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Discrete Naïve Bayes

Discrete Naïve Bayes

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Page 1: Discrete Naïve Bayes

Discrete Naïve Bayes

Page 2: Discrete Naïve Bayes

Previously on CSCI 4622

2

Probabilistic Classification:

Train model on training data

Goal: Given new data , predict it’s label

For each class c, estimate probability that belongs to class

Assign to the class that gives the highest probability

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pY Belong

ns to some

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=

Page 3: Discrete Naïve Bayes

Discriminative vs Generative Classifiers

3

How do we model/estimate these conditional probabilities?

Discriminative Classifiers: o Model the conditional relationship between data and label directly:

o Example: Logistic Regression models this relationship as

Generative Classifiers:o Model the joint probability distribution

o Make assumptions about the relationship between and

o Make assumptions about the data itself

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Page 4: Discrete Naïve Bayes

Naïve Bayes

4

Naïve Bayes is a popular off-the-shelf classifier

o Known for being simple but very effective in some cases

o Very popular in text learning

o High Bias method

o For small training sets, outperforms many more sophisticated methods

o Usually a good thing to try right off the bat, just to see if it works

Page 5: Discrete Naïve Bayes

Motivation: SPAM Classification

5

Suppose you have the following (small) set of labeled emails (with stop-words removed):

Let be a Bag-of-Words feature vector representing frequency counts of words in vocab

We want to estimate the probabilities and

We then assign the email to class for which the probability is bigger

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HAM SPAM SPAM SPAM HAM

work nigeria fly money fly

buy opportunity buy buy home

money viagra nigeria fly nigeria<latexit sha1_base64="Fk40qlV9xSoCXT3K4KQdocdHpiQ=">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</latexit><latexit sha1_base64="Fk40qlV9xSoCXT3K4KQdocdHpiQ=">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</latexit><latexit sha1_base64="Fk40qlV9xSoCXT3K4KQdocdHpiQ=">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</latexit><latexit sha1_base64="Fk40qlV9xSoCXT3K4KQdocdHpiQ=">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</latexit>

p(y = SPAM | x)<latexit sha1_base64="RoubPElvdEbV+CcllNSjRjZz2Mw=">AAACDHicbVDLSsNAFJ34rPEVdelmsBHqpiRu1IVQceNGqGi10JQymU7aoTNJmLmRltAfcOOvuHGh4tYPcOffOH0stHrgwuGce7n3njAVXIPnfVlz8wuLS8uFFXt1bX1j09navtVJpiir0UQkqh4SzQSPWQ04CFZPFSMyFOwu7J2P/Lt7pjRP4hsYpKwpSSfmEacEjNRyXNe109LgNADWByXz6+rZ5TCQvI3zIIxwf3hgu27LKXplbwz8l/hTUkRTVFvOZ9BOaCZZDFQQrRu+l0IzJwo4FWxoB5lmKaE90mENQ2MimW7m42+GeN8obRwlylQMeKz+nMiJ1HogQ9MpCXT1rDcS//MaGUTHzZzHaQYsppNFUSYwJHgUDW5zxSiIgSGEKm5uxbRLFKFgArRNCP7sy39J7bB8UvavDouV42kaBbSL9lAJ+egIVdAFqqIaougBPaEX9Go9Ws/Wm/U+aZ2zpjM76Besj2+e25mJ</latexit><latexit sha1_base64="RoubPElvdEbV+CcllNSjRjZz2Mw=">AAACDHicbVDLSsNAFJ34rPEVdelmsBHqpiRu1IVQceNGqGi10JQymU7aoTNJmLmRltAfcOOvuHGh4tYPcOffOH0stHrgwuGce7n3njAVXIPnfVlz8wuLS8uFFXt1bX1j09navtVJpiir0UQkqh4SzQSPWQ04CFZPFSMyFOwu7J2P/Lt7pjRP4hsYpKwpSSfmEacEjNRyXNe109LgNADWByXz6+rZ5TCQvI3zIIxwf3hgu27LKXplbwz8l/hTUkRTVFvOZ9BOaCZZDFQQrRu+l0IzJwo4FWxoB5lmKaE90mENQ2MimW7m42+GeN8obRwlylQMeKz+nMiJ1HogQ9MpCXT1rDcS//MaGUTHzZzHaQYsppNFUSYwJHgUDW5zxSiIgSGEKm5uxbRLFKFgArRNCP7sy39J7bB8UvavDouV42kaBbSL9lAJ+egIVdAFqqIaougBPaEX9Go9Ws/Wm/U+aZ2zpjM76Besj2+e25mJ</latexit><latexit sha1_base64="RoubPElvdEbV+CcllNSjRjZz2Mw=">AAACDHicbVDLSsNAFJ34rPEVdelmsBHqpiRu1IVQceNGqGi10JQymU7aoTNJmLmRltAfcOOvuHGh4tYPcOffOH0stHrgwuGce7n3njAVXIPnfVlz8wuLS8uFFXt1bX1j09navtVJpiir0UQkqh4SzQSPWQ04CFZPFSMyFOwu7J2P/Lt7pjRP4hsYpKwpSSfmEacEjNRyXNe109LgNADWByXz6+rZ5TCQvI3zIIxwf3hgu27LKXplbwz8l/hTUkRTVFvOZ9BOaCZZDFQQrRu+l0IzJwo4FWxoB5lmKaE90mENQ2MimW7m42+GeN8obRwlylQMeKz+nMiJ1HogQ9MpCXT1rDcS//MaGUTHzZzHaQYsppNFUSYwJHgUDW5zxSiIgSGEKm5uxbRLFKFgArRNCP7sy39J7bB8UvavDouV42kaBbSL9lAJ+egIVdAFqqIaougBPaEX9Go9Ws/Wm/U+aZ2zpjM76Besj2+e25mJ</latexit><latexit sha1_base64="RoubPElvdEbV+CcllNSjRjZz2Mw=">AAACDHicbVDLSsNAFJ34rPEVdelmsBHqpiRu1IVQceNGqGi10JQymU7aoTNJmLmRltAfcOOvuHGh4tYPcOffOH0stHrgwuGce7n3njAVXIPnfVlz8wuLS8uFFXt1bX1j09navtVJpiir0UQkqh4SzQSPWQ04CFZPFSMyFOwu7J2P/Lt7pjRP4hsYpKwpSSfmEacEjNRyXNe109LgNADWByXz6+rZ5TCQvI3zIIxwf3hgu27LKXplbwz8l/hTUkRTVFvOZ9BOaCZZDFQQrRu+l0IzJwo4FWxoB5lmKaE90mENQ2MimW7m42+GeN8obRwlylQMeKz+nMiJ1HogQ9MpCXT1rDcS//MaGUTHzZzHaQYsppNFUSYwJHgUDW5zxSiIgSGEKm5uxbRLFKFgArRNCP7sy39J7bB8UvavDouV42kaBbSL9lAJ+egIVdAFqqIaougBPaEX9Go9Ws/Wm/U+aZ2zpjM76Besj2+e25mJ</latexit>

p(y = HAM | x)<latexit sha1_base64="+Uk5L5SjMtDFL+/APaBNDtJF7x4=">AAACC3icbVC7TgJBFJ31iesLtbSZCCbYkF0asTDB2NCYYCJCwhIyO8zChJndzcxdA9nwATb+io2FGlt/wM6/cXgUCp5kkpNzzs2de/xYcA2O822trK6tb2xmtuztnd29/ezB4b2OEkVZnUYiUk2faCZ4yOrAQbBmrBiRvmANf3A98RsPTGkehXcwillbkl7IA04JGKmTzeXzdlwYXXrAhqBkWr26GXuSd3Hq+QEejs/sfN6knKIzBV4m7pzk0By1TvbL60Y0kSwEKojWLdeJoZ0SBZwKNra9RLOY0AHpsZahIZFMt9PpMWN8apQuDiJlXgh4qv6eSInUeiR9k5QE+nrRm4j/ea0EgnI75WGcAAvpbFGQCAwRnjSDu1wxCmJkCKGKm79i2ieKUDD92aYEd/HkZVIvFS+K7m0pVynP28igY3SCCshF56iCqqiG6oiiR/SMXtGb9WS9WO/Wxyy6Ys1njtAfWJ8/48aZJA==</latexit><latexit sha1_base64="+Uk5L5SjMtDFL+/APaBNDtJF7x4=">AAACC3icbVC7TgJBFJ31iesLtbSZCCbYkF0asTDB2NCYYCJCwhIyO8zChJndzcxdA9nwATb+io2FGlt/wM6/cXgUCp5kkpNzzs2de/xYcA2O822trK6tb2xmtuztnd29/ezB4b2OEkVZnUYiUk2faCZ4yOrAQbBmrBiRvmANf3A98RsPTGkehXcwillbkl7IA04JGKmTzeXzdlwYXXrAhqBkWr26GXuSd3Hq+QEejs/sfN6knKIzBV4m7pzk0By1TvbL60Y0kSwEKojWLdeJoZ0SBZwKNra9RLOY0AHpsZahIZFMt9PpMWN8apQuDiJlXgh4qv6eSInUeiR9k5QE+nrRm4j/ea0EgnI75WGcAAvpbFGQCAwRnjSDu1wxCmJkCKGKm79i2ieKUDD92aYEd/HkZVIvFS+K7m0pVynP28igY3SCCshF56iCqqiG6oiiR/SMXtGb9WS9WO/Wxyy6Ys1njtAfWJ8/48aZJA==</latexit><latexit sha1_base64="+Uk5L5SjMtDFL+/APaBNDtJF7x4=">AAACC3icbVC7TgJBFJ31iesLtbSZCCbYkF0asTDB2NCYYCJCwhIyO8zChJndzcxdA9nwATb+io2FGlt/wM6/cXgUCp5kkpNzzs2de/xYcA2O822trK6tb2xmtuztnd29/ezB4b2OEkVZnUYiUk2faCZ4yOrAQbBmrBiRvmANf3A98RsPTGkehXcwillbkl7IA04JGKmTzeXzdlwYXXrAhqBkWr26GXuSd3Hq+QEejs/sfN6knKIzBV4m7pzk0By1TvbL60Y0kSwEKojWLdeJoZ0SBZwKNra9RLOY0AHpsZahIZFMt9PpMWN8apQuDiJlXgh4qv6eSInUeiR9k5QE+nrRm4j/ea0EgnI75WGcAAvpbFGQCAwRnjSDu1wxCmJkCKGKm79i2ieKUDD92aYEd/HkZVIvFS+K7m0pVynP28igY3SCCshF56iCqqiG6oiiR/SMXtGb9WS9WO/Wxyy6Ys1njtAfWJ8/48aZJA==</latexit><latexit sha1_base64="+Uk5L5SjMtDFL+/APaBNDtJF7x4=">AAACC3icbVC7TgJBFJ31iesLtbSZCCbYkF0asTDB2NCYYCJCwhIyO8zChJndzcxdA9nwATb+io2FGlt/wM6/cXgUCp5kkpNzzs2de/xYcA2O822trK6tb2xmtuztnd29/ezB4b2OEkVZnUYiUk2faCZ4yOrAQbBmrBiRvmANf3A98RsPTGkehXcwillbkl7IA04JGKmTzeXzdlwYXXrAhqBkWr26GXuSd3Hq+QEejs/sfN6knKIzBV4m7pzk0By1TvbL60Y0kSwEKojWLdeJoZ0SBZwKNra9RLOY0AHpsZahIZFMt9PpMWN8apQuDiJlXgh4qv6eSInUeiR9k5QE+nrRm4j/ea0EgnI75WGcAAvpbFGQCAwRnjSDu1wxCmJkCKGKm79i2ieKUDD92aYEd/HkZVIvFS+K7m0pVynP28igY3SCCshF56iCqqiG6oiiR/SMXtGb9WS9WO/Wxyy6Ys1njtAfWJ8/48aZJA==</latexit>

x<latexit sha1_base64="ezxhI2prtFY0vXyy/yEAlvhFBlU=">AAAB83icbVBNT8JAEJ3iF9Yv1KOXjdTEE2m5+HEi8eIREysk0JDtsoUN223d3RJJw+/w4kGNV/+MN/+NC/Sg4EsmeXlvJjPzwpQzpV332yqtrW9sbpW37Z3dvf2DyuHRg0oySahPEp7IdogV5UxQXzPNaTuVFMchp61wdDPzW2MqFUvEvZ6kNIjxQLCIEayNFDiOnXfDCD1NbcfpVapuzZ0DrRKvIFUo0OxVvrr9hGQxFZpwrFTHc1Md5FhqRjid2t1M0RSTER7QjqECx1QF+fzoKTozSh9FiTQlNJqrvydyHCs1iUPTGWM9VMveTPzP62Q6ugxyJtJMU0EWi6KMI52gWQKozyQlmk8MwUQycysiQywx0SYn24TgLb+8Svx67arm3dWrjesijTKcwCmcgwcX0IBbaIIPBB7hGV7hzRpbL9a79bFoLVnFzDH8gfX5A3nNkB4=</latexit><latexit sha1_base64="ezxhI2prtFY0vXyy/yEAlvhFBlU=">AAAB83icbVBNT8JAEJ3iF9Yv1KOXjdTEE2m5+HEi8eIREysk0JDtsoUN223d3RJJw+/w4kGNV/+MN/+NC/Sg4EsmeXlvJjPzwpQzpV332yqtrW9sbpW37Z3dvf2DyuHRg0oySahPEp7IdogV5UxQXzPNaTuVFMchp61wdDPzW2MqFUvEvZ6kNIjxQLCIEayNFDiOnXfDCD1NbcfpVapuzZ0DrRKvIFUo0OxVvrr9hGQxFZpwrFTHc1Md5FhqRjid2t1M0RSTER7QjqECx1QF+fzoKTozSh9FiTQlNJqrvydyHCs1iUPTGWM9VMveTPzP62Q6ugxyJtJMU0EWi6KMI52gWQKozyQlmk8MwUQycysiQywx0SYn24TgLb+8Svx67arm3dWrjesijTKcwCmcgwcX0IBbaIIPBB7hGV7hzRpbL9a79bFoLVnFzDH8gfX5A3nNkB4=</latexit><latexit sha1_base64="ezxhI2prtFY0vXyy/yEAlvhFBlU=">AAAB83icbVBNT8JAEJ3iF9Yv1KOXjdTEE2m5+HEi8eIREysk0JDtsoUN223d3RJJw+/w4kGNV/+MN/+NC/Sg4EsmeXlvJjPzwpQzpV332yqtrW9sbpW37Z3dvf2DyuHRg0oySahPEp7IdogV5UxQXzPNaTuVFMchp61wdDPzW2MqFUvEvZ6kNIjxQLCIEayNFDiOnXfDCD1NbcfpVapuzZ0DrRKvIFUo0OxVvrr9hGQxFZpwrFTHc1Md5FhqRjid2t1M0RSTER7QjqECx1QF+fzoKTozSh9FiTQlNJqrvydyHCs1iUPTGWM9VMveTPzP62Q6ugxyJtJMU0EWi6KMI52gWQKozyQlmk8MwUQycysiQywx0SYn24TgLb+8Svx67arm3dWrjesijTKcwCmcgwcX0IBbaIIPBB7hGV7hzRpbL9a79bFoLVnFzDH8gfX5A3nNkB4=</latexit><latexit sha1_base64="ezxhI2prtFY0vXyy/yEAlvhFBlU=">AAAB83icbVBNT8JAEJ3iF9Yv1KOXjdTEE2m5+HEi8eIREysk0JDtsoUN223d3RJJw+/w4kGNV/+MN/+NC/Sg4EsmeXlvJjPzwpQzpV332yqtrW9sbpW37Z3dvf2DyuHRg0oySahPEp7IdogV5UxQXzPNaTuVFMchp61wdDPzW2MqFUvEvZ6kNIjxQLCIEayNFDiOnXfDCD1NbcfpVapuzZ0DrRKvIFUo0OxVvrr9hGQxFZpwrFTHc1Md5FhqRjid2t1M0RSTER7QjqECx1QF+fzoKTozSh9FiTQlNJqrvydyHCs1iUPTGWM9VMveTPzP62Q6ugxyJtJMU0EWi6KMI52gWQKozyQlmk8MwUQycysiQywx0SYn24TgLb+8Svx67arm3dWrjesijTKcwCmcgwcX0IBbaIIPBB7hGV7hzRpbL9a79bFoLVnFzDH8gfX5A3nNkB4=</latexit>

