{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Семинар 6. Теория вероятности.\n",
    "\n",
    "### Что было на лекции\n",
    "\n",
    "Вероятность — функция P, определенная на событиях. Значение функции отражает представление о том, с какой вероятностью данное событие случается: 0 - не случается вовсе, 1 - случается всегда.\n",
    "\n",
    "### Дискретные случайные величины\n",
    "\n",
    "Распределение Бернулли:\n",
    "\n",
    "$\\xi = 1$ — успех (выпадение орла/правильный ответ на вопрос и пр.)\n",
    "\n",
    "$\\xi \\sim Ber(p)$\n",
    "\n",
    "$P(\\xi = 1) = p$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 134,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "import pandas as pd\n",
    "import scipy as sp\n",
    "\n",
    "import pylab as plt\n",
    "\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 135,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def plot_hist(sample, title, xlabel='value', ylabel='freq', bins=10):\n",
    "    plt.hist(sample, bins, normed=True, facecolor='green', alpha=0.75)\n",
    "\n",
    "    plt.xlabel(xlabel)\n",
    "    plt.ylabel(ylabel)\n",
    "    plt.title(title)\n",
    "    plt.grid(True)\n",
    "\n",
    "    plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 136,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "from scipy.stats import bernoulli"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 137,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x109c58510>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "p = 0.3\n",
    "sample = bernoulli.rvs(p, size=1000)\n",
    "plot_hist(sample, 'Bernoulli')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Биномиальное распределение\n",
    "\n",
    "$\\xi_i \\sim Ber(p)$\n",
    "\n",
    "$\\eta = \\sum_{i=1}^n\\xi_i \\sim Bin(n, p)$\n",
    "\n",
    "$P(\\eta = k) = C^n_kp^k(1-p)^{n-k}$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 138,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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pThhmZlbK/wdz/S2IEXKduQAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x109c58310>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "n, p = 100, .3\n",
    "sample = np.random.binomial(n, p, 1000)\n",
    "plot_hist(sample, 'Binomial')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Распределение Пуассона\n",
    "\n",
    "$\\xi_i$ — количество раз некоторого редкого события (встретить слово в тексте)\n",
    "\n",
    "$P(\\xi_i = k) = \\frac{\\lambda^k e^{-\\lambda}}{k!}, \\lambda > 0$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 139,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<matplotlib.figure.Figure at 0x109dee710>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sample = np.random.poisson(0.1, 1000)\n",
    "plot_hist(sample, 'Poison, $\\lambda$ = 0.1')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 140,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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njYeHhzsqnjLHw8PDHRXPVI5rtRqDg4MA9PT00C5FxMQTpGEa7Z47I+JN2bb7IuL1k+5c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\nwMzMcDEwMzPg/wNotICkZnHuVgAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x109fb7e50>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sample = np.random.poisson(1, 1000)\n",
    "plot_hist(sample, 'Poison, $\\lambda$ = 1')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 141,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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3exnXky7s+1yHu9QRkk4kO6sIsosxb51JsZD0dWAJ8M+AbcBVwF3AHcA84Bng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\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x10a1e3d50>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "sample = np.random.poisson(10, 1000)\n",
