFzrdD8f/n9frci4inlRc6fVzZjBP/PefwO9OgS6Pw4GpEQflXPnmScWVPttzYPp/YOq9UKlqMNdJq0v8iXjnSgBPKq50+fFLGNMXVnwOR3aBcx6C/Q+JOirnXMiTiisdtm2GaQ/Cx4/BPjXhghehebeoo3LO5eNJxZV838+A9H7w8zdwzMVw1r+gWs2oo3LOxeFJxZVcWzbC5IEwcwgcUB8ufR0anx51VM65XfCk4kqmzMkwtj+sWxbcItzxTqiyf9RROecK4UnFlSy/rA4KQM59GWo1gT9NgMPaRR2Vc66IPKm4kmN+elAA8pdV0OEmOPnW4JZh51yp4UnFRW/DTzD+ZliQDoe0DMZO6raMOirn3G7wpOKiYwafvwLv3Abbfg0mzjrxOi8A6Vwp5knFRWPNUhjXHxZNgYbtoOt/oHaTqKNyzu2hCsnsXFInSQslZUoaEGd7FUmjwu0zJKXGbLstXL9Q0lkx65dI+lLSXEkZMetrSnpP0rfh9wOTeWxuN+Xmwoxn4Kl2sGxm8ER8n/GeUJwrI5KWVCSlAE8CZwPNgYskNc/X7ApgjZk1Bh4F7g/3bU4wtfBRQCfgqbC/PKeaWat8M48NACabWRNgcrjsSpLsb+D5TjDhFmjYFq6dHlYUTurfNs65vSiZ/5uPBzLNbLGZbQVGAvnranQDhoevRwMdJSlcP9LMtpjZd0Bm2N+uxPY1HDgvAcfgEmH7NvjwYXi6A2QvhPMGB4PxNRpGHZlzLsGSOaZSD1gWs5wFnFBQGzPLkbQOqBWu/zTfvvXC1wa8K8mAZ8xsSLj+YDNbEfa1QtJB8YKSdBVwFUDDhv5LLelWfA5j+sGPXwS1us55CPaL+9E458qAZCaVeHXIrYhtdrVvezNbHiaN9yR9bWbTihpUmISGAKSlpeWPxyXKtl/hg/vh4ydg39pwwQho3jXqqJxzSZbMpJIFNIhZrg8sL6BNlqSKwAHA6l3ta2Z531dKepPgstg04CdJdcOzlLrAysQfkiuSpdODApCrMqH1pXDmPbCP3zfhXHmQzDGVWUATSY0kVSYYeE/P1yYd6B2+7g5MMTML1/cM7w5rBDQBZkraV9L+AJL2Bc4EvorTV29gTJKOyxVkywZ4++ZgMH77VrjsrWACLU8ozpUbSTtTCcdI+gETgRRgmJnNkzQQyDCzdGAoMEJSJsEZSs9w33mSXgXmAzlAXzPbLulg4M1gLJ+KwH/N7J3wLe8DXpV0BfA90CNZx+bi+HZS8NzJuiw44c9w2h1QZb+oo3LO7WUKTgzKp7S0NMvIyCi8oSvYL6th4u3Bk/G1jwgeYmyY/34M51xZIml2vkc6fuNP1LvdYwbzxwQ1u35dA7//a/BVsUrUkTnnIuRJxRXfhh+DasJfj4O6reCyN+GQFlFH5ZwrATypuKIzC+Y5mXg75GyB0/8B7fpBiv8zcs4F/LeBK5o1S2DsDbD4fWh4YlgAsnHUUTnnShhPKm7XcrcHc8RPHgiqEDwRn3aF1+tyzsXlScUVbOXXkH4dZM2ExmdA50ehRoPC93POlVueVNzOtm+Djx6DaQ9A5X3h/GehRQ9QvOo5zjn3P55U3I6WzwkKQP70FRx1Ppz9AOxXJ+qonHOlhCcVF9j2K7x/L3zyH9j3IOj5X2h2btRROedKGU8qDpZ8FIydrF4Mx/aCM/4J+9SIOirnXCnkSaU827weJt0FGcPgwFToNQZ+d0rEQTnnSjNPKuXVN+8GBSA3rAgeYDz19mBQ3jnn9oAnlfJm0yp4ZwB8+SrUaQYXvAj149aFc865YvOkUl6YwVevw4RbYfNaOHkAnHSTF4B0ziWUJ5XyYP3yoADkwvFw6LHQLR0OPirqqJxzZZAnlbLMDD4bDu/eETzQeOY90PZaqJASdWTOuTIqqQWcJHWStFBSpqQBcbZXkTQq3D5DUmrMttvC9QslnRWuayBpqqQFkuZJuiGm/d2SfpA0N/w6J5nHVuKtXgzDuwRFIOseA9d8DCde5wnFOZdUSTtTkZQCPAmcAWQBsySlm9n8mGZXAGvMrLGknsD9wIWSmhNMLXwUcCgwSVJTgqmF/2Jmn4Vz1c+W9F5Mn4+a2UPJOqZSIXc7fDoYptwDKZWg82NwbG8vAOmc2yuSefnreCDTzBYDSBoJdCOYdz5PN+Du8PVoYJCCCei7ASPNbAvwXTiH/fFmNh1YAWBmGyQtAOrl67P8+mk+pPeDH2ZD005w7iNwQL2oo3LOlSPJ/PO1HrAsZjkrXBe3jZnlAOuAWkXZN7xU1hqYEbO6n6QvJA2TdGC8oCRdJSlDUkZ2dnZxj6lkytkK798Hz/w+mPfkj0PhopGeUJxze10yk0q8krZWxDa73FfSfsDrQH8zWx+uHgwcDrQiOJt5OF5QZjbEzNLMLK1OnTJQKDFrNgw5OajbddR50HcWtOjuFYWdc5FI5uWvLCB28o36wPIC2mRJqggcAKze1b6SKhEklJfN7I28Bmb2U95rSc8C4xJ2JCXR1l9g6r/g06dgv0PgolFwRKeoo3LOlXPJPFOZBTSR1EhSZYKB9/R8bdKB3uHr7sAUM7Nwfc/w7rBGQBNgZjjeMhRYYGaPxHYkqW7M4h+ArxJ+RCXFd9NgcDuYPigYhO/7qScU51yJkLQzFTPLkdQPmAikAMPMbJ6kgUCGmaUTJIgR4UD8aoLEQ9juVYIB+Bygr5ltl9QBuAz4UtLc8K1uN7PxwAOSWhFcJlsCXJ2sY4vM5nXBMyefDYeav4Pe46DRSVFH5Zxzv1FwYlA+paWlWUZGRtRhFM3CCTDuRtj4E7TrC6fcDpWrRR2Vc64ckjTbzOIWDfQn6ku6TT/DhFuCul0HHRVMnlXv2Kijcs65uDyplFRm8OXoIKFs2QCn/g3a94eKlaOOzDnnCuRJpSRalwXjboJvJ0L9NtB1EBzULOqonHOuUJ5USpLcXPjsBXj3TrDtcNa9cMLVXq/LOVdqeFIpKVYtgvTrYelH0Ohk6PI41GwUdVTOOVcsnlSitj0HPn0Spv4bUqoEl7paX+pPxDvnSiVPKlH68augAOTyOXDEuXDuw1C9buH7OedcCeVJJQo5W2DaQ/DRI7DPgdDjBWh+np+dOOdKPU8qe9uyWcHZSfbX0LIndLoXqtWMOirnnEsITyp7y9ZNMPmfMONpqF4PLhkNTc6IOirnnEsoTyp7w6KpMPZ6WPs9tLkSOt4FVatHHZVzziWcJ5Vk+nUtvPs3mPMS1Dwc+oyH1PZRR+Wcc0njSSVZvn47eCp+UzZ0uBFOHgCVqkYdlXPOJZUnlUTbuDKo1zXvTTikBVw8Cg5tFXVUzjm3V3hSSRQz+OJVeOfWYFD+tDug/Q2QUinqyJxzbq/xpJIIa5cFc51kvgcNTgieiq/TNOqonHNur0vmdMJI6iRpoaRMSQPibK8iaVS4fYak1Jhtt4XrF0o6q7A+w2mLZ0j6Nuwz+TXic3Nh1nPwVFtY+gmc/QD8aYInFOdcuZW0pCIpBXgSOBtoDlwkqXm+ZlcAa8ysMfAocH+4b3OCqYWPAjoBT0lKKaTP+4FHzawJsCbsO3l+/hZeOBfe/ktQnv7a6V5R2DlX7iXzTOV4INPMFpvZVmAk0C1fm27A8PD1aKCjJIXrR5rZFjP7DsgM+4vbZ7jPaWEfhH2el7Qj+2wEDG4PK+dBt6fgsjfhwMOS9nbOOVdaJDOp1AOWxSxnhevitjGzHGAdUGsX+xa0vhawNuyjoPcCQNJVkjIkZWRnZ+/GYQG1GkPTs6DvTGh9idfscs65UDIH6uP9prUitilofbwkuKv2O680GwIMAUhLS4vbplCHtQu+nHPO7SCZZypZQIOY5frA8oLaSKoIHACs3sW+Ba3/GagR9lHQeznnnEuyZCaVWUCT8K6sygQD7+n52qQDvcPX3YEpZmbh+p7h3WGNgCbAzIL6DPeZGvZB2OeYJB6bc865OJJ2+cvMciT1AyYCKcAwM5snaSCQYWbpwFBghKRMgjOUnuG+8yS9CswHcoC+ZrYdIF6f4VveCoyUdA8wJ+zbOefcXqTgj/zyKS0tzTIyMqIOwznnShVJs80sLd62pD786JxzrnzxpOKccy5hPKk455xLGE8qzjnnEqZcD9RLygaW7ubutQmejylP/JjLBz/m8mFPjvkwM6sTb0O5Tip7QlJGQXc/lFV+zOWDH3P5kKxj9stfzjnnEsaTinPOuYTxpLL7hkQdQAT8mMsHP+byISnH7GMqzjnnEsbPVJxzziWMJxXnnHMJ40llN0jqJGmhpExJA6KOJxkkNZA0VdICSfMk3RCurynpPUnfht8PjDrWRJKUImmOpHHhciNJM8LjHRVOuVBmSKohabSkr8PPul05+IxvDP9NfyXpFUlVy9rnLGmYpJWSvopZF/dzVeCJ8PfZF5KO3ZP39qRSTJJSgCeBs4HmwEWSmkcbVVLkAH8xsyOBtkDf8DgHAJPNrAkwOVwuS24AFsQs3w88Gh7vGuCKSKJKnseBd8ysGXAMwbGX2c9YUj3geiDNzI4mmEKjJ2Xvc34B6JRvXUGf69kEc1Y1Aa4CBu/JG3tSKb7jgUwzW2xmW4GRQLeIY0o4M1thZp+FrzcQ/LKpR3Csw8Nmw4Hzookw8STVB84FnguXBZwGjA6blLXjrQ78nnDuITPbamZrKcOfcagisE84U2w1YAVl7HM2s2kEc1TFKuhz7Qa8aIFPCWbRrbu77+1JpfjqActilrPCdWWWpFSgNTADONjMVkCQeICDooss4R4DbgFyw+VawFozywmXy9pn/TsgG3g+vOT3nKR9KcOfsZn9ADwEfE+QTNYBsynbn3Oegj7XhP5O86RSfIqzrszely1pP+B1oL+ZrY86nmSR1BlYaWazY1fHaVqWPuuKwLHAYDNrDWyiDF3qiiccR+gGNAIOBfYluPyTX1n6nAuT0H/nnlSKLwtoELNcH1geUSxJJakSQUJ52czeCFf/lHdqHH5fGVV8CdYe6CppCcElzdMIzlxqhJdJoOx91llAlpnNCJdHEySZsvoZA5wOfGdm2Wa2DXgDOJGy/TnnKehzTejvNE8qxTcLaBLeLVKZYJAvPeKYEi4cTxgKLDCzR2I2pQO9w9e9gTF7O7ZkMLPbzKy+maUSfKZTzOwSYCrQPWxWZo4XwMx+BJZJOiJc1RGYTxn9jEPfA20lVQv/jecdc5n9nGMU9LmmA73Cu8DaAuvyLpPtDn+ifjdIOofgr9gUYJiZ/SvikBJOUgfgQ+BL/jfGcDvBuMqrQEOC/6A9zCz/gGCpJukU4GYz6yzpdwRnLjWBOcClZrYlyvgSSVIrghsTKgOLgT8R/LFZZj9jSf8ALiS4w3EOcCXBGEKZ+ZwlvQKcQlDe/ifgLuAt4nyuYXIdRHC32C/An8wsY7ff25OKc865RPHLX8455xLGk4pzzrmE8aTinHMuYTypOOecSxhPKs455xKmYuFNnHPFJWk7we3YArYD/czskz3sc6OZ7beL7fcCE4EaQDMzu29P3s+53eFnKs4lx69m1srMjgFuA+7dC+95AsFzRCcTPGPk3F7nScW55KtOUE49b+6KB8O5PL6UdGG4/ilJXcPXb0oaFr6+QtI9u+o87O8LoA0wneBhvsGS7kziMTkXl1/+ci459pE0F6gK1CWoJQZwPtCKYO6S2sAsSdOAacBJBCUz6oX7AHQgeNK7QGb2V0mvAZcBNwHvm1n7xB6Oc0XjZyrOJUfe5a9mBOUvXgzLYXQAXjGz7Wb2E/ABwRnGh8BJ4URo8/lf8b92QFHGYloDc4Fm4f7ORcLPVJxLMjObLqk2UIf4ZcYxsx/CsuydCM5aagIXABvDSdLiCmt3vUBQWfZngkmnFJ4ltTOzXxN5LM4Vxs9UnEsySc0Iio+uIkgYF0pKkVSHYObFmWHT6UD/sM2HwM0UMuBuZnPNrBXwDcH01lOAs8KzJE8obq/zMxXnkiNvTAWCs5PeZrZd0psEl7Q+J5gI6ZawBD0ECeRMM8uUtJTgbKXQu7jC5LTGzHIlNTMzv/zlIuNVip1zziWMX/5yzjmXMJ5UnHPOJYwnFeeccwnjScU551zCeFJxzjmXMJ5UnHPOJYwnFeeccwnz/86r4aWP9rrtAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_table(table)\n",
    "plt.title('One cookie');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The prior probabilities are uniform; that is, they are the same for every bowl.\n",
    "\n",
    "The posterior probabilities increase linearly; Bowl 0 is the least likely (actually impossible), and Bowl 100 is the most likely."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Two cookies\n",
    "\n",