Page 6: Discrete Naïve Bayes

Bayesian Classifiers

6

We can estimate these probabilities from training data using Bayes Rule

p(y | x) = p(x | y)p(y)p(x)

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Page 7: Discrete Naïve Bayes

Bayesian Classifiers

7

We can estimate these probabilities from training data using Bayes Rule

Bayesian methods give special names to each of these probabilities

Let’s look of each of these one-by-one

p(y | x) = p(x | y)p(y)p(x)

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posterior =class-conditional likelihood⇥ prior

evidence<latexit sha1_base64="ZY2mUp0loJMtM/evU3SqCX2fFUY=">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</latexit><latexit sha1_base64="ZY2mUp0loJMtM/evU3SqCX2fFUY=">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</latexit><latexit sha1_base64="ZY2mUp0loJMtM/evU3SqCX2fFUY=">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</latexit><latexit sha1_base64="ZY2mUp0loJMtM/evU3SqCX2fFUY=">AAACWXicbVBdSxtBFJ3dVhO3fkR99GVoUvDFsJsXFRQEX/powVQhG8Ls7F0zZHZmnbkrhmX/pC+l0L9S6GSN0ia9MHA4Z879OEkhhcUw/On5Hz5ubLbaW8Gn7Z3dvc7+wXerS8NhyLXU5j5hFqRQMESBEu4LAyxPJNwls+uFfvcExgqtbnFewDhnD0pkgjN01KTz2OsFMcIzmrwqtEUwQpuaXtI4zQzj1ZvGJbP2hGuVioWRSSrFzE2dap3WNEaRg6XvjZom9bsZnkQKikNdB73epNMN+2FTdB1ES9Aly7qZdF7iVPMyB4XNFqMoLHBcMYOCS6iDuLRQMD5jDzByUDG3yrhqoqnpF8ekNNPGPYW0Yf92VCy3dp4n7mfOcGpXtQX5P21UYnY2roQqSnSnvQ7KSklR00XONBUGOMq5A4wbFxqnfMpcpC5iG7gQotWT18Fw0D/vR98G3auLZRptckQ+k2MSkVNyRb6SGzIknPwgv71Nr+X98n2/7QevX31v6Tkk/5R/+AeaZbdj</latexit>

Page 8: Discrete Naïve Bayes

Bayesian Classifiers

8

The Posterior Probability:

General Interpretation: The probability that a particular object belongs to a particular class given its observed features

Concretely: The probability an email is SPAM given the words in the email

posterior =class-likelihood⇥ prior

evidence<latexit sha1_base64="Mw7+2IQ2YZmlQoq3Tf5rKrHKkJk=">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</latexit><latexit sha1_base64="Mw7+2IQ2YZmlQoq3Tf5rKrHKkJk=">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</latexit><latexit sha1_base64="Mw7+2IQ2YZmlQoq3Tf5rKrHKkJk=">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</latexit><latexit sha1_base64="Mw7+2IQ2YZmlQoq3Tf5rKrHKkJk=">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</latexit>

p(y | x) = p(x | y)p(y)p(x)

<latexit sha1_base64="rx3ngS8jQjLkF8TBFaNOULN3afo=">AAACKnicbVDLSsNAFJ3UV42vqks3g63QbkrSjXUhFN24rGBtoQllMpm0QycPZiZiCPkfN/6KC11oceuHOE0DauuBgcM553LnHidiVEjDmGmltfWNza3ytr6zu7d/UDk8uhdhzDHp4ZCFfOAgQRgNSE9Sycgg4gT5DiN9Z3o99/sPhAsaBncyiYjto3FAPYqRVNKoclWr6VE9sXzqwtRyPPiYNeAltFyPI5xG9UKDeSBpqGgj+5EbmV6rjSpVo2nkgKvELEgVFOiOKq+WG+LYJ4HEDAkxNI1I2inikmJGMt2KBYkQnqIxGSoaIJ8IO81vzeCZUlzohVy9QMJc/T2RIl+IxHdU0kdyIpa9ufifN4yl17ZTGkSxJAFeLPJiBmUI58VBl3KCJUsUQZhT9VeIJ0i1JFW9uirBXD55lfRazYumeduqdtpFG2VwAk5BHZjgHHTADeiCHsDgCbyAd/ChPWtv2kz7XERLWjFzDP5A+/oGxwmkiw==</latexit><latexit sha1_base64="rx3ngS8jQjLkF8TBFaNOULN3afo=">AAACKnicbVDLSsNAFJ3UV42vqks3g63QbkrSjXUhFN24rGBtoQllMpm0QycPZiZiCPkfN/6KC11oceuHOE0DauuBgcM553LnHidiVEjDmGmltfWNza3ytr6zu7d/UDk8uhdhzDHp4ZCFfOAgQRgNSE9Sycgg4gT5DiN9Z3o99/sPhAsaBncyiYjto3FAPYqRVNKoclWr6VE9sXzqwtRyPPiYNeAltFyPI5xG9UKDeSBpqGgj+5EbmV6rjSpVo2nkgKvELEgVFOiOKq+WG+LYJ4HEDAkxNI1I2inikmJGMt2KBYkQnqIxGSoaIJ8IO81vzeCZUlzohVy9QMJc/T2RIl+IxHdU0kdyIpa9ufifN4yl17ZTGkSxJAFeLPJiBmUI58VBl3KCJUsUQZhT9VeIJ0i1JFW9uirBXD55lfRazYumeduqdtpFG2VwAk5BHZjgHHTADeiCHsDgCbyAd/ChPWtv2kz7XERLWjFzDP5A+/oGxwmkiw==</latexit><latexit sha1_base64="rx3ngS8jQjLkF8TBFaNOULN3afo=">AAACKnicbVDLSsNAFJ3UV42vqks3g63QbkrSjXUhFN24rGBtoQllMpm0QycPZiZiCPkfN/6KC11oceuHOE0DauuBgcM553LnHidiVEjDmGmltfWNza3ytr6zu7d/UDk8uhdhzDHp4ZCFfOAgQRgNSE9Sycgg4gT5DiN9Z3o99/sPhAsaBncyiYjto3FAPYqRVNKoclWr6VE9sXzqwtRyPPiYNeAltFyPI5xG9UKDeSBpqGgj+5EbmV6rjSpVo2nkgKvELEgVFOiOKq+WG+LYJ4HEDAkxNI1I2inikmJGMt2KBYkQnqIxGSoaIJ8IO81vzeCZUlzohVy9QMJc/T2RIl+IxHdU0kdyIpa9ufifN4yl17ZTGkSxJAFeLPJiBmUI58VBl3KCJUsUQZhT9VeIJ0i1JFW9uirBXD55lfRazYumeduqdtpFG2VwAk5BHZjgHHTADeiCHsDgCbyAd/ChPWtv2kz7XERLWjFzDP5A+/oGxwmkiw==</latexit><latexit sha1_base64="rx3ngS8jQjLkF8TBFaNOULN3afo=">AAACKnicbVDLSsNAFJ3UV42vqks3g63QbkrSjXUhFN24rGBtoQllMpm0QycPZiZiCPkfN/6KC11oceuHOE0DauuBgcM553LnHidiVEjDmGmltfWNza3ytr6zu7d/UDk8uhdhzDHp4ZCFfOAgQRgNSE9Sycgg4gT5DiN9Z3o99/sPhAsaBncyiYjto3FAPYqRVNKoclWr6VE9sXzqwtRyPPiYNeAltFyPI5xG9UKDeSBpqGgj+5EbmV6rjSpVo2nkgKvELEgVFOiOKq+WG+LYJ4HEDAkxNI1I2inikmJGMt2KBYkQnqIxGSoaIJ8IO81vzeCZUlzohVy9QMJc/T2RIl+IxHdU0kdyIpa9ufifN4yl17ZTGkSxJAFeLPJiBmUI58VBl3KCJUsUQZhT9VeIJ0i1JFW9uirBXD55lfRazYumeduqdtpFG2VwAk5BHZjgHHTADeiCHsDgCbyAd/ChPWtv2kz7XERLWjFzDP5A+/oGxwmkiw==</latexit>

-

= =

- -

x

Page 9: Discrete Naïve Bayes

Bayesian Classifiers

9

The Class-Conditional Likelihood:

General Interpretation: Given a class , the probability that is observed

Concretely: Given assumptions about the nature of SPAM email, the probability that we observe this particular email

Example:

y = c<latexit sha1_base64="FwBWK4sj7Hla1kslHdAnS0C0cjY=">AAAB8XicbVBNS8NAEJ3Urxq/qh69LDaCp5L0Yj0IBS8eKxhbSEPZbDft0s0m7G6EUPozvHhQ8eq/8ea/cdvmoK0PBh7vzTAzL8o4U9p1v63KxubW9k51197bPzg8qh2fPKo0l4T6JOWp7EVYUc4E9TXTnPYySXEScdqNJrdzv/tEpWKpeNBFRsMEjwSLGcHaSIHj2AW6QcR2nEGt7jbcBdA68UpShxKdQe2rP0xJnlChCcdKBZ6b6XCKpWaE05ndzxXNMJngEQ0MFTihKpwuTp6hC6MMUZxKU0Kjhfp7YooTpYokMp0J1mO16s3F/7wg13ErnDKR5ZoKslwU5xzpFM3/R0MmKdG8MAQTycytiIyxxESblGwTgrf68jrxm43rhnffrLdbZRpVOINzuAQPrqANd9ABHwik8Ayv8GZp68V6tz6WrRWrnDmFP7A+fwAYnY6t</latexit><latexit sha1_base64="FwBWK4sj7Hla1kslHdAnS0C0cjY=">AAAB8XicbVBNS8NAEJ3Urxq/qh69LDaCp5L0Yj0IBS8eKxhbSEPZbDft0s0m7G6EUPozvHhQ8eq/8ea/cdvmoK0PBh7vzTAzL8o4U9p1v63KxubW9k51197bPzg8qh2fPKo0l4T6JOWp7EVYUc4E9TXTnPYySXEScdqNJrdzv/tEpWKpeNBFRsMEjwSLGcHaSIHj2AW6QcR2nEGt7jbcBdA68UpShxKdQe2rP0xJnlChCcdKBZ6b6XCKpWaE05ndzxXNMJngEQ0MFTihKpwuTp6hC6MMUZxKU0Kjhfp7YooTpYokMp0J1mO16s3F/7wg13ErnDKR5ZoKslwU5xzpFM3/R0MmKdG8MAQTycytiIyxxESblGwTgrf68jrxm43rhnffrLdbZRpVOINzuAQPrqANd9ABHwik8Ayv8GZp68V6tz6WrRWrnDmFP7A+fwAYnY6t</latexit><latexit sha1_base64="FwBWK4sj7Hla1kslHdAnS0C0cjY=">AAAB8XicbVBNS8NAEJ3Urxq/qh69LDaCp5L0Yj0IBS8eKxhbSEPZbDft0s0m7G6EUPozvHhQ8eq/8ea/cdvmoK0PBh7vzTAzL8o4U9p1v63KxubW9k51197bPzg8qh2fPKo0l4T6JOWp7EVYUc4E9TXTnPYySXEScdqNJrdzv/tEpWKpeNBFRsMEjwSLGcHaSIHj2AW6QcR2nEGt7jbcBdA68UpShxKdQe2rP0xJnlChCcdKBZ6b6XCKpWaE05ndzxXNMJngEQ0MFTihKpwuTp6hC6MMUZxKU0Kjhfp7YooTpYokMp0J1mO16s3F/7wg13ErnDKR5ZoKslwU5xzpFM3/R0MmKdG8MAQTycytiIyxxESblGwTgrf68jrxm43rhnffrLdbZRpVOINzuAQPrqANd9ABHwik8Ayv8GZp68V6tz6WrRWrnDmFP7A+fwAYnY6t</latexit><latexit sha1_base64="FwBWK4sj7Hla1kslHdAnS0C0cjY=">AAAB8XicbVBNS8NAEJ3Urxq/qh69LDaCp5L0Yj0IBS8eKxhbSEPZbDft0s0m7G6EUPozvHhQ8eq/8ea/cdvmoK0PBh7vzTAzL8o4U9p1v63KxubW9k51197bPzg8qh2fPKo0l4T6JOWp7EVYUc4E9TXTnPYySXEScdqNJrdzv/tEpWKpeNBFRsMEjwSLGcHaSIHj2AW6QcR2nEGt7jbcBdA68UpShxKdQe2rP0xJnlChCcdKBZ6b6XCKpWaE05ndzxXNMJngEQ0MFTihKpwuTp6hC6MMUZxKU0Kjhfp7YooTpYokMp0J1mO16s3F/7wg13ErnDKR5ZoKslwU5xzpFM3/R0MmKdG8MAQTycytiIyxxESblGwTgrf68jrxm43rhnffrLdbZRpVOINzuAQPrqANd9ABHwik8Ayv8GZp68V6tz6WrRWrnDmFP7A+fwAYnY6t</latexit>

p(y | x) = p(x | y)p(y)p(x)

