workshop03.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Bayesian Bandits from Scratch\n",
    "===\n",
    "\n",
    "Code from a talk presented to the Boston Bayesian meetup, July 2016.\n",
    "\n",
    "Copyright 2016 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 numpy as np\n",
    "\n",
    "from thinkbayes2 import Suite\n",
    "import thinkplot\n",
    "\n",
    "import warnings\n",
    "warnings.filterwarnings('ignore')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The cookie problem\n",
    "\n",
    "Create a Suite with two equally likely hypotheses.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Bowl 1 0.5\n",
      "Bowl 2 0.5\n"
     ]
    }
   ],
   "source": [
    "cookie = Suite(['Bowl 1', 'Bowl 2'])\n",
    "cookie.Print()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Update each hypothesis with the likelihood of the data (a vanilla cookie)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.625"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "cookie['Bowl 1'] *= 0.75\n",
    "cookie['Bowl 2'] *= 0.5\n",
    "cookie.Normalize()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Print the posterior probabilities."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Bowl 1 0.6\n",
      "Bowl 2 0.4\n"
     ]
    }
   ],
   "source": [
    "cookie.Print()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Suppose we put the first cookie back, stir, choose again from the same bowl, and get a chocolate cookie."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Bowl 1 0.428571428571\n",
      "Bowl 2 0.571428571429\n"
     ]
    }
   ],
   "source": [
    "cookie['Bowl 1'] *= 0.25\n",
    "cookie['Bowl 2'] *= 0.5\n",
    "cookie.Normalize()\n",
    "cookie.Print()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## The red dice problem\n",
    "\n",
    "Create a Suite to represent six-sided dice with different numbers of red sides."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0 0.25\n",
      "2 0.25\n",
      "4 0.25\n",
      "6 0.25\n"
     ]
    }
   ],
   "source": [
    "suite = Suite([0, 2, 4, 6])\n",
    "suite.Print()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can do the update by hand like this:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0 0.0\n",
      "2 0.166666666667\n",
      "4 0.333333333333\n",
      "6 0.5\n"
     ]
    }
   ],
   "source": [
    "suite[0] *= 0\n",
    "suite[2] *= 2/6\n",
    "suite[4] *= 4/6\n",
    "suite[6] *= 6/6\n",
    "\n",
    "suite.Normalize()\n",
    "suite.Print()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now I'll do the same calculation using `Suite.Update`.\n",
    "\n",
    "I'll define a new class called `RedDice` that extends `Suite`.\n",
    "\n",
    "It provides a method called `Likelihood` that takes `data` and `hypo` and returns the probability of the data (the outcome of rolling the die) for a given hypothesis (number of red sides on the die)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [],
   "source": [
    "class RedDice(Suite):\n",
    "    # hypo is the number of red sides on the die\n",
    "    # data is the outcome, either 'Red' or 'Blue'\n",
    "    def Likelihood(self, data, hypo):\n",
    "        p = hypo / 6\n",
    "        if data == 'Red':\n",
    "            return p\n",
    "        else:\n",
    "            return 1 - p"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we can create a `RedDice` object and update it."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0 0.0\n",
      "2 0.166666666667\n",
      "4 0.333333333333\n",
      "6 0.5\n"
     ]
    }
   ],
   "source": [
    "dice = RedDice([0, 2, 4, 6])\n",
    "dice.Update('Red')\n",
    "dice.Print()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "If we get more data, we can perform more updates."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "for roll in ['Red', 'Blue', 'Red']:\n",
    "    dice.Update(roll)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here are the results."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0 0.0\n",
      "2 0.2\n",
      "4 0.8\n",
      "6 0.0\n"
     ]
    }
   ],
   "source": [
    "dice.Print()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Bayesian bandits\n",
    "\n",
    "\n",