    "Suppose we put the first cookie back, stir the bowl thoroughly, and draw another cookie from the same bowl.  It turns out to be another vanilla cookie.\n",
    "\n",
    "Now what is the probability that we are drawing from Bowl $x$?\n",
    "\n",
    "To answer this question, we can use the posterior probabilities from the previous problem as prior probabilities for a new Bayes table, and then update with the new data."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "xs2 = xs\n",
    "prior2 = table['posterior']\n",
    "likelihood2 = likelihood\n",
    "\n",
    "table2 = make_bayes_table(xs2, prior2, likelihood2)\n",
    "plot_table(table2)\n",
    "plt.title('Two cookies');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The blue line shows the posterior after one cookie, which is the prior before the second cookie.\n",
    "\n",
    "The orange line shows the posterior after two cookies, which curves upward.  Having see two vanilla cookies, the high-numbered bowls are more likely; the low-numbered bowls are less likely.\n",
    "\n",
    "I bet you can guess what's coming next."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Three cookies\n",
    "\n",
    "Suppose we put the cookie back, stir, draw another cookie, and get a chocolate cookie.\n",
    "\n",
    "And let's assume that there are only two kinds of cookies, vanilla and chocolate.\n",
    "\n",
    "What do you think the posterior distribution looks like after these three cookies?\n",
    "\n",
    "Hint: what's the probability that the chocolate cookie came from Bowl 100?\n",
    "\n",
    "Here's how we can compute the results:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "xs3 = xs\n",
    "prior3 = table2['posterior']"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The index for `table3` is the same as the others.\n",
    "\n",
    "The prior for `table3` is the posterior from `table2`.\n",
    "\n",
    "But what about the likelihoods?  Remember that the probability of a vanilla cookie from Bowl $x$ is $x/100$.  So the probability of a chocolate cookie is $(1 - x/100)$, which we can compute like this."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "likelihood3 = 1 - likelihood"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "That's it.  Everything else is the same."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "table3 = make_bayes_table(xs3, prior3, likelihood3)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And here are the results"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot_table(table3)\n",
    "plt.title('Three cookies');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So far we have seen two vanilla cookies and one chocolate, so in the dataset, $2/3$ of the cookies are vanilla.\n",
    "\n",
    "And which bowl do you think has the maximum posterior probability?\n",
    "\n",
    "We can use `idxmax` to find it: "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "67"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "table3['posterior'].idxmax()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This value has a name; it is the **MAP**, which stands for \"Maximum Aposteori Probability\" (\"aposteori\" is Latin for posterior).\n",
    "\n",