<latexit sha1_base64="rx3ngS8jQjLkF8TBFaNOULN3afo=">AAACKnicbVDLSsNAFJ3UV42vqks3g63QbkrSjXUhFN24rGBtoQllMpm0QycPZiZiCPkfN/6KC11oceuHOE0DauuBgcM553LnHidiVEjDmGmltfWNza3ytr6zu7d/UDk8uhdhzDHp4ZCFfOAgQRgNSE9Sycgg4gT5DiN9Z3o99/sPhAsaBncyiYjto3FAPYqRVNKoclWr6VE9sXzqwtRyPPiYNeAltFyPI5xG9UKDeSBpqGgj+5EbmV6rjSpVo2nkgKvELEgVFOiOKq+WG+LYJ4HEDAkxNI1I2inikmJGMt2KBYkQnqIxGSoaIJ8IO81vzeCZUlzohVy9QMJc/T2RIl+IxHdU0kdyIpa9ufifN4yl17ZTGkSxJAFeLPJiBmUI58VBl3KCJUsUQZhT9VeIJ0i1JFW9uirBXD55lfRazYumeduqdtpFG2VwAk5BHZjgHHTADeiCHsDgCbyAd/ChPWtv2kz7XERLWjFzDP5A+/oGxwmkiw==</latexit><latexit sha1_base64="rx3ngS8jQjLkF8TBFaNOULN3afo=">AAACKnicbVDLSsNAFJ3UV42vqks3g63QbkrSjXUhFN24rGBtoQllMpm0QycPZiZiCPkfN/6KC11oceuHOE0DauuBgcM553LnHidiVEjDmGmltfWNza3ytr6zu7d/UDk8uhdhzDHp4ZCFfOAgQRgNSE9Sycgg4gT5DiN9Z3o99/sPhAsaBncyiYjto3FAPYqRVNKoclWr6VE9sXzqwtRyPPiYNeAltFyPI5xG9UKDeSBpqGgj+5EbmV6rjSpVo2nkgKvELEgVFOiOKq+WG+LYJ4HEDAkxNI1I2inikmJGMt2KBYkQnqIxGSoaIJ8IO81vzeCZUlzohVy9QMJc/T2RIl+IxHdU0kdyIpa9ufifN4yl17ZTGkSxJAFeLPJiBmUI58VBl3KCJUsUQZhT9VeIJ0i1JFW9uirBXD55lfRazYumeduqdtpFG2VwAk5BHZjgHHTADeiCHsDgCbyAd/ChPWtv2kz7XERLWjFzDP5A+/oGxwmkiw==</latexit><latexit sha1_base64="rx3ngS8jQjLkF8TBFaNOULN3afo=">AAACKnicbVDLSsNAFJ3UV42vqks3g63QbkrSjXUhFN24rGBtoQllMpm0QycPZiZiCPkfN/6KC11oceuHOE0DauuBgcM553LnHidiVEjDmGmltfWNza3ytr6zu7d/UDk8uhdhzDHp4ZCFfOAgQRgNSE9Sycgg4gT5DiN9Z3o99/sPhAsaBncyiYjto3FAPYqRVNKoclWr6VE9sXzqwtRyPPiYNeAltFyPI5xG9UKDeSBpqGgj+5EbmV6rjSpVo2nkgKvELEgVFOiOKq+WG+LYJ4HEDAkxNI1I2inikmJGMt2KBYkQnqIxGSoaIJ8IO81vzeCZUlzohVy9QMJc/T2RIl+IxHdU0kdyIpa9ufifN4yl17ZTGkSxJAFeLPJiBmUI58VBl3KCJUsUQZhT9VeIJ0i1JFW9uirBXD55lfRazYumeduqdtpFG2VwAk5BHZjgHHTADeiCHsDgCbyAd/ChPWtv2kz7XERLWjFzDP5A+/oGxwmkiw==</latexit><latexit sha1_base64="rx3ngS8jQjLkF8TBFaNOULN3afo=">AAACKnicbVDLSsNAFJ3UV42vqks3g63QbkrSjXUhFN24rGBtoQllMpm0QycPZiZiCPkfN/6KC11oceuHOE0DauuBgcM553LnHidiVEjDmGmltfWNza3ytr6zu7d/UDk8uhdhzDHp4ZCFfOAgQRgNSE9Sycgg4gT5DiN9Z3o99/sPhAsaBncyiYjto3FAPYqRVNKoclWr6VE9sXzqwtRyPPiYNeAltFyPI5xG9UKDeSBpqGgj+5EbmV6rjSpVo2nkgKvELEgVFOiOKq+WG+LYJ4HEDAkxNI1I2inikmJGMt2KBYkQnqIxGSoaIJ8IO81vzeCZUlzohVy9QMJc/T2RIl+IxHdU0kdyIpa9ufifN4yl17ZTGkSxJAFeLPJiBmUI58VBl3KCJUsUQZhT9VeIJ0i1JFW9uirBXD55lfRazYumeduqdtpFG2VwAk5BHZjgHHTADeiCHsDgCbyAd/ChPWtv2kz7XERLWjFzDP5A+/oGxwmkiw==</latexit>

posterior =class-likelihood⇥ prior

evidence<latexit sha1_base64="FPyo09EVttLZURXNoBbAoQ/K6CY=">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</latexit><latexit sha1_base64="FPyo09EVttLZURXNoBbAoQ/K6CY=">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</latexit><latexit sha1_base64="FPyo09EVttLZURXNoBbAoQ/K6CY=">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</latexit><latexit sha1_base64="FPyo09EVttLZURXNoBbAoQ/K6CY=">AAACU3icbVBNS8NAEN3Erxq/qh69LLaCF0viRT0IghePFawKTSmbzcQu3WTD7kQsIT9SD4K/xIsHt7GKXwMLj/fm7cy8KJfCoO+/OO7c/MLiUmPZW1ldW99obm5dG1VoDj2upNK3ETMgRQY9FCjhNtfA0kjCTTQ+n+o396CNUNkVTnIYpOwuE4ngDC01bI7bbS9EeECdlrkyCFooXdFTGsaJZrz81MowSiiXzJgDKcZ23EipuKpoiCIFQ7++qO3Vlw3uRQwZh6ry2u1hs+V3/LroXxDMQIvMqjtsPoWx4kUKGdaj+4Gf46BkGgWXUHlhYSBnfMzuoG9hxuwqg7IOpaJ7lolporR9GdKa/e4oWWrMJI1sZ8pwZH5rU/I/rV9gcjwoRZYXaE/7GJQUkqKi04RpLDRwlBMLGNfC7kr5iNkwbbjGsyEEv0/+C3qHnZNOcHnYOjuepdEgO2SX7JOAHJEzckG6pEc4eSSvDnEc59l5c113/qPVdWaebfKj3LV3Fk21qg==</latexit>

x<latexit sha1_base64="ezxhI2prtFY0vXyy/yEAlvhFBlU=">AAAB83icbVBNT8JAEJ3iF9Yv1KOXjdTEE2m5+HEi8eIREysk0JDtsoUN223d3RJJw+/w4kGNV/+MN/+NC/Sg4EsmeXlvJjPzwpQzpV332yqtrW9sbpW37Z3dvf2DyuHRg0oySahPEp7IdogV5UxQXzPNaTuVFMchp61wdDPzW2MqFUvEvZ6kNIjxQLCIEayNFDiOnXfDCD1NbcfpVapuzZ0DrRKvIFUo0OxVvrr9hGQxFZpwrFTHc1Md5FhqRjid2t1M0RSTER7QjqECx1QF+fzoKTozSh9FiTQlNJqrvydyHCs1iUPTGWM9VMveTPzP62Q6ugxyJtJMU0EWi6KMI52gWQKozyQlmk8MwUQycysiQywx0SYn24TgLb+8Svx67arm3dWrjesijTKcwCmcgwcX0IBbaIIPBB7hGV7hzRpbL9a79bFoLVnFzDH8gfX5A3nNkB4=</latexit><latexit sha1_base64="ezxhI2prtFY0vXyy/yEAlvhFBlU=">AAAB83icbVBNT8JAEJ3iF9Yv1KOXjdTEE2m5+HEi8eIREysk0JDtsoUN223d3RJJw+/w4kGNV/+MN/+NC/Sg4EsmeXlvJjPzwpQzpV332yqtrW9sbpW37Z3dvf2DyuHRg0oySahPEp7IdogV5UxQXzPNaTuVFMchp61wdDPzW2MqFUvEvZ6kNIjxQLCIEayNFDiOnXfDCD1NbcfpVapuzZ0DrRKvIFUo0OxVvrr9hGQxFZpwrFTHc1Md5FhqRjid2t1M0RSTER7QjqECx1QF+fzoKTozSh9FiTQlNJqrvydyHCs1iUPTGWM9VMveTPzP62Q6ugxyJtJMU0EWi6KMI52gWQKozyQlmk8MwUQycysiQywx0SYn24TgLb+8Svx67arm3dWrjesijTKcwCmcgwcX0IBbaIIPBB7hGV7hzRpbL9a79bFoLVnFzDH8gfX5A3nNkB4=</latexit><latexit sha1_base64="ezxhI2prtFY0vXyy/yEAlvhFBlU=">AAAB83icbVBNT8JAEJ3iF9Yv1KOXjdTEE2m5+HEi8eIREysk0JDtsoUN223d3RJJw+/w4kGNV/+MN/+NC/Sg4EsmeXlvJjPzwpQzpV332yqtrW9sbpW37Z3dvf2DyuHRg0oySahPEp7IdogV5UxQXzPNaTuVFMchp61wdDPzW2MqFUvEvZ6kNIjxQLCIEayNFDiOnXfDCD1NbcfpVapuzZ0DrRKvIFUo0OxVvrr9hGQxFZpwrFTHc1Md5FhqRjid2t1M0RSTER7QjqECx1QF+fzoKTozSh9FiTQlNJqrvydyHCs1iUPTGWM9VMveTPzP62Q6ugxyJtJMU0EWi6KMI52gWQKozyQlmk8MwUQycysiQywx0SYn24TgLb+8Svx67arm3dWrjesijTKcwCmcgwcX0IBbaIIPBB7hGV7hzRpbL9a79bFoLVnFzDH8gfX5A3nNkB4=</latexit><latexit sha1_base64="ezxhI2prtFY0vXyy/yEAlvhFBlU=">AAAB83icbVBNT8JAEJ3iF9Yv1KOXjdTEE2m5+HEi8eIREysk0JDtsoUN223d3RJJw+/w4kGNV/+MN/+NC/Sg4EsmeXlvJjPzwpQzpV332yqtrW9sbpW37Z3dvf2DyuHRg0oySahPEp7IdogV5UxQXzPNaTuVFMchp61wdDPzW2MqFUvEvZ6kNIjxQLCIEayNFDiOnXfDCD1NbcfpVapuzZ0DrRKvIFUo0OxVvrr9hGQxFZpwrFTHc1Md5FhqRjid2t1M0RSTER7QjqECx1QF+fzoKTozSh9FiTQlNJqrvydyHCs1iUPTGWM9VMveTPzP62Q6ugxyJtJMU0EWi6KMI52gWQKozyQlmk8MwUQycysiQywx0SYn24TgLb+8Svx67arm3dWrjesijTKcwCmcgwcX0IBbaIIPBB7hGV7hzRpbL9a79bFoLVnFzDH8gfX5A3nNkB4=</latexit>

p(x = [buy, viagra] | SPAM)<latexit sha1_base64="5hUhGr6RblYk5Vcdhk/ntEIzge4=">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</latexit><latexit sha1_base64="5hUhGr6RblYk5Vcdhk/ntEIzge4=">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</latexit><latexit sha1_base64="5hUhGr6RblYk5Vcdhk/ntEIzge4=">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</latexit><latexit sha1_base64="5hUhGr6RblYk5Vcdhk/ntEIzge4=">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</latexit>

Page 10: Discrete Naïve Bayes

Bayesian Classifiers

10

The Class Prior:

General Interpretation: The probability that any object belongs to class

Concretely: The probability that any new email is SPAM/HAM

p(y | x) = p(x | y)p(y)p(x)