    "Here's a definition for `Bandit`, which extends `Suite` and defines a likelihood function that computes the probability of the data (win or lose) for a given value of `x` (the probability of win).\n",
    "\n",
    "Note that `hypo` is in the range 0 to 100."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "class Bandit(Suite):\n",
    "    \n",
    "    def Likelihood(self, data, hypo):\n",
    "        \"\"\" \n",
    "        hypo is the prob of win (0-100)\n",
    "        data is a string, either 'W' or 'L'\n",
    "        \"\"\"\n",
    "        x = hypo / 100\n",
    "        if data == 'W':\n",
    "            return x\n",
    "        else:\n",
    "            return 1-x"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We'll start with a uniform distribution from 0 to 100."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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TBo4kqQkDR5LUhIEjSWrCwJEkNWHgSJKaMHAkSU0YOJKkJgwcSVITBo4kqQkDR5LUhIEj\nSWrCwJEkNWHgSJKa6D1wkpySZGuS+5NcOE2dK5JsS3JnkpfO1jbJCUk+n+SuJDcmOWA32+7ptu/T\n7xFKkkbRa+AkWQZcCZwMHA+sSXLcUJ1TgSOr6mhgLXDVCG3/GLigql4CfAq4oGvzNOB64Ner6kXA\nBLCzz2OUJI2m7xHOicC2qnqoqnYCNwCrh+qsBq4DqKpbgRVJVs7S9piquqVb3gyc2S2/Drirqu7p\n9vdoVVVPxyZJmoO+A+cQ4OEp6490ZaPUmantPUnO6JbPAg7tlo8BSPKZJLcn+e09PgJJ0oJYvtgd\n2I2MUOetwBVJ3gVsAH7QlS8HXgm8Avh/wF8kub2qPju8g3Xr1j25PDExwcTExJ71WpKWmMnJSSYn\nJxdsf30Hznbg8Cnrh3Zlw3UO202dfaZrW1X3Mbi3Q5KjgdO7Oo8Af1lVj3bbNgIvB2YMHEnSPzb8\nn/FLL710j/bX9yW124CjkhzRzRY7m8GIZKoNwJsBkqwCHquqHTO1TfLs7ucy4HfoJhoANwEvTrJv\nkuXAq4F7+zxASdJoeh3hVNUTSc4HNjEIt/VVtSXJ2sHmurqqNiY5LckDwPeAc2dq2+16TZLzgAI+\nWVUf7No8luTdwO3Aj4A/q6o/7/MYJUmjyThO4kri5DVJmqMkVNUo99l3yycNSJKaMHAkSU0YOJKk\nJgwcSVITBo4kqQkDR5LUhIEjSWrCwJEkNWHgSJKaMHAkSU0YOJKkJgwcSVITBo4kqQkDR5LUhIEj\nSWrCwJEkNWHgSJKaMHAkSU0YOJKkJgwcSVITBo4kqQkDR5LUhIEjSWrCwJEkNWHgSJKaMHAkSU0Y\nOJKkJgwcSVITBo4kqQkDR5LUhIEjSWrCwJEkNWHgSJKaMHAkSU0YOJKkJnoPnCSnJNma5P4kF05T\n54ok25LcmeSls7VNckKSzye5K8mNSQ4Y2t/hSR5P8o7+jkySNBe9Bk6SZcCVwMnA8cCaJMcN1TkV\nOLKqjgbWAleN0PaPgQuq6iXAp4ALht76D4CNvRzUEjM5ObnYXdhreC6e4rl4iudi4fQ9wjkR2FZV\nD1XVTuAGYPVQndXAdQBVdSuwIsnKWdoeU1W3dMubgTN37SzJauBB4Cs9HdOS4j+mp3gunuK5eIrn\nYuH0HTiHAA9PWX+kKxulzkxt70lyRrd8FnAoQHdp7QLgUiAL0H9J0gLZGycNjBIUbwXOS3IbsD/w\ng678EuA9VfX9OexLktRCVfX2AlYBn5myfhFw4VCdq4BfnrK+FVg5Stuu/GjgC93yXzK4nPYg8Cjw\nHeBtu2lTvnz58uVr7q89yYTl9Os24KgkRwDfAM4G1gzV2QCcB3wkySrgsarakeQ707VN8uyq+nY3\nseB36CYaVNWrdu00ySXA41X1/uFOVZUjH0lqrNfAqaonkpwPbGJw+W59VW1Jsnawua6uqo1JTkvy\nAPA94NyZ2na7XpPkPAaJ+8mq+mCfxyFJ2nPpLjFJktSrvXHSQK9G+SDqUpXk0CQ3J/lKki8n+c2u\n/MAkm5Lcl+SmJCsWu68tJFmW5I4kG7r1sTwPAElWJPlYki3d78dJ43g+kvz7JPckuTvJh5LsM07n\nIcn6JDuS3D2lbNrjT3Jx96H9LUleN9v+xypwRvkg6hL3Q+AdVXU88HMMZvodx2BCxuaqOha4Gbh4\nEfvY0tuBe6esj+t5AHgfsLGqXgC8hMHknbE6H0meC/wG8PKqOoHBLYc1jNd5uIbB38epdnv8SV7I\n4GMpLwBOBd6fZMb742MVOIz2QdQlq6q+WVV3dst/B2xh8Bmm1cC1XbVrgTcsTg/bSXIocBqDp1bs\nMnbnASDJM4F/XlXXAFTVD6vqbxnP8/E0YP8ky4H9gO2M0XnoPlD/6FDxdMd/BnBD9/vyVWAbg7+x\n0xq3wBnlg6hjIcnPAC8FvgCsrKodMAgl4ODF61kz7wF+m8HEk13G8TwAPA/4TpJrukuMVyd5BmN2\nPqrq6wwei/U1BkHzt1W1mTE7D7tx8DTHP/z3dDuz/D0dt8ARTz6R4ePA27uRzvDMkSU9kyTJ6cCO\nbrQ30yWAJX0eplgOvBz4w6p6OYPZohcxfr8Xz2Lwv/kjgOcyGOn8CmN2HkYw7+Mft8DZDhw+Zf3Q\nrmxsdJcKPg5cX1U3dsU7uufXkeQ5wLcWq3+NvBI4I8mDwP8A/mWS64Fvjtl52OUR4OGqur1b/wSD\nABq334vXAg9W1Xer6gkGDwb+ecbvPAyb7vi3A4dNqTfr39NxC5wnP4iaZB8GHybdsMh9au2/A/dW\n1fumlG0AzumW3wLcONxoKamqd1bV4VX1fAa/AzdX1ZuATzNG52GX7nLJw0mO6Ypew+Dht2P1e8Hg\nUtqqJPt2N79fw2BSybidh/DjI//pjn8DcHY3k+95wFHAF2fc8bh9DifJKQxm5Oz6MOlli9ylZpK8\nksHjf77MU4+qeCeDX5KPMvjfykPAWVX12GL1s6UkrwZ+q6rOSPLTjO95eAmDCRRPZ/BoqHMZ3EAf\nq/PRPaHkbGAn8CXg3wA/xZichyQfBiaAg4AdDJ5P+T+Bj7Gb409yMYNnW+5kcIl+04z7H7fAkSQt\njnG7pCZJWiQGjiSpCQNHktSEgSNJasLAkSQ1YeBIkpowcCRJTRg4kqQmDBxpL5DkFUnu6h4Tsn/3\nJWAvXOx+SQvJJw1Ie4kkv8vgO1j2Y/AwzcsXuUvSgjJwpL1EkqczeMDs3wM/X/7j1BLjJTVp7/FP\ngQMYPCxy30Xui7TgHOFIe4kkNzL4fp7nAc+tqt9Y5C5JC2r5YndAEiR5E/CDqrohyTLgr5JMVNXk\nIndNWjCOcCRJTXgPR5LUhIEjSWrCwJEkNWHgSJKaMHAkSU0YOJKkJgwcSVITBo4kqYn/D0+Ah3NI\nMeViAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f7b96e8c650>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "bandit = Bandit(range(101))\n",
    "thinkplot.Pdf(bandit)\n",
