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    "# Applying Naive Bayes classification to spam filtering"
   ]
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   "source": [
    "Let's pretend we have an email with three words: \"Send money now.\" We'll use Naive Bayes to classify it as **ham or spam.**\n",
    "\n",
    "$$P(spam \\ | \\ \\text{send money now}) = \\frac {P(\\text{send money now} \\ | \\ spam) \\times P(spam)} {P(\\text{send money now})}$$\n",
    "\n",
    "By assuming that the features (the words) are **conditionally independent**, we can simplify the likelihood function:\n",
    "\n",
    "$$P(spam \\ | \\ \\text{send money now}) \\approx \\frac {P(\\text{send} \\ | \\ spam) \\times P(\\text{money} \\ | \\ spam) \\times P(\\text{now} \\ | \\ spam) \\times P(spam)} {P(\\text{send money now})}$$\n",
    "\n",
    "We can calculate all of the values in the numerator by examining a corpus of **spam email**:\n",
    "\n",
    "$$P(spam \\ | \\ \\text{send money now}) \\approx \\frac {0.2 \\times 0.1 \\times 0.1 \\times 0.9} {P(\\text{send money now})} = \\frac {0.0018} {P(\\text{send money now})}$$\n",
    "\n",
    "We would repeat this process with a corpus of **ham email**:\n",
    "\n",
    "$$P(ham \\ | \\ \\text{send money now}) \\approx \\frac {0.05 \\times 0.01 \\times 0.1 \\times 0.1} {P(\\text{send money now})} = \\frac {0.000005} {P(\\text{send money now})}$$\n",
    "\n",
    "All we care about is whether spam or ham has the **higher probability**, and so we predict that the email is **spam**."
   ]
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   "source": [
    "## Key takeaways\n",
    "\n",
    "- The **\"naive\" assumption** of Naive Bayes (that the features are conditionally independent) is critical to making these calculations simple.\n",
    "- The **normalization constant** (the denominator) can be ignored since it's the same for all classes.\n",
    "- The **prior probability** is much less relevant once you have a lot of features."
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Comparing Naive Bayes with other models\n",
    "\n",
    "Advantages of Naive Bayes:\n",
    "\n",
    "- Model training and prediction are very fast\n",
    "- Somewhat interpretable\n",
    "- No tuning is required\n",
    "- Features don't need scaling\n",
    "- Insensitive to irrelevant features (with enough observations)\n",
    "- Performs better than logistic regression when the training set is very small\n",
    "\n",
    "Disadvantages of Naive Bayes:\n",
    "\n",
    "- Predicted probabilities are not well-calibrated\n",
    "- Correlated features can be problematic (due to the independence assumption)\n",
    "- Can't handle negative features (with Multinomial Naive Bayes)\n",
    "- Has a higher \"asymptotic error\" than logistic regression"
   ]
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