<latexit sha1_base64="rx3ngS8jQjLkF8TBFaNOULN3afo=">AAACKnicbVDLSsNAFJ3UV42vqks3g63QbkrSjXUhFN24rGBtoQllMpm0QycPZiZiCPkfN/6KC11oceuHOE0DauuBgcM553LnHidiVEjDmGmltfWNza3ytr6zu7d/UDk8uhdhzDHp4ZCFfOAgQRgNSE9Sycgg4gT5DiN9Z3o99/sPhAsaBncyiYjto3FAPYqRVNKoclWr6VE9sXzqwtRyPPiYNeAltFyPI5xG9UKDeSBpqGgj+5EbmV6rjSpVo2nkgKvELEgVFOiOKq+WG+LYJ4HEDAkxNI1I2inikmJGMt2KBYkQnqIxGSoaIJ8IO81vzeCZUlzohVy9QMJc/T2RIl+IxHdU0kdyIpa9ufifN4yl17ZTGkSxJAFeLPJiBmUI58VBl3KCJUsUQZhT9VeIJ0i1JFW9uirBXD55lfRazYumeduqdtpFG2VwAk5BHZjgHHTADeiCHsDgCbyAd/ChPWtv2kz7XERLWjFzDP5A+/oGxwmkiw==</latexit><latexit sha1_base64="rx3ngS8jQjLkF8TBFaNOULN3afo=">AAACKnicbVDLSsNAFJ3UV42vqks3g63QbkrSjXUhFN24rGBtoQllMpm0QycPZiZiCPkfN/6KC11oceuHOE0DauuBgcM553LnHidiVEjDmGmltfWNza3ytr6zu7d/UDk8uhdhzDHp4ZCFfOAgQRgNSE9Sycgg4gT5DiN9Z3o99/sPhAsaBncyiYjto3FAPYqRVNKoclWr6VE9sXzqwtRyPPiYNeAltFyPI5xG9UKDeSBpqGgj+5EbmV6rjSpVo2nkgKvELEgVFOiOKq+WG+LYJ4HEDAkxNI1I2inikmJGMt2KBYkQnqIxGSoaIJ8IO81vzeCZUlzohVy9QMJc/T2RIl+IxHdU0kdyIpa9ufifN4yl17ZTGkSxJAFeLPJiBmUI58VBl3KCJUsUQZhT9VeIJ0i1JFW9uirBXD55lfRazYumeduqdtpFG2VwAk5BHZjgHHTADeiCHsDgCbyAd/ChPWtv2kz7XERLWjFzDP5A+/oGxwmkiw==</latexit><latexit sha1_base64="rx3ngS8jQjLkF8TBFaNOULN3afo=">AAACKnicbVDLSsNAFJ3UV42vqks3g63QbkrSjXUhFN24rGBtoQllMpm0QycPZiZiCPkfN/6KC11oceuHOE0DauuBgcM553LnHidiVEjDmGmltfWNza3ytr6zu7d/UDk8uhdhzDHp4ZCFfOAgQRgNSE9Sycgg4gT5DiN9Z3o99/sPhAsaBncyiYjto3FAPYqRVNKoclWr6VE9sXzqwtRyPPiYNeAltFyPI5xG9UKDeSBpqGgj+5EbmV6rjSpVo2nkgKvELEgVFOiOKq+WG+LYJ4HEDAkxNI1I2inikmJGMt2KBYkQnqIxGSoaIJ8IO81vzeCZUlzohVy9QMJc/T2RIl+IxHdU0kdyIpa9ufifN4yl17ZTGkSxJAFeLPJiBmUI58VBl3KCJUsUQZhT9VeIJ0i1JFW9uirBXD55lfRazYumeduqdtpFG2VwAk5BHZjgHHTADeiCHsDgCbyAd/ChPWtv2kz7XERLWjFzDP5A+/oGxwmkiw==</latexit><latexit sha1_base64="rx3ngS8jQjLkF8TBFaNOULN3afo=">AAACKnicbVDLSsNAFJ3UV42vqks3g63QbkrSjXUhFN24rGBtoQllMpm0QycPZiZiCPkfN/6KC11oceuHOE0DauuBgcM553LnHidiVEjDmGmltfWNza3ytr6zu7d/UDk8uhdhzDHp4ZCFfOAgQRgNSE9Sycgg4gT5DiN9Z3o99/sPhAsaBncyiYjto3FAPYqRVNKoclWr6VE9sXzqwtRyPPiYNeAltFyPI5xG9UKDeSBpqGgj+5EbmV6rjSpVo2nkgKvELEgVFOiOKq+WG+LYJ4HEDAkxNI1I2inikmJGMt2KBYkQnqIxGSoaIJ8IO81vzeCZUlzohVy9QMJc/T2RIl+IxHdU0kdyIpa9ufifN4yl17ZTGkSxJAFeLPJiBmUI58VBl3KCJUsUQZhT9VeIJ0i1JFW9uirBXD55lfRazYumeduqdtpFG2VwAk5BHZjgHHTADeiCHsDgCbyAd/ChPWtv2kz7XERLWjFzDP5A+/oGxwmkiw==</latexit>

posterior =class-likelihood⇥ prior

evidence<latexit sha1_base64="eV4psQv3vEsvhGcmsw7AF/uvh8c=">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</latexit><latexit sha1_base64="eV4psQv3vEsvhGcmsw7AF/uvh8c=">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</latexit><latexit sha1_base64="eV4psQv3vEsvhGcmsw7AF/uvh8c=">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</latexit><latexit sha1_base64="eV4psQv3vEsvhGcmsw7AF/uvh8c=">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</latexit>

y = c<latexit sha1_base64="FwBWK4sj7Hla1kslHdAnS0C0cjY=">AAAB8XicbVBNS8NAEJ3Urxq/qh69LDaCp5L0Yj0IBS8eKxhbSEPZbDft0s0m7G6EUPozvHhQ8eq/8ea/cdvmoK0PBh7vzTAzL8o4U9p1v63KxubW9k51197bPzg8qh2fPKo0l4T6JOWp7EVYUc4E9TXTnPYySXEScdqNJrdzv/tEpWKpeNBFRsMEjwSLGcHaSIHj2AW6QcR2nEGt7jbcBdA68UpShxKdQe2rP0xJnlChCcdKBZ6b6XCKpWaE05ndzxXNMJngEQ0MFTihKpwuTp6hC6MMUZxKU0Kjhfp7YooTpYokMp0J1mO16s3F/7wg13ErnDKR5ZoKslwU5xzpFM3/R0MmKdG8MAQTycytiIyxxESblGwTgrf68jrxm43rhnffrLdbZRpVOINzuAQPrqANd9ABHwik8Ayv8GZp68V6tz6WrRWrnDmFP7A+fwAYnY6t</latexit><latexit sha1_base64="FwBWK4sj7Hla1kslHdAnS0C0cjY=">AAAB8XicbVBNS8NAEJ3Urxq/qh69LDaCp5L0Yj0IBS8eKxhbSEPZbDft0s0m7G6EUPozvHhQ8eq/8ea/cdvmoK0PBh7vzTAzL8o4U9p1v63KxubW9k51197bPzg8qh2fPKo0l4T6JOWp7EVYUc4E9TXTnPYySXEScdqNJrdzv/tEpWKpeNBFRsMEjwSLGcHaSIHj2AW6QcR2nEGt7jbcBdA68UpShxKdQe2rP0xJnlChCcdKBZ6b6XCKpWaE05ndzxXNMJngEQ0MFTihKpwuTp6hC6MMUZxKU0Kjhfp7YooTpYokMp0J1mO16s3F/7wg13ErnDKR5ZoKslwU5xzpFM3/R0MmKdG8MAQTycytiIyxxESblGwTgrf68jrxm43rhnffrLdbZRpVOINzuAQPrqANd9ABHwik8Ayv8GZp68V6tz6WrRWrnDmFP7A+fwAYnY6t</latexit><latexit sha1_base64="FwBWK4sj7Hla1kslHdAnS0C0cjY=">AAAB8XicbVBNS8NAEJ3Urxq/qh69LDaCp5L0Yj0IBS8eKxhbSEPZbDft0s0m7G6EUPozvHhQ8eq/8ea/cdvmoK0PBh7vzTAzL8o4U9p1v63KxubW9k51197bPzg8qh2fPKo0l4T6JOWp7EVYUc4E9TXTnPYySXEScdqNJrdzv/tEpWKpeNBFRsMEjwSLGcHaSIHj2AW6QcR2nEGt7jbcBdA68UpShxKdQe2rP0xJnlChCcdKBZ6b6XCKpWaE05ndzxXNMJngEQ0MFTihKpwuTp6hC6MMUZxKU0Kjhfp7YooTpYokMp0J1mO16s3F/7wg13ErnDKR5ZoKslwU5xzpFM3/R0MmKdG8MAQTycytiIyxxESblGwTgrf68jrxm43rhnffrLdbZRpVOINzuAQPrqANd9ABHwik8Ayv8GZp68V6tz6WrRWrnDmFP7A+fwAYnY6t</latexit><latexit sha1_base64="FwBWK4sj7Hla1kslHdAnS0C0cjY=">AAAB8XicbVBNS8NAEJ3Urxq/qh69LDaCp5L0Yj0IBS8eKxhbSEPZbDft0s0m7G6EUPozvHhQ8eq/8ea/cdvmoK0PBh7vzTAzL8o4U9p1v63KxubW9k51197bPzg8qh2fPKo0l4T6JOWp7EVYUc4E9TXTnPYySXEScdqNJrdzv/tEpWKpeNBFRsMEjwSLGcHaSIHj2AW6QcR2nEGt7jbcBdA68UpShxKdQe2rP0xJnlChCcdKBZ6b6XCKpWaE05ndzxXNMJngEQ0MFTihKpwuTp6hC6MMUZxKU0Kjhfp7YooTpYokMp0J1mO16s3F/7wg13ErnDKR5ZoKslwU5xzpFM3/R0MmKdG8MAQTycytiIyxxESblGwTgrf68jrxm43rhnffrLdbZRpVOINzuAQPrqANd9ABHwik8Ayv8GZp68V6tz6WrRWrnDmFP7A+fwAYnY6t</latexit>

Page 11: Discrete Naïve Bayes

Bayesian Classifiers

11

The Evidence:

General Interpretation: The probability we encounter data independent of class

Or in concrete terms: The probability that of observing a particular email in general

p(y | x) = p(x | y)p(y)p(x)

<latexit sha1_base64="rx3ngS8jQjLkF8TBFaNOULN3afo=">AAACKnicbVDLSsNAFJ3UV42vqks3g63QbkrSjXUhFN24rGBtoQllMpm0QycPZiZiCPkfN/6KC11oceuHOE0DauuBgcM553LnHidiVEjDmGmltfWNza3ytr6zu7d/UDk8uhdhzDHp4ZCFfOAgQRgNSE9Sycgg4gT5DiN9Z3o99/sPhAsaBncyiYjto3FAPYqRVNKoclWr6VE9sXzqwtRyPPiYNeAltFyPI5xG9UKDeSBpqGgj+5EbmV6rjSpVo2nkgKvELEgVFOiOKq+WG+LYJ4HEDAkxNI1I2inikmJGMt2KBYkQnqIxGSoaIJ8IO81vzeCZUlzohVy9QMJc/T2RIl+IxHdU0kdyIpa9ufifN4yl17ZTGkSxJAFeLPJiBmUI58VBl3KCJUsUQZhT9VeIJ0i1JFW9uirBXD55lfRazYumeduqdtpFG2VwAk5BHZjgHHTADeiCHsDgCbyAd/ChPWtv2kz7XERLWjFzDP5A+/oGxwmkiw==</latexit><latexit sha1_base64="rx3ngS8jQjLkF8TBFaNOULN3afo=">AAACKnicbVDLSsNAFJ3UV42vqks3g63QbkrSjXUhFN24rGBtoQllMpm0QycPZiZiCPkfN/6KC11oceuHOE0DauuBgcM553LnHidiVEjDmGmltfWNza3ytr6zu7d/UDk8uhdhzDHp4ZCFfOAgQRgNSE9Sycgg4gT5DiN9Z3o99/sPhAsaBncyiYjto3FAPYqRVNKoclWr6VE9sXzqwtRyPPiYNeAltFyPI5xG9UKDeSBpqGgj+5EbmV6rjSpVo2nkgKvELEgVFOiOKq+WG+LYJ4HEDAkxNI1I2inikmJGMt2KBYkQnqIxGSoaIJ8IO81vzeCZUlzohVy9QMJc/T2RIl+IxHdU0kdyIpa9ufifN4yl17ZTGkSxJAFeLPJiBmUI58VBl3KCJUsUQZhT9VeIJ0i1JFW9uirBXD55lfRazYumeduqdtpFG2VwAk5BHZjgHHTADeiCHsDgCbyAd/ChPWtv2kz7XERLWjFzDP5A+/oGxwmkiw==</latexit><latexit sha1_base64="rx3ngS8jQjLkF8TBFaNOULN3afo=">AAACKnicbVDLSsNAFJ3UV42vqks3g63QbkrSjXUhFN24rGBtoQllMpm0QycPZiZiCPkfN/6KC11oceuHOE0DauuBgcM553LnHidiVEjDmGmltfWNza3ytr6zu7d/UDk8uhdhzDHp4ZCFfOAgQRgNSE9Sycgg4gT5DiN9Z3o99/sPhAsaBncyiYjto3FAPYqRVNKoclWr6VE9sXzqwtRyPPiYNeAltFyPI5xG9UKDeSBpqGgj+5EbmV6rjSpVo2nkgKvELEgVFOiOKq+WG+LYJ4HEDAkxNI1I2inikmJGMt2KBYkQnqIxGSoaIJ8IO81vzeCZUlzohVy9QMJc/T2RIl+IxHdU0kdyIpa9ufifN4yl17ZTGkSxJAFeLPJiBmUI58VBl3KCJUsUQZhT9VeIJ0i1JFW9uirBXD55lfRazYumeduqdtpFG2VwAk5BHZjgHHTADeiCHsDgCbyAd/ChPWtv2kz7XERLWjFzDP5A+/oGxwmkiw==</latexit><latexit sha1_base64="rx3ngS8jQjLkF8TBFaNOULN3afo=">AAACKnicbVDLSsNAFJ3UV42vqks3g63QbkrSjXUhFN24rGBtoQllMpm0QycPZiZiCPkfN/6KC11oceuHOE0DauuBgcM553LnHidiVEjDmGmltfWNza3ytr6zu7d/UDk8uhdhzDHp4ZCFfOAgQRgNSE9Sycgg4gT5DiN9Z3o99/sPhAsaBncyiYjto3FAPYqRVNKoclWr6VE9sXzqwtRyPPiYNeAltFyPI5xG9UKDeSBpqGgj+5EbmV6rjSpVo2nkgKvELEgVFOiOKq+WG+LYJ4HEDAkxNI1I2inikmJGMt2KBYkQnqIxGSoaIJ8IO81vzeCZUlzohVy9QMJc/T2RIl+IxHdU0kdyIpa9ufifN4yl17ZTGkSxJAFeLPJiBmUI58VBl3KCJUsUQZhT9VeIJ0i1JFW9uirBXD55lfRazYumeduqdtpFG2VwAk5BHZjgHHTADeiCHsDgCbyAd/ChPWtv2kz7XERLWjFzDP5A+/oGxwmkiw==</latexit>

posterior =class-likelihood⇥ prior

evidence<latexit sha1_base64="FHrbvCdHaqwPSRG6U7ENLnJ/gk8=">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</latexit><latexit sha1_base64="FHrbvCdHaqwPSRG6U7ENLnJ/gk8=">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</latexit><latexit sha1_base64="FHrbvCdHaqwPSRG6U7ENLnJ/gk8=">AAACU3icbVBNS8NAEN3Erxq/qh69LLaCF0viRT0IghePFawKTSmbzcQu3WTD7kQsIT9SD4K/xIsHt7GKXwMDj/fmMTMvyqUw6Psvjjs3v7C41Fj2VlbX1jeam1vXRhWaQ48rqfRtxAxIkUEPBUq4zTWwNJJwE43Pp/rNPWgjVHaFkxwGKbvLRCI4Q0sNm+N22wsRHlCnZa4MghZKV/SUhnGiGS8/NS6ZMQdSjO2mkVJxRUMUKRj6Za6N1ZehDKOEwr2IIeNQVZXXbg+bLb/j10X/gmAGWmRW3WHzKYwVL1LIsN7fD/wcByXTKLiEygsLAznjY3YHfQszZg8alHUoFd2zTEwTpW1nSGv2u6NkqTGTNLKTKcOR+a1Nyf+0foHJ8aAUWV6g/e1jUVJIiopOE6ax0MBRTixgXAt7K+UjZsO04RrPhhD8fvkv6B12TjrB5WHr7HiWRoPskF2yTwJyRM7IBemSHuHkkbw6xHGcZ+fNdd35j1HXmXm2yY9y194BIuu1qg==</latexit><latexit sha1_base64="FHrbvCdHaqwPSRG6U7ENLnJ/gk8=">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</latexit>