    "thinkplot.Config(xlabel='x', ylabel='Probability')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we can update with a single loss:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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ULCKl53xmNpO/WM7XCzZQ8F9Vp+ua8lRqMvVrV/OtNyldChYPChaR0rdp+34m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\nRK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f7b949a7290>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "bandit.Update('L')\n",
    "thinkplot.Pdf(bandit)\n",
    "thinkplot.Config(xlabel='x', ylabel='Probability')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Another loss:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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GAA8RCrEn3f1+MxtH6Mjl8XCfScAIQhNe3uzuS8LtzwPZQBMgD7jX3Z82s8bA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\nRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f7b946b05d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "bandit.Update('L')\n",
    "thinkplot.Pdf(bandit)\n",
    "thinkplot.Config(xlabel='x', ylabel='Probability')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And a win:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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mQkRJEySXeP+dhWdtsEPAQWC+95gpxIr1Ozh+8gwA1auUtyVcTMSpX6sKt1/b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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f7b945bf350>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "bandit.Update('W')\n",
    "thinkplot.Pdf(bandit)\n",
    "thinkplot.Config(xlabel='x', ylabel='Probability')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Starting over, here's what it looks like after 1 win and 9 losses."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "10"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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H10xDmvvGqjRG45xrquImFkV6t98CJgDPAs8Al5hZQjPvmxqfdd8wUy4fSfWN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\n/w+y6Slh3TwqjgAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f7b96e8c6d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "bandit = Bandit(range(101))\n",
    "\n",
    "for outcome in 'LLWLLLLLLL':\n",
    "    bandit.Update(outcome)\n",
    "\n",
    "thinkplot.Pdf(bandit)\n",
    "thinkplot.Config(xlabel='x', ylabel='Probability')\n",
    "bandit.MaximumLikelihood()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The posterior mean is about 17%"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "16.68194469884906"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "bandit.Mean()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The most likely value is the observed proportion 1/10"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "10"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "bandit.MaximumLikelihood()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The posterior credible interval has a 90% chance of containing the true value (provided that the prior distribution truly represents our background knowledge)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(3, 36)"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "bandit.CredibleInterval(90)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Bayesian A/B testing"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now suppose we have several bandits and we want to decide which one to play."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For this example, we have 4 machines with these probabilities:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "actual_probs = [10, 20, 30, 40]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The following function simulates playing one machine once."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from random import random\n",
    "from collections import Counter\n",
    "\n",
    "counter = Counter()\n",
    "\n",
    "def flip(p):\n",
    "    return random() < p\n",
    "\n",
    "def play(i):\n",
    "    counter[i] += 1\n",
    "    p = actual_probs[i] / 100\n",
    "    if flip(p):\n",
    "        return 'W'\n",
    "    else:\n",
    "        return 'L'"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here's a test, playing machine 3 twenty times:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "W W W W W W L L L L W L W L L W W L L L "
     ]
    }
   ],
   "source": [
    "for i in range(20):\n",
    "    result = play(3)\n",