    "plot_hist(sample, 'Poison, $\\lambda$ = 10')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Независимые события**\n",
    "\n",
    "События A и B называются независимыми, если выполнено соотношение:\n",
    "\n",
    "$P(AB) = P(A)P(B)$\n",
    "\n",
    "где P(AB) — вероятность что произошло и A и B.\n",
    "\n",
    "Определение продолжается очевидным образом: n событий $A_1,...,A_n$ называются независимыми, если выполнено\n",
    "\n",
    "$P(A_1,...,A_n) = P(A_1) \\cdots P(A_n)$\n",
    "\n",
    "**Условная вероятность**\n",
    "\n",
    "Допустим есть два события: A и B. Если известно, что B произошло, то вероятность A можно вычислить по формуле:\n",
    "\n",
    "$P(A|B) = \\dfrac{P(AB)}{P(B)}$\n",
    "\n",
    "Отсюда получаем что \n",
    "\n",
    "$P(AB) = P(A|B)P(B)$ или $P(AB) = P(B|A)P(A)$\n",
    "\n",
    "Отсюда получаем **формулу Байеса**\n",
    "\n",
    "$P(A|B) = \\dfrac{P(B|A)P(A)}{P(B)}$\n",
    "\n",
    "*Как из этого построить классификатор?*\n",
    "\n",
    "Пусть у нас есть некоторая обучающая выборка: $X^l = (x_i, y_i)_{i=1}^l$\n",
    "\n",
    "Будем считать что событие A — \"объект принадлежит классу $\\dot{y}$\", а событие B — \"объект описывается набором признаков $(x^1, \\cdots, x^n)$\". Тогда можно переписать формулу в следующем виде:\n",
    "\n",
    "$P(\\dot{y}~|~(x^1, \\cdots, x^n)) = \\dfrac{P((x^1, \\cdots, x^n)~|~\\dot{y})P(\\dot{y})}{P((x^1, \\cdots, x^n))}$\n",
    "\n",
    "А теперь сделаем \"наивное\" предположение: пусть все признаки независимы. Тогда:\n",
    "\n",
    "$P(\\dot{y}~|~(x^1, \\cdots, x^n)) = \\dfrac{\\prod_j{P(x^j~|~\\dot{y}})P(\\dot{y})}{P(x^1, \\cdots, x^n)}$\n",
    "\n",
    "Откуда получаем сам классификатор:\n",
    "\n",
    "$a(x) = \\arg\\max_{\\dot{y}} P(\\dot{y}~|~(x^1, \\cdots, x^n)) = \\arg\\max_{\\dot{y}} \\prod_j{P(x^j~|~\\dot{y}})P(\\dot{y}) $"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Байесовский классификатор\n",
    "\n",
    "$a(x) = \\arg\\max_{y \\in Y} P(y) \\prod_j P(x^j|y)$\n",
    "\n",
    "**Небольшой спам фильтр своими руками**\n",
    "\n",
    "Разберем работу наивного байесовского классификатора на примере. В качестве данных взяли набор sms-сообщений, нужно определить, является ли данное сообщение спамом.\n",
    "\n",
    "Объекты — сообщения\n",
    "\n",
    "Признаки — слова сообщения\n",
    "\n",
    "Целевая метка — спам/не спам"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "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>label</th>\n",
       "      <th>text</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>ham</td>\n",
       "      <td>Go until jurong point, crazy.. Available only ...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>ham</td>\n",
       "      <td>Ok lar... Joking wif u oni...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>spam</td>\n",
       "      <td>Free entry in 2 a wkly comp to win FA Cup fina...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>ham</td>\n",
       "      <td>U dun say so early hor... U c already then say...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>ham</td>\n",
       "      <td>Nah I don't think he goes to usf, he lives aro...</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "  label                                               text\n",
       "0   ham  Go until jurong point, crazy.. Available only ...\n",
       "1   ham                      Ok lar... Joking wif u oni...\n",
       "2  spam  Free entry in 2 a wkly comp to win FA Cup fina...\n",
       "3   ham  U dun say so early hor... U c already then say...\n",
       "4   ham  Nah I don't think he goes to usf, he lives aro..."