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Page 12: Discrete Naïve Bayes

Bayesian SPAM Classification

12

The Plan: o Estimate and compare and from data via Bayes

o Predict label for based on which probability estimate is larger

Question: Does the denominator add any information to this decision?

p(y | x) = p(x | y)p(y)p(x)

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posterior =class-likelihood⇥ prior

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p(y = SPAM | x)<latexit sha1_base64="RoubPElvdEbV+CcllNSjRjZz2Mw=">AAACDHicbVDLSsNAFJ34rPEVdelmsBHqpiRu1IVQceNGqGi10JQymU7aoTNJmLmRltAfcOOvuHGh4tYPcOffOH0stHrgwuGce7n3njAVXIPnfVlz8wuLS8uFFXt1bX1j09navtVJpiir0UQkqh4SzQSPWQ04CFZPFSMyFOwu7J2P/Lt7pjRP4hsYpKwpSSfmEacEjNRyXNe109LgNADWByXz6+rZ5TCQvI3zIIxwf3hgu27LKXplbwz8l/hTUkRTVFvOZ9BOaCZZDFQQrRu+l0IzJwo4FWxoB5lmKaE90mENQ2MimW7m42+GeN8obRwlylQMeKz+nMiJ1HogQ9MpCXT1rDcS//MaGUTHzZzHaQYsppNFUSYwJHgUDW5zxSiIgSGEKm5uxbRLFKFgArRNCP7sy39J7bB8UvavDouV42kaBbSL9lAJ+egIVdAFqqIaougBPaEX9Go9Ws/Wm/U+aZ2zpjM76Besj2+e25mJ</latexit><latexit sha1_base64="RoubPElvdEbV+CcllNSjRjZz2Mw=">AAACDHicbVDLSsNAFJ34rPEVdelmsBHqpiRu1IVQceNGqGi10JQymU7aoTNJmLmRltAfcOOvuHGh4tYPcOffOH0stHrgwuGce7n3njAVXIPnfVlz8wuLS8uFFXt1bX1j09navtVJpiir0UQkqh4SzQSPWQ04CFZPFSMyFOwu7J2P/Lt7pjRP4hsYpKwpSSfmEacEjNRyXNe109LgNADWByXz6+rZ5TCQvI3zIIxwf3hgu27LKXplbwz8l/hTUkRTVFvOZ9BOaCZZDFQQrRu+l0IzJwo4FWxoB5lmKaE90mENQ2MimW7m42+GeN8obRwlylQMeKz+nMiJ1HogQ9MpCXT1rDcS//MaGUTHzZzHaQYsppNFUSYwJHgUDW5zxSiIgSGEKm5uxbRLFKFgArRNCP7sy39J7bB8UvavDouV42kaBbSL9lAJ+egIVdAFqqIaougBPaEX9Go9Ws/Wm/U+aZ2zpjM76Besj2+e25mJ</latexit><latexit sha1_base64="RoubPElvdEbV+CcllNSjRjZz2Mw=">AAACDHicbVDLSsNAFJ34rPEVdelmsBHqpiRu1IVQceNGqGi10JQymU7aoTNJmLmRltAfcOOvuHGh4tYPcOffOH0stHrgwuGce7n3njAVXIPnfVlz8wuLS8uFFXt1bX1j09navtVJpiir0UQkqh4SzQSPWQ04CFZPFSMyFOwu7J2P/Lt7pjRP4hsYpKwpSSfmEacEjNRyXNe109LgNADWByXz6+rZ5TCQvI3zIIxwf3hgu27LKXplbwz8l/hTUkRTVFvOZ9BOaCZZDFQQrRu+l0IzJwo4FWxoB5lmKaE90mENQ2MimW7m42+GeN8obRwlylQMeKz+nMiJ1HogQ9MpCXT1rDcS//MaGUTHzZzHaQYsppNFUSYwJHgUDW5zxSiIgSGEKm5uxbRLFKFgArRNCP7sy39J7bB8UvavDouV42kaBbSL9lAJ+egIVdAFqqIaougBPaEX9Go9Ws/Wm/U+aZ2zpjM76Besj2+e25mJ</latexit><latexit sha1_base64="RoubPElvdEbV+CcllNSjRjZz2Mw=">AAACDHicbVDLSsNAFJ34rPEVdelmsBHqpiRu1IVQceNGqGi10JQymU7aoTNJmLmRltAfcOOvuHGh4tYPcOffOH0stHrgwuGce7n3njAVXIPnfVlz8wuLS8uFFXt1bX1j09navtVJpiir0UQkqh4SzQSPWQ04CFZPFSMyFOwu7J2P/Lt7pjRP4hsYpKwpSSfmEacEjNRyXNe109LgNADWByXz6+rZ5TCQvI3zIIxwf3hgu27LKXplbwz8l/hTUkRTVFvOZ9BOaCZZDFQQrRu+l0IzJwo4FWxoB5lmKaE90mENQ2MimW7m42+GeN8obRwlylQMeKz+nMiJ1HogQ9MpCXT1rDcS//MaGUTHzZzHaQYsppNFUSYwJHgUDW5zxSiIgSGEKm5uxbRLFKFgArRNCP7sy39J7bB8UvavDouV42kaBbSL9lAJ+egIVdAFqqIaougBPaEX9Go9Ws/Wm/U+aZ2zpjM76Besj2+e25mJ</latexit>

p(y = HAM | x)<latexit sha1_base64="+Uk5L5SjMtDFL+/APaBNDtJF7x4=">AAACC3icbVC7TgJBFJ31iesLtbSZCCbYkF0asTDB2NCYYCJCwhIyO8zChJndzcxdA9nwATb+io2FGlt/wM6/cXgUCp5kkpNzzs2de/xYcA2O822trK6tb2xmtuztnd29/ezB4b2OEkVZnUYiUk2faCZ4yOrAQbBmrBiRvmANf3A98RsPTGkehXcwillbkl7IA04JGKmTzeXzdlwYXXrAhqBkWr26GXuSd3Hq+QEejs/sfN6knKIzBV4m7pzk0By1TvbL60Y0kSwEKojWLdeJoZ0SBZwKNra9RLOY0AHpsZahIZFMt9PpMWN8apQuDiJlXgh4qv6eSInUeiR9k5QE+nrRm4j/ea0EgnI75WGcAAvpbFGQCAwRnjSDu1wxCmJkCKGKm79i2ieKUDD92aYEd/HkZVIvFS+K7m0pVynP28igY3SCCshF56iCqqiG6oiiR/SMXtGb9WS9WO/Wxyy6Ys1njtAfWJ8/48aZJA==</latexit><latexit sha1_base64="+Uk5L5SjMtDFL+/APaBNDtJF7x4=">AAACC3icbVC7TgJBFJ31iesLtbSZCCbYkF0asTDB2NCYYCJCwhIyO8zChJndzcxdA9nwATb+io2FGlt/wM6/cXgUCp5kkpNzzs2de/xYcA2O822trK6tb2xmtuztnd29/ezB4b2OEkVZnUYiUk2faCZ4yOrAQbBmrBiRvmANf3A98RsPTGkehXcwillbkl7IA04JGKmTzeXzdlwYXXrAhqBkWr26GXuSd3Hq+QEejs/sfN6knKIzBV4m7pzk0By1TvbL60Y0kSwEKojWLdeJoZ0SBZwKNra9RLOY0AHpsZahIZFMt9PpMWN8apQuDiJlXgh4qv6eSInUeiR9k5QE+nrRm4j/ea0EgnI75WGcAAvpbFGQCAwRnjSDu1wxCmJkCKGKm79i2ieKUDD92aYEd/HkZVIvFS+K7m0pVynP28igY3SCCshF56iCqqiG6oiiR/SMXtGb9WS9WO/Wxyy6Ys1njtAfWJ8/48aZJA==</latexit><latexit sha1_base64="+Uk5L5SjMtDFL+/APaBNDtJF7x4=">AAACC3icbVC7TgJBFJ31iesLtbSZCCbYkF0asTDB2NCYYCJCwhIyO8zChJndzcxdA9nwATb+io2FGlt/wM6/cXgUCp5kkpNzzs2de/xYcA2O822trK6tb2xmtuztnd29/ezB4b2OEkVZnUYiUk2faCZ4yOrAQbBmrBiRvmANf3A98RsPTGkehXcwillbkl7IA04JGKmTzeXzdlwYXXrAhqBkWr26GXuSd3Hq+QEejs/sfN6knKIzBV4m7pzk0By1TvbL60Y0kSwEKojWLdeJoZ0SBZwKNra9RLOY0AHpsZahIZFMt9PpMWN8apQuDiJlXgh4qv6eSInUeiR9k5QE+nrRm4j/ea0EgnI75WGcAAvpbFGQCAwRnjSDu1wxCmJkCKGKm79i2ieKUDD92aYEd/HkZVIvFS+K7m0pVynP28igY3SCCshF56iCqqiG6oiiR/SMXtGb9WS9WO/Wxyy6Ys1njtAfWJ8/48aZJA==</latexit><latexit sha1_base64="+Uk5L5SjMtDFL+/APaBNDtJF7x4=">AAACC3icbVC7TgJBFJ31iesLtbSZCCbYkF0asTDB2NCYYCJCwhIyO8zChJndzcxdA9nwATb+io2FGlt/wM6/cXgUCp5kkpNzzs2de/xYcA2O822trK6tb2xmtuztnd29/ezB4b2OEkVZnUYiUk2faCZ4yOrAQbBmrBiRvmANf3A98RsPTGkehXcwillbkl7IA04JGKmTzeXzdlwYXXrAhqBkWr26GXuSd3Hq+QEejs/sfN6knKIzBV4m7pzk0By1TvbL60Y0kSwEKojWLdeJoZ0SBZwKNra9RLOY0AHpsZahIZFMt9PpMWN8apQuDiJlXgh4qv6eSInUeiR9k5QE+nrRm4j/ea0EgnI75WGcAAvpbFGQCAwRnjSDu1wxCmJkCKGKm79i2ieKUDD92aYEd/HkZVIvFS+K7m0pVynP28igY3SCCshF56iCqqiG6oiiR/SMXtGb9WS9WO/Wxyy6Ys1njtAfWJ8/48aZJA==</latexit>

x<latexit sha1_base64="ezxhI2prtFY0vXyy/yEAlvhFBlU=">AAAB83icbVBNT8JAEJ3iF9Yv1KOXjdTEE2m5+HEi8eIREysk0JDtsoUN223d3RJJw+/w4kGNV/+MN/+NC/Sg4EsmeXlvJjPzwpQzpV332yqtrW9sbpW37Z3dvf2DyuHRg0oySahPEp7IdogV5UxQXzPNaTuVFMchp61wdDPzW2MqFUvEvZ6kNIjxQLCIEayNFDiOnXfDCD1NbcfpVapuzZ0DrRKvIFUo0OxVvrr9hGQxFZpwrFTHc1Md5FhqRjid2t1M0RSTER7QjqECx1QF+fzoKTozSh9FiTQlNJqrvydyHCs1iUPTGWM9VMveTPzP62Q6ugxyJtJMU0EWi6KMI52gWQKozyQlmk8MwUQycysiQywx0SYn24TgLb+8Svx67arm3dWrjesijTKcwCmcgwcX0IBbaIIPBB7hGV7hzRpbL9a79bFoLVnFzDH8gfX5A3nNkB4=</latexit><latexit sha1_base64="ezxhI2prtFY0vXyy/yEAlvhFBlU=">AAAB83icbVBNT8JAEJ3iF9Yv1KOXjdTEE2m5+HEi8eIREysk0JDtsoUN223d3RJJw+/w4kGNV/+MN/+NC/Sg4EsmeXlvJjPzwpQzpV332yqtrW9sbpW37Z3dvf2DyuHRg0oySahPEp7IdogV5UxQXzPNaTuVFMchp61wdDPzW2MqFUvEvZ6kNIjxQLCIEayNFDiOnXfDCD1NbcfpVapuzZ0DrRKvIFUo0OxVvrr9hGQxFZpwrFTHc1Md5FhqRjid2t1M0RSTER7QjqECx1QF+fzoKTozSh9FiTQlNJqrvydyHCs1iUPTGWM9VMveTPzP62Q6ugxyJtJMU0EWi6KMI52gWQKozyQlmk8MwUQycysiQywx0SYn24TgLb+8Svx67arm3dWrjesijTKcwCmcgwcX0IBbaIIPBB7hGV7hzRpbL9a79bFoLVnFzDH8gfX5A3nNkB4=</latexit><latexit sha1_base64="ezxhI2prtFY0vXyy/yEAlvhFBlU=">AAAB83icbVBNT8JAEJ3iF9Yv1KOXjdTEE2m5+HEi8eIREysk0JDtsoUN223d3RJJw+/w4kGNV/+MN/+NC/Sg4EsmeXlvJjPzwpQzpV332yqtrW9sbpW37Z3dvf2DyuHRg0oySahPEp7IdogV5UxQXzPNaTuVFMchp61wdDPzW2MqFUvEvZ6kNIjxQLCIEayNFDiOnXfDCD1NbcfpVapuzZ0DrRKvIFUo0OxVvrr9hGQxFZpwrFTHc1Md5FhqRjid2t1M0RSTER7QjqECx1QF+fzoKTozSh9FiTQlNJqrvydyHCs1iUPTGWM9VMveTPzP62Q6ugxyJtJMU0EWi6KMI52gWQKozyQlmk8MwUQycysiQywx0SYn24TgLb+8Svx67arm3dWrjesijTKcwCmcgwcX0IBbaIIPBB7hGV7hzRpbL9a79bFoLVnFzDH8gfX5A3nNkB4=</latexit><latexit sha1_base64="ezxhI2prtFY0vXyy/yEAlvhFBlU=">AAAB83icbVBNT8JAEJ3iF9Yv1KOXjdTEE2m5+HEi8eIREysk0JDtsoUN223d3RJJw+/w4kGNV/+MN/+NC/Sg4EsmeXlvJjPzwpQzpV332yqtrW9sbpW37Z3dvf2DyuHRg0oySahPEp7IdogV5UxQXzPNaTuVFMchp61wdDPzW2MqFUvEvZ6kNIjxQLCIEayNFDiOnXfDCD1NbcfpVapuzZ0DrRKvIFUo0OxVvrr9hGQxFZpwrFTHc1Md5FhqRjid2t1M0RSTER7QjqECx1QF+fzoKTozSh9FiTQlNJqrvydyHCs1iUPTGWM9VMveTPzP62Q6ugxyJtJMU0EWi6KMI52gWQKozyQlmk8MwUQycysiQywx0SYn24TgLb+8Svx67arm3dWrjesijTKcwCmcgwcX0IBbaIIPBB7hGV7hzRpbL9a79bFoLVnFzDH8gfX5A3nNkB4=</latexit>

p(x | SPAM) · p(SPAM)

p(x)vs

p(x | HAM) · p(HAM)

p(x)<latexit sha1_base64="YIJLkefZt741SRJB1+sC9eEWtBs=">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</latexit><latexit sha1_base64="YIJLkefZt741SRJB1+sC9eEWtBs=">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</latexit><latexit sha1_base64="YIJLkefZt741SRJB1+sC9eEWtBs=">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</latexit><latexit sha1_base64="YIJLkefZt741SRJB1+sC9eEWtBs=">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</latexit>

tag -3

Page 13: Discrete Naïve Bayes

Bayesian SPAM Classification

13

The Plan: o Estimate and compare and from data via Bayes

o Predict label for based on which probability estimate is larger

Answer: Nope, so we compare numerators. Note, no longer probabilities. Think of as scores

p(y | x) = p(x | y)p(y)p(x)