    "In this example, the MAP is the same as the proportion of vanilla cookies in the dataset."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise:** Let's do a version of the dice problem where we roll the die more than once.  Here's the statement of the problem again:\n",
    "\n",
    "> Suppose you have a 4-sided, 6-sided, 8-sided, and 12-sided die.  You choose one at random, roll it and get a 1. What is the probability that the die you rolled is 4-sided?  What are the posterior probabilities for the other dice?\n",
    "\n",
    "And here's a solution using a Bayes table:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>prior</th>\n",
       "      <th>likelihood</th>\n",
       "      <th>unnorm</th>\n",
       "      <th>posterior</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>H4</th>\n",
       "      <td>0.25</td>\n",
       "      <td>0.250000</td>\n",
       "      <td>0.062500</td>\n",
       "      <td>0.400000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>H6</th>\n",
       "      <td>0.25</td>\n",
       "      <td>0.166667</td>\n",
       "      <td>0.041667</td>\n",
       "      <td>0.266667</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>H8</th>\n",
       "      <td>0.25</td>\n",
       "      <td>0.125000</td>\n",
       "      <td>0.031250</td>\n",
       "      <td>0.200000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>H12</th>\n",
       "      <td>0.25</td>\n",
       "      <td>0.083333</td>\n",
       "      <td>0.020833</td>\n",
       "      <td>0.133333</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "     prior  likelihood    unnorm  posterior\n",
       "H4    0.25    0.250000  0.062500   0.400000\n",
       "H6    0.25    0.166667  0.041667   0.266667\n",
       "H8    0.25    0.125000  0.031250   0.200000\n",
       "H12   0.25    0.083333  0.020833   0.133333"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "xs = ['H4', 'H6', 'H8', 'H12']\n",
    "prior = 1/4\n",
    "likelihood = 1/4, 1/6, 1/8, 1/12\n",
    "\n",
    "table = make_bayes_table(xs, prior, likelihood)\n",
    "table"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now suppose you roll the same die again and get a 6.  What are the posterior probabilities after the second roll?\n",
    "\n",
    "Use `idxmax` to find the MAP."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<style scoped>\n",
       "    .dataframe tbody tr th:only-of-type {\n",
       "        vertical-align: middle;\n",
       "    }\n",
       "\n",
       "    .dataframe tbody tr th {\n",
       "        vertical-align: top;\n",
       "    }\n",
       "\n",
       "    .dataframe thead th {\n",
       "        text-align: right;\n",
       "    }\n",
       "</style>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>prior</th>\n",
       "      <th>likelihood</th>\n",
       "      <th>unnorm</th>\n",
       "      <th>posterior</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>H4</th>\n",
       "      <td>0.400000</td>\n",
       "      <td>0.000000</td>\n",
       "      <td>0.000000</td>\n",
       "      <td>0.000000</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>H6</th>\n",
       "      <td>0.266667</td>\n",
       "      <td>0.166667</td>\n",
       "      <td>0.044444</td>\n",