    "    print(result, end=' ')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now I'll make 4 `Bandit` objects to represent our beliefs about the 4 machines."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "prior = range(101)\n",
    "beliefs = [Bandit(prior) for i in range(4)]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The core of the algorithm is this process for choosing which machine to play:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [],
   "source": [
    "def choose(beliefs):\n",
    "    ps = [b.Random() for b in beliefs]\n",
    "    return np.argmax(ps)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here's an example."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "choose(beliefs)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This function updates our beliefs about one of the machines based on one outcome."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def update(beliefs, i, outcome):\n",
    "    beliefs[i].Update(outcome)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Finally, this function chooses a machine, plays once, and updates `beliefs`:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def choose_play_update(beliefs, verbose=False):\n",
    "    i = choose(beliefs)\n",
    "    outcome = play(i)\n",
    "    update(beliefs, i, outcome)\n",
    "    if verbose:\n",
    "        print(i, outcome, beliefs[i].Mean())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here's an example"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0 L 33.0\n"
     ]
    }
   ],
   "source": [
    "counter = Counter()\n",
    "choose_play_update(beliefs, verbose=True)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This function displays the four posterior distributions"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {},
   "outputs": [],
   "source": [
    "options = dict(yticklabels='invisible')\n",
    "\n",
    "def plot(beliefs):\n",
    "    thinkplot.preplot(rows=2, cols=2)\n",
    "    for i, b in enumerate(beliefs):\n",
    "        thinkplot.subplot(i+1)\n",
    "        thinkplot.Pdf(b, label=i)\n",
    "        thinkplot.config(**options)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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PPl+gB5+brn37CCUAAMobkeRjxhjddMX56tOjg7v20Rff68Hn3iGUAAAoZ0SSzxljdMMv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\nAABgQSQBAABY/H8fPpcSeNLUhwAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f7b94591310>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "plot(beliefs)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now we can play a few times and see how `beliefs` gets updated:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 32,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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fr4IkSfqRl16lyPLAjmeOndfZkSmXVwQAFRMBbNou2e0YKklfEoLPd0FSd0eL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\nIiKiJhgkERERETXx/9Pkc5RPgCcaAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f7b945039d0>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "for i in range(200):\n",
    "    choose_play_update(beliefs)\n",
    "    \n",
    "plot(beliefs)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can summarize `beliefs` by printing the posterior mean and credible interval:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "9.55671624054 (0, 28)\n",
      "22.2222222206 (12, 34)\n",
      "26.3157894737 (15, 39)\n",
      "35.2 (28, 42)\n"
     ]
    }
   ],
   "source": [
    "for i, b in enumerate(beliefs):\n",
    "    print(b.Mean(), b.CredibleInterval(90))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The credible intervals usually contain the true value, but the estimates are still rough, especially for the lower-probability machines.\n",
    "\n",
    "But that's a feature, not a bug: the goal is to play the high-probability machines most often.  Making the estimates more precise is a means to that end, but not an end itself."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0 8\n",
      "1 34\n",
      "2 36\n",
      "3 123\n"
     ]
    }
   ],
   "source": [
    "for machine, count in sorted(counter.items()):\n",
    "    print(machine, count)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": []
  }
 ],
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