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posterior =class-likelihood⇥ prior

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p(y = SPAM | x)<latexit sha1_base64="RoubPElvdEbV+CcllNSjRjZz2Mw=">AAACDHicbVDLSsNAFJ34rPEVdelmsBHqpiRu1IVQceNGqGi10JQymU7aoTNJmLmRltAfcOOvuHGh4tYPcOffOH0stHrgwuGce7n3njAVXIPnfVlz8wuLS8uFFXt1bX1j09navtVJpiir0UQkqh4SzQSPWQ04CFZPFSMyFOwu7J2P/Lt7pjRP4hsYpKwpSSfmEacEjNRyXNe109LgNADWByXz6+rZ5TCQvI3zIIxwf3hgu27LKXplbwz8l/hTUkRTVFvOZ9BOaCZZDFQQrRu+l0IzJwo4FWxoB5lmKaE90mENQ2MimW7m42+GeN8obRwlylQMeKz+nMiJ1HogQ9MpCXT1rDcS//MaGUTHzZzHaQYsppNFUSYwJHgUDW5zxSiIgSGEKm5uxbRLFKFgArRNCP7sy39J7bB8UvavDouV42kaBbSL9lAJ+egIVdAFqqIaougBPaEX9Go9Ws/Wm/U+aZ2zpjM76Besj2+e25mJ</latexit><latexit sha1_base64="RoubPElvdEbV+CcllNSjRjZz2Mw=">AAACDHicbVDLSsNAFJ34rPEVdelmsBHqpiRu1IVQceNGqGi10JQymU7aoTNJmLmRltAfcOOvuHGh4tYPcOffOH0stHrgwuGce7n3njAVXIPnfVlz8wuLS8uFFXt1bX1j09navtVJpiir0UQkqh4SzQSPWQ04CFZPFSMyFOwu7J2P/Lt7pjRP4hsYpKwpSSfmEacEjNRyXNe109LgNADWByXz6+rZ5TCQvI3zIIxwf3hgu27LKXplbwz8l/hTUkRTVFvOZ9BOaCZZDFQQrRu+l0IzJwo4FWxoB5lmKaE90mENQ2MimW7m42+GeN8obRwlylQMeKz+nMiJ1HogQ9MpCXT1rDcS//MaGUTHzZzHaQYsppNFUSYwJHgUDW5zxSiIgSGEKm5uxbRLFKFgArRNCP7sy39J7bB8UvavDouV42kaBbSL9lAJ+egIVdAFqqIaougBPaEX9Go9Ws/Wm/U+aZ2zpjM76Besj2+e25mJ</latexit><latexit sha1_base64="RoubPElvdEbV+CcllNSjRjZz2Mw=">AAACDHicbVDLSsNAFJ34rPEVdelmsBHqpiRu1IVQceNGqGi10JQymU7aoTNJmLmRltAfcOOvuHGh4tYPcOffOH0stHrgwuGce7n3njAVXIPnfVlz8wuLS8uFFXt1bX1j09navtVJpiir0UQkqh4SzQSPWQ04CFZPFSMyFOwu7J2P/Lt7pjRP4hsYpKwpSSfmEacEjNRyXNe109LgNADWByXz6+rZ5TCQvI3zIIxwf3hgu27LKXplbwz8l/hTUkRTVFvOZ9BOaCZZDFQQrRu+l0IzJwo4FWxoB5lmKaE90mENQ2MimW7m42+GeN8obRwlylQMeKz+nMiJ1HogQ9MpCXT1rDcS//MaGUTHzZzHaQYsppNFUSYwJHgUDW5zxSiIgSGEKm5uxbRLFKFgArRNCP7sy39J7bB8UvavDouV42kaBbSL9lAJ+egIVdAFqqIaougBPaEX9Go9Ws/Wm/U+aZ2zpjM76Besj2+e25mJ</latexit><latexit sha1_base64="RoubPElvdEbV+CcllNSjRjZz2Mw=">AAACDHicbVDLSsNAFJ34rPEVdelmsBHqpiRu1IVQceNGqGi10JQymU7aoTNJmLmRltAfcOOvuHGh4tYPcOffOH0stHrgwuGce7n3njAVXIPnfVlz8wuLS8uFFXt1bX1j09navtVJpiir0UQkqh4SzQSPWQ04CFZPFSMyFOwu7J2P/Lt7pjRP4hsYpKwpSSfmEacEjNRyXNe109LgNADWByXz6+rZ5TCQvI3zIIxwf3hgu27LKXplbwz8l/hTUkRTVFvOZ9BOaCZZDFQQrRu+l0IzJwo4FWxoB5lmKaE90mENQ2MimW7m42+GeN8obRwlylQMeKz+nMiJ1HogQ9MpCXT1rDcS//MaGUTHzZzHaQYsppNFUSYwJHgUDW5zxSiIgSGEKm5uxbRLFKFgArRNCP7sy39J7bB8UvavDouV42kaBbSL9lAJ+egIVdAFqqIaougBPaEX9Go9Ws/Wm/U+aZ2zpjM76Besj2+e25mJ</latexit>

p(y = HAM | x)<latexit sha1_base64="+Uk5L5SjMtDFL+/APaBNDtJF7x4=">AAACC3icbVC7TgJBFJ31iesLtbSZCCbYkF0asTDB2NCYYCJCwhIyO8zChJndzcxdA9nwATb+io2FGlt/wM6/cXgUCp5kkpNzzs2de/xYcA2O822trK6tb2xmtuztnd29/ezB4b2OEkVZnUYiUk2faCZ4yOrAQbBmrBiRvmANf3A98RsPTGkehXcwillbkl7IA04JGKmTzeXzdlwYXXrAhqBkWr26GXuSd3Hq+QEejs/sfN6knKIzBV4m7pzk0By1TvbL60Y0kSwEKojWLdeJoZ0SBZwKNra9RLOY0AHpsZahIZFMt9PpMWN8apQuDiJlXgh4qv6eSInUeiR9k5QE+nrRm4j/ea0EgnI75WGcAAvpbFGQCAwRnjSDu1wxCmJkCKGKm79i2ieKUDD92aYEd/HkZVIvFS+K7m0pVynP28igY3SCCshF56iCqqiG6oiiR/SMXtGb9WS9WO/Wxyy6Ys1njtAfWJ8/48aZJA==</latexit><latexit sha1_base64="+Uk5L5SjMtDFL+/APaBNDtJF7x4=">AAACC3icbVC7TgJBFJ31iesLtbSZCCbYkF0asTDB2NCYYCJCwhIyO8zChJndzcxdA9nwATb+io2FGlt/wM6/cXgUCp5kkpNzzs2de/xYcA2O822trK6tb2xmtuztnd29/ezB4b2OEkVZnUYiUk2faCZ4yOrAQbBmrBiRvmANf3A98RsPTGkehXcwillbkl7IA04JGKmTzeXzdlwYXXrAhqBkWr26GXuSd3Hq+QEejs/sfN6knKIzBV4m7pzk0By1TvbL60Y0kSwEKojWLdeJoZ0SBZwKNra9RLOY0AHpsZahIZFMt9PpMWN8apQuDiJlXgh4qv6eSInUeiR9k5QE+nrRm4j/ea0EgnI75WGcAAvpbFGQCAwRnjSDu1wxCmJkCKGKm79i2ieKUDD92aYEd/HkZVIvFS+K7m0pVynP28igY3SCCshF56iCqqiG6oiiR/SMXtGb9WS9WO/Wxyy6Ys1njtAfWJ8/48aZJA==</latexit><latexit sha1_base64="+Uk5L5SjMtDFL+/APaBNDtJF7x4=">AAACC3icbVC7TgJBFJ31iesLtbSZCCbYkF0asTDB2NCYYCJCwhIyO8zChJndzcxdA9nwATb+io2FGlt/wM6/cXgUCp5kkpNzzs2de/xYcA2O822trK6tb2xmtuztnd29/ezB4b2OEkVZnUYiUk2faCZ4yOrAQbBmrBiRvmANf3A98RsPTGkehXcwillbkl7IA04JGKmTzeXzdlwYXXrAhqBkWr26GXuSd3Hq+QEejs/sfN6knKIzBV4m7pzk0By1TvbL60Y0kSwEKojWLdeJoZ0SBZwKNra9RLOY0AHpsZahIZFMt9PpMWN8apQuDiJlXgh4qv6eSInUeiR9k5QE+nrRm4j/ea0EgnI75WGcAAvpbFGQCAwRnjSDu1wxCmJkCKGKm79i2ieKUDD92aYEd/HkZVIvFS+K7m0pVynP28igY3SCCshF56iCqqiG6oiiR/SMXtGb9WS9WO/Wxyy6Ys1njtAfWJ8/48aZJA==</latexit><latexit sha1_base64="+Uk5L5SjMtDFL+/APaBNDtJF7x4=">AAACC3icbVC7TgJBFJ31iesLtbSZCCbYkF0asTDB2NCYYCJCwhIyO8zChJndzcxdA9nwATb+io2FGlt/wM6/cXgUCp5kkpNzzs2de/xYcA2O822trK6tb2xmtuztnd29/ezB4b2OEkVZnUYiUk2faCZ4yOrAQbBmrBiRvmANf3A98RsPTGkehXcwillbkl7IA04JGKmTzeXzdlwYXXrAhqBkWr26GXuSd3Hq+QEejs/sfN6knKIzBV4m7pzk0By1TvbL60Y0kSwEKojWLdeJoZ0SBZwKNra9RLOY0AHpsZahIZFMt9PpMWN8apQuDiJlXgh4qv6eSInUeiR9k5QE+nrRm4j/ea0EgnI75WGcAAvpbFGQCAwRnjSDu1wxCmJkCKGKm79i2ieKUDD92aYEd/HkZVIvFS+K7m0pVynP28igY3SCCshF56iCqqiG6oiiR/SMXtGb9WS9WO/Wxyy6Ys1njtAfWJ8/48aZJA==</latexit>

x<latexit sha1_base64="ezxhI2prtFY0vXyy/yEAlvhFBlU=">AAAB83icbVBNT8JAEJ3iF9Yv1KOXjdTEE2m5+HEi8eIREysk0JDtsoUN223d3RJJw+/w4kGNV/+MN/+NC/Sg4EsmeXlvJjPzwpQzpV332yqtrW9sbpW37Z3dvf2DyuHRg0oySahPEp7IdogV5UxQXzPNaTuVFMchp61wdDPzW2MqFUvEvZ6kNIjxQLCIEayNFDiOnXfDCD1NbcfpVapuzZ0DrRKvIFUo0OxVvrr9hGQxFZpwrFTHc1Md5FhqRjid2t1M0RSTER7QjqECx1QF+fzoKTozSh9FiTQlNJqrvydyHCs1iUPTGWM9VMveTPzP62Q6ugxyJtJMU0EWi6KMI52gWQKozyQlmk8MwUQycysiQywx0SYn24TgLb+8Svx67arm3dWrjesijTKcwCmcgwcX0IBbaIIPBB7hGV7hzRpbL9a79bFoLVnFzDH8gfX5A3nNkB4=</latexit><latexit sha1_base64="ezxhI2prtFY0vXyy/yEAlvhFBlU=">AAAB83icbVBNT8JAEJ3iF9Yv1KOXjdTEE2m5+HEi8eIREysk0JDtsoUN223d3RJJw+/w4kGNV/+MN/+NC/Sg4EsmeXlvJjPzwpQzpV332yqtrW9sbpW37Z3dvf2DyuHRg0oySahPEp7IdogV5UxQXzPNaTuVFMchp61wdDPzW2MqFUvEvZ6kNIjxQLCIEayNFDiOnXfDCD1NbcfpVapuzZ0DrRKvIFUo0OxVvrr9hGQxFZpwrFTHc1Md5FhqRjid2t1M0RSTER7QjqECx1QF+fzoKTozSh9FiTQlNJqrvydyHCs1iUPTGWM9VMveTPzP62Q6ugxyJtJMU0EWi6KMI52gWQKozyQlmk8MwUQycysiQywx0SYn24TgLb+8Svx67arm3dWrjesijTKcwCmcgwcX0IBbaIIPBB7hGV7hzRpbL9a79bFoLVnFzDH8gfX5A3nNkB4=</latexit><latexit sha1_base64="ezxhI2prtFY0vXyy/yEAlvhFBlU=">AAAB83icbVBNT8JAEJ3iF9Yv1KOXjdTEE2m5+HEi8eIREysk0JDtsoUN223d3RJJw+/w4kGNV/+MN/+NC/Sg4EsmeXlvJjPzwpQzpV332yqtrW9sbpW37Z3dvf2DyuHRg0oySahPEp7IdogV5UxQXzPNaTuVFMchp61wdDPzW2MqFUvEvZ6kNIjxQLCIEayNFDiOnXfDCD1NbcfpVapuzZ0DrRKvIFUo0OxVvrr9hGQxFZpwrFTHc1Md5FhqRjid2t1M0RSTER7QjqECx1QF+fzoKTozSh9FiTQlNJqrvydyHCs1iUPTGWM9VMveTPzP62Q6ugxyJtJMU0EWi6KMI52gWQKozyQlmk8MwUQycysiQywx0SYn24TgLb+8Svx67arm3dWrjesijTKcwCmcgwcX0IBbaIIPBB7hGV7hzRpbL9a79bFoLVnFzDH8gfX5A3nNkB4=</latexit><latexit sha1_base64="ezxhI2prtFY0vXyy/yEAlvhFBlU=">AAAB83icbVBNT8JAEJ3iF9Yv1KOXjdTEE2m5+HEi8eIREysk0JDtsoUN223d3RJJw+/w4kGNV/+MN/+NC/Sg4EsmeXlvJjPzwpQzpV332yqtrW9sbpW37Z3dvf2DyuHRg0oySahPEp7IdogV5UxQXzPNaTuVFMchp61wdDPzW2MqFUvEvZ6kNIjxQLCIEayNFDiOnXfDCD1NbcfpVapuzZ0DrRKvIFUo0OxVvrr9hGQxFZpwrFTHc1Md5FhqRjid2t1M0RSTER7QjqECx1QF+fzoKTozSh9FiTQlNJqrvydyHCs1iUPTGWM9VMveTPzP62Q6ugxyJtJMU0EWi6KMI52gWQKozyQlmk8MwUQycysiQywx0SYn24TgLb+8Svx67arm3dWrjesijTKcwCmcgwcX0IBbaIIPBB7hGV7hzRpbL9a79bFoLVnFzDH8gfX5A3nNkB4=</latexit>

p(x | SPAM) · p(SPAM) vs p(x | HAM) · p(HAM)<latexit sha1_base64="tuNxsFOIVlOyWp7e34N26MB/ljo=">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</latexit><latexit sha1_base64="tuNxsFOIVlOyWp7e34N26MB/ljo=">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</latexit><latexit sha1_base64="tuNxsFOIVlOyWp7e34N26MB/ljo=">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</latexit><latexit sha1_base64="tuNxsFOIVlOyWp7e34N26MB/ljo=">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</latexit>