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "import pandas as pd\n",
    "import numpy as np\n",
    "from __future__ import division\n",
    "\n",
    "data = pd.read_csv('sms_data.txt', sep='\\t', names=('label', 'text'))\n",
    "data.head()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Разобъем выборку на 2: обучающую и тестовую"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "data = data.reindex(np.random.permutation(data.index))\n",
    "\n",
    "data_train = data[:4000]\n",
    "data_test = data[4000:]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Напишем вспомогательные функции:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "['go',\n",
       " 'until',\n",
       " 'jurong',\n",
       " 'point',\n",
       " 'crazy',\n",
       " 'available',\n",
       " 'only',\n",
       " 'in',\n",
       " 'bugis',\n",
       " 'n',\n",
       " 'great',\n",
       " 'world',\n",
       " 'la',\n",
       " 'e',\n",
       " 'buffet',\n",
       " 'cine',\n",
       " 'there',\n",
       " 'got',\n",
       " 'amore',\n",
       " 'wat']"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import re\n",
    "\n",
    "def split_message(message):\n",
    "    return re.sub(\"[^a-zA-Z]\", \" \",  message.lower()).split()\n",
    "\n",
    "split_message(data.text[0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stderr",
     "output_type": "stream",
     "text": [
      "/usr/local/lib/python2.7/site-packages/pandas/core/indexing.py:499: SettingWithCopyWarning: \n",
      "A value is trying to be set on a copy of a slice from a DataFrame.\n",
      "Try using .loc[row_indexer,col_indexer] = value instead\n",
      "\n",
      "See the caveats in the documentation: http://pandas.pydata.org/pandas-docs/stable/indexing.html#indexing-view-versus-copy\n",
      "  self.obj[item_labels[indexer[info_axis]]] = value\n"
     ]
    },
    {
     "data": {
      "text/html": [
       "<div>\n",
       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>label</th>\n",
       "      <th>text</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>3229</th>\n",
       "      <td>spam</td>\n",
       "      <td>[six, chances, to, win, cash, from, to, pounds...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2301</th>\n",
       "      <td>ham</td>\n",
       "      <td>[nothin, comes, to, my, mind, help, me, buy, h...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>378</th>\n",
       "      <td>ham</td>\n",
       "      <td>[well, there, s, not, a, lot, of, things, happ...</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>5298</th>\n",
       "      <td>ham</td>\n",
       "      <td>[i, ll, hand, her, my, phone, to, chat, wit, u]</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>639</th>\n",
       "      <td>ham</td>\n",
       "      <td>[what, will, we, do, in, the, shower, baby]</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
       "</div>"
      ],
      "text/plain": [
       "     label                                               text\n",
       "3229  spam  [six, chances, to, win, cash, from, to, pounds...\n",
       "2301   ham  [nothin, comes, to, my, mind, help, me, buy, h...\n",
       "378    ham  [well, there, s, not, a, lot, of, things, happ...\n",
       "5298   ham    [i, ll, hand, her, my, phone, to, chat, wit, u]\n",
       "639    ham        [what, will, we, do, in, the, shower, baby]"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "train_messages = data_train.apply(lambda row: split_message(row['text']), axis=1)\n",
    "data_train.iloc[:, 1] = train_messages\n",
    "test_messages = data_test.apply(lambda row: split_message(row['text']), axis=1)\n",
    "data_test.iloc[:, 1] = test_messages\n",
    "\n",
    "data_train.head()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Теперь есть две задачи:\n",
    "1. Оценить по выборке $P(y)$ — вероятность класса (спам/не спам)\n",
    "2. Оценить по выборке $P(x^j|y)$ — вероятность признака для данного класса (вероятность встретить некоторое слово в сообщении типа спам/не спам)\n",
    "\n",
    "Обратите внимание, что все оценки будут вычисляться только по обучающей выборке!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0.12925 0.87075\n"
     ]
    }
   ],
   "source": [
    "n = data_train.shape[0]\n",
    "n_spam = data_train[data_train['label'] == 'spam'].shape[0]\n",
    "n_ham = data_train[data_train['label'] == 'ham'].shape[0]\n",
    "\n",
    "p_spam = n_spam / n\n",
    "p_ham = n_ham / n\n",
    "\n",
    "print p_spam, p_ham"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false
   },
   "source": [
    "Как будем оценивать вероятности слов? \n",
    "\n",
    "1. Найдем все уникальные слова.\n",
    "2. Для каждого слова посчитаем в скольки spam/ham-сообщениях оно встречалось\n",
    "3. Посчитаем для каждого класса: $P(x^j~|~y) = \\dfrac{count(x^j, x^j \\in y)}{count(y)}$\n",
    "\n",
    "Однако вычисление вероятности по нижней формуле плохо, так как если слово не встречалось в каком-либо из классов, соответствующая его вероятность будет равна 0, отсюда все выражение $P(y) \\prod_j P(x^j|y)$ будет равно 0. Поэтому немного изменим нашу формулу:\n",