       "      <td>0.551724</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>H8</th>\n",
       "      <td>0.200000</td>\n",
       "      <td>0.125000</td>\n",
       "      <td>0.025000</td>\n",
       "      <td>0.310345</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>H12</th>\n",
       "      <td>0.133333</td>\n",
       "      <td>0.083333</td>\n",
       "      <td>0.011111</td>\n",
       "      <td>0.137931</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "        prior  likelihood    unnorm  posterior\n",
       "H4   0.400000    0.000000  0.000000   0.000000\n",
       "H6   0.266667    0.166667  0.044444   0.551724\n",
       "H8   0.200000    0.125000  0.025000   0.310345\n",
       "H12  0.133333    0.083333  0.011111   0.137931"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "xs2 = xs\n",
    "prior2 = table['posterior']\n",
    "likelihood2 = 0, 1/6, 1/8, 1/12\n",
    "\n",
    "table2 = make_bayes_table(xs2, prior2, likelihood2)\n",
    "table2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'H6'"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "table2['posterior'].idxmax()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Probability Mass Functions\n",
    "\n",
    "When we do more than one update, we don't always want to keep the whole Bayes table.  In this section we'll replace the Bayes table with a more compact representation, using a Pandas Series.\n",
    "\n",
    "Here's a function that takes a sequence of values, `xs`, and their corresponding probabilities, `ps`, and returns a Pandas Series:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 36,
   "metadata": {},
   "outputs": [],
   "source": [
    "def make_pmf(xs, ps, **options):\n",
    "    \"\"\"Make a Series that represents a PMF.\n",
    "    \n",
    "    xs: sequence of values\n",
    "    ps: sequence of probabilities\n",
    "    options: keyword arguments passed to Series constructor\n",
    "    \n",
    "    returns: Pandas Series\n",
    "    \"\"\"\n",
    "    pmf = pd.Series(ps, index=xs, **options)\n",
    "    return pmf"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And here's an example that represents the prior from the 101 Bowls problem."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0    0.009901\n",
       "1    0.009901\n",
       "2    0.009901\n",
       "3    0.009901\n",
       "4    0.009901\n",
       "dtype: float64"
      ]
     },
     "execution_count": 21,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "xs = np.arange(101)\n",
    "prior = 1/101\n",
    "\n",
    "pmf = make_pmf(xs, prior)\n",
    "pmf.head()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The probabilities in this series are sometimes called \"probability masses\", which distinguishes them from \"probability densities\", which we'll learn about later.\n",
    "\n",
    "And the relationship between values and their probabilities is called a \"probability mass function\", or PMF.\n",
    "\n",
    "So this Series represents a PMF, and this PMF contains the prior probabilities for the 101 Bowls problem.\n",
    "\n",
    "To compute posterior probabilities, we can use the following function, which takes a PMF and a sequence of likelihoods, updates the PMF, and returns the total probability of the data."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {},