Page 14: Discrete Naïve Bayes

The Naïve Bayes Assumption

14

OK, so we need a way to estimate these SPAM and HAM scores from the data for email

Consider again the following training data.

How can we estimate ?

p(SPAM | x) / p(x | SPAM)·p(SPAM) vs p(x | HAM)·p(HAM) / p(HAM | x)<latexit sha1_base64="Cwo4bp3EalNu74+sVTIdJuDbjxs=">AAACuHicdZFNaxsxEIa127RNtx9x22MvInbBvZjdHPJBLim95FJwSNwEvMZotbOOiLRSpdkQs+x/LL3131S2lyZ20gHByzPvjEajzEjhMI7/BOGzrecvXm6/il6/eftup/P+ww+nK8thxLXU9ipjDqQoYYQCJVwZC0xlEi6zm2+L/OUtWCd0eYFzAxPFZqUoBGfo0bTzq9eLTD9FuEOr6vPh1+8NTZXIaZ1mBb1rvtDUWG1QU9Nv0Sq/VuJdPNdINzp5/LNi995b16xI9J9mp0/2aum/OR7y9WGjXm/a6caDeBn0sUha0SVtDKed32mueaWgRC6Zc+MkNjipmUXBJTRRWjkwjN+wGYy9LJkCN6mXm2/oZ09yWmjrT4l0SR9W1Ew5N1eZdyqG124zt4BP5cYVFoeTWpSmQij56qKiktQvYPGNNBcWOMq5F4xb4Wel/JpZxtF/duSXkGw++bEY7Q2OBsnZXvfksN3GNvlEdkmfJOSAnJBTMiQjwoP9IA0gKMLjkIWzUKysYdDWfCRrEdq/f/TVOA==</latexit><latexit sha1_base64="Cwo4bp3EalNu74+sVTIdJuDbjxs=">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</latexit><latexit sha1_base64="Cwo4bp3EalNu74+sVTIdJuDbjxs=">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</latexit><latexit sha1_base64="Cwo4bp3EalNu74+sVTIdJuDbjxs=">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</latexit>

x<latexit sha1_base64="ezxhI2prtFY0vXyy/yEAlvhFBlU=">AAAB83icbVBNT8JAEJ3iF9Yv1KOXjdTEE2m5+HEi8eIREysk0JDtsoUN223d3RJJw+/w4kGNV/+MN/+NC/Sg4EsmeXlvJjPzwpQzpV332yqtrW9sbpW37Z3dvf2DyuHRg0oySahPEp7IdogV5UxQXzPNaTuVFMchp61wdDPzW2MqFUvEvZ6kNIjxQLCIEayNFDiOnXfDCD1NbcfpVapuzZ0DrRKvIFUo0OxVvrr9hGQxFZpwrFTHc1Md5FhqRjid2t1M0RSTER7QjqECx1QF+fzoKTozSh9FiTQlNJqrvydyHCs1iUPTGWM9VMveTPzP62Q6ugxyJtJMU0EWi6KMI52gWQKozyQlmk8MwUQycysiQywx0SYn24TgLb+8Svx67arm3dWrjesijTKcwCmcgwcX0IBbaIIPBB7hGV7hzRpbL9a79bFoLVnFzDH8gfX5A3nNkB4=</latexit><latexit sha1_base64="ezxhI2prtFY0vXyy/yEAlvhFBlU=">AAAB83icbVBNT8JAEJ3iF9Yv1KOXjdTEE2m5+HEi8eIREysk0JDtsoUN223d3RJJw+/w4kGNV/+MN/+NC/Sg4EsmeXlvJjPzwpQzpV332yqtrW9sbpW37Z3dvf2DyuHRg0oySahPEp7IdogV5UxQXzPNaTuVFMchp61wdDPzW2MqFUvEvZ6kNIjxQLCIEayNFDiOnXfDCD1NbcfpVapuzZ0DrRKvIFUo0OxVvrr9hGQxFZpwrFTHc1Md5FhqRjid2t1M0RSTER7QjqECx1QF+fzoKTozSh9FiTQlNJqrvydyHCs1iUPTGWM9VMveTPzP62Q6ugxyJtJMU0EWi6KMI52gWQKozyQlmk8MwUQycysiQywx0SYn24TgLb+8Svx67arm3dWrjesijTKcwCmcgwcX0IBbaIIPBB7hGV7hzRpbL9a79bFoLVnFzDH8gfX5A3nNkB4=</latexit><latexit sha1_base64="ezxhI2prtFY0vXyy/yEAlvhFBlU=">AAAB83icbVBNT8JAEJ3iF9Yv1KOXjdTEE2m5+HEi8eIREysk0JDtsoUN223d3RJJw+/w4kGNV/+MN/+NC/Sg4EsmeXlvJjPzwpQzpV332yqtrW9sbpW37Z3dvf2DyuHRg0oySahPEp7IdogV5UxQXzPNaTuVFMchp61wdDPzW2MqFUvEvZ6kNIjxQLCIEayNFDiOnXfDCD1NbcfpVapuzZ0DrRKvIFUo0OxVvrr9hGQxFZpwrFTHc1Md5FhqRjid2t1M0RSTER7QjqECx1QF+fzoKTozSh9FiTQlNJqrvydyHCs1iUPTGWM9VMveTPzP62Q6ugxyJtJMU0EWi6KMI52gWQKozyQlmk8MwUQycysiQywx0SYn24TgLb+8Svx67arm3dWrjesijTKcwCmcgwcX0IBbaIIPBB7hGV7hzRpbL9a79bFoLVnFzDH8gfX5A3nNkB4=</latexit><latexit sha1_base64="ezxhI2prtFY0vXyy/yEAlvhFBlU=">AAAB83icbVBNT8JAEJ3iF9Yv1KOXjdTEE2m5+HEi8eIREysk0JDtsoUN223d3RJJw+/w4kGNV/+MN/+NC/Sg4EsmeXlvJjPzwpQzpV332yqtrW9sbpW37Z3dvf2DyuHRg0oySahPEp7IdogV5UxQXzPNaTuVFMchp61wdDPzW2MqFUvEvZ6kNIjxQLCIEayNFDiOnXfDCD1NbcfpVapuzZ0DrRKvIFUo0OxVvrr9hGQxFZpwrFTHc1Md5FhqRjid2t1M0RSTER7QjqECx1QF+fzoKTozSh9FiTQlNJqrvydyHCs1iUPTGWM9VMveTPzP62Q6ugxyJtJMU0EWi6KMI52gWQKozyQlmk8MwUQycysiQywx0SYn24TgLb+8Svx67arm3dWrjesijTKcwCmcgwcX0IBbaIIPBB7hGV7hzRpbL9a79bFoLVnFzDH8gfX5A3nNkB4=</latexit>

HAM SPAM SPAM SPAM HAM

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p(x = [buy, viagra] | SPAM)<latexit sha1_base64="5hUhGr6RblYk5Vcdhk/ntEIzge4=">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</latexit><latexit sha1_base64="5hUhGr6RblYk5Vcdhk/ntEIzge4=">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</latexit><latexit sha1_base64="5hUhGr6RblYk5Vcdhk/ntEIzge4=">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</latexit><latexit sha1_base64="5hUhGr6RblYk5Vcdhk/ntEIzge4=">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</latexit>

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Page 15: Discrete Naïve Bayes

The Naïve Bayes Assumption

15

OK, so we need a way to estimate these SPAM and HAM scores from the data for email

Consider again the following training data.

How can we estimate ?

Joint probabilities are hard to estimate from data. Here is where we make our naïve assumption

Naïve Bayes Assumption: The features of are independent given the class label

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me

Page 16: Discrete Naïve Bayes

The Naïve Bayes Assumption

16

Naïve Bayes Assumption: The features of are independent given the class label

Conditional Independence Example: You have two (possibly different) weighted coins. Pick a coin at random and flip it three times.

Note: Flips are independent given knowledge of the choice of coin, but NOT without.

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p(x = [H, H, T] | C1) = p(H | C1) · p(H | C1) · p(T | C1)<latexit sha1_base64="ZPAXrS9UmI7p8FYERexJpPAjE1A=">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</latexit><latexit sha1_base64="ZPAXrS9UmI7p8FYERexJpPAjE1A=">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</latexit><latexit sha1_base64="ZPAXrS9UmI7p8FYERexJpPAjE1A=">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</latexit><latexit sha1_base64="ZPAXrS9UmI7p8FYERexJpPAjE1A=">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</latexit>

p(x = [H, H, T]) 6= p(H) · p(H) · p(T )<latexit sha1_base64="lfIW/EvgywyY0ib0g8LCnNOZJmE=">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</latexit><latexit sha1_base64="lfIW/EvgywyY0ib0g8LCnNOZJmE=">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</latexit><latexit sha1_base64="lfIW/EvgywyY0ib0g8LCnNOZJmE=">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</latexit><latexit sha1_base64="lfIW/EvgywyY0ib0g8LCnNOZJmE=">AAACLnicbZDNSgMxFIUz/jv+VV26CbaCBSkzbtSFIAjSpUKrhc5QMumdGsxkxuSOWIa+kRtfxY2Iilsfw7R2odVLAh/n3EtyT5RJYdDzXpyp6ZnZufmFRXdpeWV1rbS+cWnSXHNo8lSmuhUxA1IoaKJACa1MA0siCVfRzenQv7oDbUSqGtjPIExYT4lYcIZW6pTOKhU32y2CKKb3A3pMA4R71EnRru9RexrhoEoDBbfUdtUt8m6K9Bc2qm6l0imVvZo3KvoX/DGUybjOO6WnoJvyPAGFXDJj2r6XYVgwjYJLGLhBbiBj/Ib1oG1RsQRMWIz2HdAdq3RpnGp7FdKR+nOiYIkx/SSynQnDazPpDcX/vHaO8WFYCJXlCIp/PxTnkmJKh+HRrtDAUfYtMK6F/Svl10wzjjZi14bgT678F5r7taOaf7FfPjkcp7FAtsg22SU+OSAnpE7OSZNw8kCeyCt5cx6dZ+fd+fhunXLGM5vkVzmfX8HKo44=</latexit>

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in =for 1 COM 01,Cz

, Cz

Page 17: Discrete Naïve Bayes

The Naïve Bayes Assumption

17

Naïve Bayes Assumption: The features of are independent given the class label

The Crux: What does this assumption mean for us and Email classification?

This means that we can estimate class-conditional probabilities of an email by multiplying class-conditional probabilities of individual words

This is much easier to do!

Question: Is this assumption valid?

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p ( I PEANUT ,Butter 3 IHAM)

Page 18: Discrete Naïve Bayes

The Naïve Bayes Assumption

18

Naïve Bayes Assumption: The features of are independent given the class label

The Crux: What does this assumption mean for us and Email classification?

This means that we can estimate class-conditional probabilities of an email by multiplying class-conditional probabilities of individual words

This is much easier to do!