    "\n",
    "$P(x^j~|~y) = \\dfrac{count(x^j, x^j \\in y) + \\alpha}{count(y) + |V|\\alpha}$\n",
    "\n",
    "где $|V|$ — количество уникальных слов"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "from collections import defaultdict\n",
    "\n",
    "def find_counts(data):\n",
    "    spam_counts, ham_counts = defaultdict(int), defaultdict(int)\n",
    "    for _, (label, message) in data.iterrows():\n",
    "        if label == 'spam':\n",
    "            for word in message:\n",
    "                spam_counts[word] += 1\n",
    "        else:\n",
    "            for word in message:\n",
    "                ham_counts[word] += 1\n",
    "    return spam_counts, ham_counts"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "spam_counts, ham_counts = find_counts(data_train)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "def find_probs(spam_counts, ham_counts, alpha=0.1):\n",
    "    spam_probs, ham_probs = defaultdict(lambda: 1 / n_words), defaultdict(lambda: 1 / n_words)\n",
    "    unique_words = np.unique(spam_counts.keys() + ham_counts.keys())\n",
    "    n_words = unique_words.shape[0]\n",
    "    spam_div = np.sum(spam_counts[w] for w in spam_counts) + alpha * n_words\n",
    "    ham_div = np.sum(ham_counts[w] for w in ham_counts) + alpha * n_words\n",
    "    for word in unique_words:\n",
    "        spam_probs[word] = (spam_counts[word] + alpha) / spam_div\n",
    "        ham_probs[word] = (ham_counts[word] + alpha) / ham_div\n",
    "    return spam_probs, ham_probs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "spam_probs, ham_probs = find_probs(spam_counts, ham_counts, alpha=0.1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "def classify(p_spam, p_ham, spam_probs, ham_probs, message):\n",
    "    spam_score = p_spam * np.prod([spam_probs[word] for word in message])\n",
    "    ham_score = p_ham * np.prod([ham_probs[word] for word in message])\n",
    "    if spam_score > ham_score:\n",
    "        return 'spam'\n",
    "    return 'ham'"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Итак, посчитаем качество нашего классификатора:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.98409669211195927"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "make_classify = lambda m: classify(p_spam, p_ham, spam_probs, ham_probs, m)\n",
    "(data_test.label == data_test.text.apply(make_classify)).mean()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Как часто спам оказывается помечен как обычное сообщение?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.069565217391304349"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "(((data_test.label == 'spam') & (data_test.text.apply(make_classify) == 'ham')).mean() / \n",
    "     (data_test.label == 'spam').mean())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Что интересного можно посмотреть? Например, найдем топ10 слов, наиболее вероятных для каждого класса."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[('to', 0.03798017099506282),\n",
       " ('a', 0.02152289969092442),\n",
       " ('call', 0.020398988479910087),\n",
       " ('you', 0.01678641673022117),\n",
       " ('your', 0.014137197447115963),\n",
       " ('the', 0.011889375025087303),\n",
       " ('now', 0.011889375025087303),\n",
       " ('free', 0.011889375025087303),\n",
       " ('for', 0.011407698791795448),\n",
       " ('or', 0.0110063019307189),\n",
       " ('p', 0.010123228836350499),\n",
       " ('txt', 0.01004294946413519),\n",
       " ('u', 0.01004294946413519),\n",
       " ('is', 0.008597920764259623),\n",
       " ('have', 0.0074740095532452935),\n",
       " ('ur', 0.007393730181029984),\n",
       " ('on', 0.007313450808814675),\n",
       " ('from', 0.007152892064384056),\n",
       " ('stop', 0.006992333319953438),\n",
       " ('text', 0.006912053947738128)]"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sorted(spam_probs.iteritems(), key=lambda x: x[1], reverse=True)[:20]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[('i', 0.04134996559177304),\n",
       " ('you', 0.027547904975138843),\n",
       " ('to', 0.02231397749411183),\n",
       " ('the', 0.01543233210239113),\n",
       " ('a', 0.015180328186637978),\n",
       " ('u', 0.014346776772992934),\n",
       " ('and', 0.012001201864828977),\n",
       " ('in', 0.011322729783955106),\n",
       " ('me', 0.010954416368623574),\n",
       " ('my', 0.010799337035852404),\n",
       " ('is', 0.010644257703081233),\n",
       " ('it', 0.009810706289436191),\n",
       " ('that', 0.007872214629796556),\n",
       " ('of', 0.00758144088085061),\n",
       " ('for', 0.007058048132747909),\n",
       " ('s', 0.00641834588506683),\n",
       " ('have', 0.006243881635699263),\n",
       " ('can', 0.006108187219524488),\n",
       " ('not', 0.00599187771994611),\n",
       " ('but', 0.00599187771994611)]"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sorted(ham_probs.iteritems(), key=lambda x: x[1], reverse=True)[:20]"
   ]
  }
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