   "outputs": [],
   "source": [
    "def bayes_update(pmf, likelihood):\n",
    "    \"\"\"Do a Bayesian update.\n",
    "    \n",
    "    pmf: Series that represents the prior\n",
    "    likelihood: sequence of likelihoods\n",
    "    \n",
    "    returns: float probability of the data\n",
    "    \"\"\"\n",
    "    pmf *= likelihood\n",
    "    prob_data = pmf.sum()\n",
    "    pmf /= prob_data\n",
    "    return prob_data"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Before we can use this function, we need to compute likelihoods.\n",
    "\n",
    "Here's are the likelihoods for a vanilla cookie:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [],
   "source": [
    "likelihood_vanilla = xs / 100"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And for a chocolate cookie."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [],
   "source": [
    "likelihood_chocolate = 1 - likelihood_vanilla"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we can do the update for a vanilla cookie."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.5000000000000001"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "bayes_update(pmf, likelihood_vanilla)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The return value is the total probability of the data, which is $1/2$.\n",
    "\n",
    "And here's what the PMF looks like after the update."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "pmf.plot()\n",
    "\n",
    "plt.xlabel('Bowl #')\n",
    "plt.ylabel('Probability')\n",
    "plt.title('One cookie');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "That's consistent with what we got with the Bayes table.\n",
    "\n",
    "The advantage of using a PMF is that it is easier to do multiple updates.  The following cell starts again with the uniform prior and does updates with two vanilla cookies and one chocolate cookie:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [],
   "source": [
    "data = 'VVC'\n",
    "\n",
    "pmf = make_pmf(xs, prior)\n",
    "\n",
    "for cookie in data:\n",
    "    if cookie == 'V':\n",
    "        bayes_update(pmf, likelihood_vanilla)\n",
    "    else:\n",
    "        bayes_update(pmf, likelihood_chocolate)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here's what the results look like:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "pmf.plot()\n",
    "\n",
    "plt.xlabel('Bowl #')\n",
    "plt.ylabel('Probability')\n",
    "plt.title('Three cookies');"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Again, that's consistent with what we got with the Bayes table."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The dice problem\n",
    "\n",
    "As an exercise, let's do one more version of the dice problem:\n",
    "\n",
    "> Suppose you have a 4-sided, 6-sided, 8-sided, 12-sided, and a **20-sided die**.  You choose one at random, roll it and **get a 7**. What is the probability that the die you rolled is 4-sided?  What are the posterior probabilities for the other dice?\n",
    "\n",
    "Notice that in this version I've added a 20-sided die and the outcome is 7, not 1.\n",
    "\n",
    "Here's a PMF that represents the prior:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4     0.2\n",
       "6     0.2\n",
       "8     0.2\n",
       "12    0.2\n",
       "20    0.2\n",
       "dtype: float64"
      ]
     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "xs = np.array([4, 6, 8, 12, 20])\n",
    "prior = 1/5\n",
    "\n",
    "pmf = make_pmf(xs, prior)\n",
    "pmf"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In this version, the index contains integers rather than strings, so we can compute the likelihoods like this:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [],
   "source": [
    "likelihood = 1 / xs"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "But the outcome is 7, so any die with fewer than 7 sides has likelihood 0.\n",
    "\n",
    "We can adjust `likelihood` by making a Boolean Series:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [],
   "source": [
    "too_low = (xs < 7)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And using it to set the corresponding elements of `likelihood` to 0."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([0.        , 0.        , 0.125     , 0.08333333, 0.05      ])"
      ]
     },
     "execution_count": 32,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "likelihood[too_low] = 0\n",
    "likelihood"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we can do the update and display the results."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4     0.000000\n",
       "6     0.000000\n",
       "8     0.483871\n",
       "12    0.322581\n",
       "20    0.193548\n",
       "dtype: float64"
      ]
     },
     "execution_count": 33,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "bayes_update(pmf, likelihood)\n",
    "pmf"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The 4-sided and 6-sided dice have been eliminated.  Of the remaining dice, the 8-sided die is the most likely."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Exercise:** Suppose you have the same set of 5 die.  You choose a die, roll it six times, and get 6, 7, 2, 5, 1, and 2 again.  What is the probability that you chose the 8-sided die? "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Solution\n",
    "\n",
    "xs = np.array([4, 6, 8, 12, 20])\n",
    "prior = 1/5\n",
    "\n",
    "pmf = make_pmf(xs, prior)\n",
    "xs = pmf.index.values"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 35,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4     0.000000\n",
       "6     0.000000\n",
       "8     0.915845\n",
       "12    0.080403\n",
       "20    0.003751\n",
       "dtype: float64"
      ]
     },
     "execution_count": 35,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# Solution\n",
    "\n",
    "outcomes = [6, 7, 2, 5, 1, 2]\n",
    "\n",
    "for outcome in outcomes:\n",
    "    likelihood = 1 / xs\n",
    "    too_low = (xs < outcome)\n",
    "    likelihood[too_low] = 0\n",
    "    \n",
    "    bayes_update(pmf, likelihood)\n",
    "\n",
    "pmf\n",
    "    "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Summary\n",
    "\n",
    "In this notebook, \n",
    "\n",
    "[In the next notebook](https://colab.research.google.com/github/AllenDowney/BiteSizeBayes/blob/master/07_euro.ipynb)..."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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