Answer: Probably not! But it makes things this method tractable, so we make it anyway

Example: Consider

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p(x = [buy, viagra] | SPAM) = p(buy | SPAM) · p(viagra | SPAM)<latexit sha1_base64="w/IPFRjtij8HzK+fWd1aYegCglc=">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</latexit><latexit sha1_base64="w/IPFRjtij8HzK+fWd1aYegCglc=">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</latexit><latexit sha1_base64="w/IPFRjtij8HzK+fWd1aYegCglc=">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</latexit><latexit sha1_base64="w/IPFRjtij8HzK+fWd1aYegCglc=">AAACbHicbZFPa9swGMZlr9ta75/X0cMaBmLJRgIj2L20Oww6dtmlkLF5LcQmyLKciki2kV6XBuPbPuFu/Qq97CtMThxIk7wgeHie36s/r+JCcA2ed2fZj/YeP3m6f+A8e/7i5Sv39eFvnZeKsoDmIldXMdFM8IwFwEGwq0IxImPBLuPZtya/vGFK8zz7BfOCRZJMM55ySsBYE/dPr+cU/SqMU3xb4y84BHYLSlbjuJx/wjecTBWJahxKnuCfo68XA8MU/RVloDZbOQ1TD3BIkxzWwOVGO1mn15u4XW/oLQpvC78VXdTWaOL+DZOclpJlQAXReux7BUQVUcCpYLUTlpoVhM7IlI2NzIhkOqoW06rxB+MkOM2VWRnghbveURGp9VzGhpQErvVm1pi7snEJ6VlU8awogWV0eVBaCgw5bkaPE64YBTE3glDFzV0xvSaKUDAf5Jgh+JtP3hbByfDz0P9x0j0/a6exjzroPeojH52ic/QdjVCAKLq3XOutdWz9s4/sjv1uidpW2/MGPSj7438utLkC</latexit>

p(x = [peanut, butter] | HAM)<latexit sha1_base64="G7RBr4S920KiIrBuxmDjrfcttGM=">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</latexit><latexit sha1_base64="G7RBr4S920KiIrBuxmDjrfcttGM=">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</latexit><latexit sha1_base64="G7RBr4S920KiIrBuxmDjrfcttGM=">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</latexit><latexit sha1_base64="G7RBr4S920KiIrBuxmDjrfcttGM=">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</latexit>

Page 19: Discrete Naïve Bayes

Naïve Bayes

19

OK, so we need a way to estimate these SPAM and HAM scores from the data for email

To estimate we just have to count instances of term in Class in training data

Note: Here by words we include repeated instances of all terms

p(SPAM | x) / p(x | SPAM)·p(SPAM) vs p(x | HAM)·p(HAM) / p(HAM | x)<latexit sha1_base64="Cwo4bp3EalNu74+sVTIdJuDbjxs=">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</latexit><latexit sha1_base64="Cwo4bp3EalNu74+sVTIdJuDbjxs=">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</latexit><latexit sha1_base64="Cwo4bp3EalNu74+sVTIdJuDbjxs=">AAACuHicdZFNaxsxEIa127RNtx9x22MvInbBvZjdHPJBLim95FJwSNwEvMZotbOOiLRSpdkQs+x/LL3131S2lyZ20gHByzPvjEajzEjhMI7/BOGzrecvXm6/il6/eftup/P+ww+nK8thxLXU9ipjDqQoYYQCJVwZC0xlEi6zm2+L/OUtWCd0eYFzAxPFZqUoBGfo0bTzq9eLTD9FuEOr6vPh1+8NTZXIaZ1mBb1rvtDUWG1QU9Nv0Sq/VuJdPNdINzp5/LNi995b16xI9J9mp0/2aum/OR7y9WGjXm/a6caDeBn0sUha0SVtDKed32mueaWgRC6Zc+MkNjipmUXBJTRRWjkwjN+wGYy9LJkCN6mXm2/oZ09yWmjrT4l0SR9W1Ew5N1eZdyqG124zt4BP5cYVFoeTWpSmQij56qKiktQvYPGNNBcWOMq5F4xb4Wel/JpZxtF/duSXkGw++bEY7Q2OBsnZXvfksN3GNvlEdkmfJOSAnJBTMiQjwoP9IA0gKMLjkIWzUKysYdDWfCRrEdq/f/TVOA==</latexit><latexit sha1_base64="Cwo4bp3EalNu74+sVTIdJuDbjxs=">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</latexit>

x<latexit sha1_base64="ezxhI2prtFY0vXyy/yEAlvhFBlU=">AAAB83icbVBNT8JAEJ3iF9Yv1KOXjdTEE2m5+HEi8eIREysk0JDtsoUN223d3RJJw+/w4kGNV/+MN/+NC/Sg4EsmeXlvJjPzwpQzpV332yqtrW9sbpW37Z3dvf2DyuHRg0oySahPEp7IdogV5UxQXzPNaTuVFMchp61wdDPzW2MqFUvEvZ6kNIjxQLCIEayNFDiOnXfDCD1NbcfpVapuzZ0DrRKvIFUo0OxVvrr9hGQxFZpwrFTHc1Md5FhqRjid2t1M0RSTER7QjqECx1QF+fzoKTozSh9FiTQlNJqrvydyHCs1iUPTGWM9VMveTPzP62Q6ugxyJtJMU0EWi6KMI52gWQKozyQlmk8MwUQycysiQywx0SYn24TgLb+8Svx67arm3dWrjesijTKcwCmcgwcX0IBbaIIPBB7hGV7hzRpbL9a79bFoLVnFzDH8gfX5A3nNkB4=</latexit><latexit sha1_base64="ezxhI2prtFY0vXyy/yEAlvhFBlU=">AAAB83icbVBNT8JAEJ3iF9Yv1KOXjdTEE2m5+HEi8eIREysk0JDtsoUN223d3RJJw+/w4kGNV/+MN/+NC/Sg4EsmeXlvJjPzwpQzpV332yqtrW9sbpW37Z3dvf2DyuHRg0oySahPEp7IdogV5UxQXzPNaTuVFMchp61wdDPzW2MqFUvEvZ6kNIjxQLCIEayNFDiOnXfDCD1NbcfpVapuzZ0DrRKvIFUo0OxVvrr9hGQxFZpwrFTHc1Md5FhqRjid2t1M0RSTER7QjqECx1QF+fzoKTozSh9FiTQlNJqrvydyHCs1iUPTGWM9VMveTPzP62Q6ugxyJtJMU0EWi6KMI52gWQKozyQlmk8MwUQycysiQywx0SYn24TgLb+8Svx67arm3dWrjesijTKcwCmcgwcX0IBbaIIPBB7hGV7hzRpbL9a79bFoLVnFzDH8gfX5A3nNkB4=</latexit><latexit sha1_base64="ezxhI2prtFY0vXyy/yEAlvhFBlU=">AAAB83icbVBNT8JAEJ3iF9Yv1KOXjdTEE2m5+HEi8eIREysk0JDtsoUN223d3RJJw+/w4kGNV/+MN/+NC/Sg4EsmeXlvJjPzwpQzpV332yqtrW9sbpW37Z3dvf2DyuHRg0oySahPEp7IdogV5UxQXzPNaTuVFMchp61wdDPzW2MqFUvEvZ6kNIjxQLCIEayNFDiOnXfDCD1NbcfpVapuzZ0DrRKvIFUo0OxVvrr9hGQxFZpwrFTHc1Md5FhqRjid2t1M0RSTER7QjqECx1QF+fzoKTozSh9FiTQlNJqrvydyHCs1iUPTGWM9VMveTPzP62Q6ugxyJtJMU0EWi6KMI52gWQKozyQlmk8MwUQycysiQywx0SYn24TgLb+8Svx67arm3dWrjesijTKcwCmcgwcX0IBbaIIPBB7hGV7hzRpbL9a79bFoLVnFzDH8gfX5A3nNkB4=</latexit><latexit sha1_base64="ezxhI2prtFY0vXyy/yEAlvhFBlU=">AAAB83icbVBNT8JAEJ3iF9Yv1KOXjdTEE2m5+HEi8eIREysk0JDtsoUN223d3RJJw+/w4kGNV/+MN/+NC/Sg4EsmeXlvJjPzwpQzpV332yqtrW9sbpW37Z3dvf2DyuHRg0oySahPEp7IdogV5UxQXzPNaTuVFMchp61wdDPzW2MqFUvEvZ6kNIjxQLCIEayNFDiOnXfDCD1NbcfpVapuzZ0DrRKvIFUo0OxVvrr9hGQxFZpwrFTHc1Md5FhqRjid2t1M0RSTER7QjqECx1QF+fzoKTozSh9FiTQlNJqrvydyHCs1iUPTGWM9VMveTPzP62Q6ugxyJtJMU0EWi6KMI52gWQKozyQlmk8MwUQycysiQywx0SYn24TgLb+8Svx67arm3dWrjesijTKcwCmcgwcX0IBbaIIPBB7hGV7hzRpbL9a79bFoLVnFzDH8gfX5A3nNkB4=</latexit>

p(term | Class)<latexit sha1_base64="JZx0YEemw6ewmrbA6CkNQaZn6J8=">AAACE3icbVA9SwNBEN3zM55fUUubxUSICOEujbELpLGMYIyQhLC3mSSLu3fH7pwYjvwIG/+KjYWKrY2d/8bNh6CJDwYe783szrwglsKg5305S8srq2vrmQ13c2t7Zze7t39tokRzqPNIRvomYAakCKGOAiXcxBqYCiQ0gtvq2G/cgTYiCq9wGENbsX4oeoIztFIne5rPu3GhhXCPWqUIWo1oS4ku/ZGqkhkzOqFuPt/J5ryiNwFdJP6M5MgMtU72s9WNeKIgRD5+pul7MbZTplFwCSO3lRiIGb9lfWhaGjIFpp1OjhrRY6t0aS/StkKkE/X3RMqUMUMV2E7FcGDmvbH4n9dMsFdupyKME4SQTz/qJZJiRMcJ0a7QwFEOLWFcC7sr5QOmGbfpGNeG4M+fvEjqpeJ50b8s5SrlWRoZckiOSIH45IxUyAWpkTrh5IE8kRfy6jw6z86b8z5tXXJmMwfkD5yPbw8JnRo=</latexit><latexit sha1_base64="JZx0YEemw6ewmrbA6CkNQaZn6J8=">AAACE3icbVA9SwNBEN3zM55fUUubxUSICOEujbELpLGMYIyQhLC3mSSLu3fH7pwYjvwIG/+KjYWKrY2d/8bNh6CJDwYe783szrwglsKg5305S8srq2vrmQ13c2t7Zze7t39tokRzqPNIRvomYAakCKGOAiXcxBqYCiQ0gtvq2G/cgTYiCq9wGENbsX4oeoIztFIne5rPu3GhhXCPWqUIWo1oS4ku/ZGqkhkzOqFuPt/J5ryiNwFdJP6M5MgMtU72s9WNeKIgRD5+pul7MbZTplFwCSO3lRiIGb9lfWhaGjIFpp1OjhrRY6t0aS/StkKkE/X3RMqUMUMV2E7FcGDmvbH4n9dMsFdupyKME4SQTz/qJZJiRMcJ0a7QwFEOLWFcC7sr5QOmGbfpGNeG4M+fvEjqpeJ50b8s5SrlWRoZckiOSIH45IxUyAWpkTrh5IE8kRfy6jw6z86b8z5tXXJmMwfkD5yPbw8JnRo=</latexit><latexit sha1_base64="JZx0YEemw6ewmrbA6CkNQaZn6J8=">AAACE3icbVA9SwNBEN3zM55fUUubxUSICOEujbELpLGMYIyQhLC3mSSLu3fH7pwYjvwIG/+KjYWKrY2d/8bNh6CJDwYe783szrwglsKg5305S8srq2vrmQ13c2t7Zze7t39tokRzqPNIRvomYAakCKGOAiXcxBqYCiQ0gtvq2G/cgTYiCq9wGENbsX4oeoIztFIne5rPu3GhhXCPWqUIWo1oS4ku/ZGqkhkzOqFuPt/J5ryiNwFdJP6M5MgMtU72s9WNeKIgRD5+pul7MbZTplFwCSO3lRiIGb9lfWhaGjIFpp1OjhrRY6t0aS/StkKkE/X3RMqUMUMV2E7FcGDmvbH4n9dMsFdupyKME4SQTz/qJZJiRMcJ0a7QwFEOLWFcC7sr5QOmGbfpGNeG4M+fvEjqpeJ50b8s5SrlWRoZckiOSIH45IxUyAWpkTrh5IE8kRfy6jw6z86b8z5tXXJmMwfkD5yPbw8JnRo=</latexit><latexit sha1_base64="JZx0YEemw6ewmrbA6CkNQaZn6J8=">AAACE3icbVA9SwNBEN3zM55fUUubxUSICOEujbELpLGMYIyQhLC3mSSLu3fH7pwYjvwIG/+KjYWKrY2d/8bNh6CJDwYe783szrwglsKg5305S8srq2vrmQ13c2t7Zze7t39tokRzqPNIRvomYAakCKGOAiXcxBqYCiQ0gtvq2G/cgTYiCq9wGENbsX4oeoIztFIne5rPu3GhhXCPWqUIWo1oS4ku/ZGqkhkzOqFuPt/J5ryiNwFdJP6M5MgMtU72s9WNeKIgRD5+pul7MbZTplFwCSO3lRiIGb9lfWhaGjIFpp1OjhrRY6t0aS/StkKkE/X3RMqUMUMV2E7FcGDmvbH4n9dMsFdupyKME4SQTz/qJZJiRMcJ0a7QwFEOLWFcC7sr5QOmGbfpGNeG4M+fvEjqpeJ50b8s5SrlWRoZckiOSIH45IxUyAWpkTrh5IE8kRfy6jw6z86b8z5tXXJmMwfkD5yPbw8JnRo=</latexit>

p(term | Class) = # instances of term in Class

# total words in Class<latexit sha1_base64="bbnQ04sT4KK3Xpp9KaE1bC8h/jM=">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</latexit><latexit sha1_base64="bbnQ04sT4KK3Xpp9KaE1bC8h/jM=">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</latexit><latexit sha1_base64="bbnQ04sT4KK3Xpp9KaE1bC8h/jM=">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</latexit><latexit sha1_base64="bbnQ04sT4KK3Xpp9KaE1bC8h/jM=">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</latexit>

a = =

Page 20: Discrete Naïve Bayes

Naïve Bayes

20

To estimate we just have to count instances of term in Class in training data

Example: Estimate from the training data

p(term | Class)<latexit sha1_base64="JZx0YEemw6ewmrbA6CkNQaZn6J8=">AAACE3icbVA9SwNBEN3zM55fUUubxUSICOEujbELpLGMYIyQhLC3mSSLu3fH7pwYjvwIG/+KjYWKrY2d/8bNh6CJDwYe783szrwglsKg5305S8srq2vrmQ13c2t7Zze7t39tokRzqPNIRvomYAakCKGOAiXcxBqYCiQ0gtvq2G/cgTYiCq9wGENbsX4oeoIztFIne5rPu3GhhXCPWqUIWo1oS4ku/ZGqkhkzOqFuPt/J5ryiNwFdJP6M5MgMtU72s9WNeKIgRD5+pul7MbZTplFwCSO3lRiIGb9lfWhaGjIFpp1OjhrRY6t0aS/StkKkE/X3RMqUMUMV2E7FcGDmvbH4n9dMsFdupyKME4SQTz/qJZJiRMcJ0a7QwFEOLWFcC7sr5QOmGbfpGNeG4M+fvEjqpeJ50b8s5SrlWRoZckiOSIH45IxUyAWpkTrh5IE8kRfy6jw6z86b8z5tXXJmMwfkD5yPbw8JnRo=</latexit><latexit sha1_base64="JZx0YEemw6ewmrbA6CkNQaZn6J8=">AAACE3icbVA9SwNBEN3zM55fUUubxUSICOEujbELpLGMYIyQhLC3mSSLu3fH7pwYjvwIG/+KjYWKrY2d/8bNh6CJDwYe783szrwglsKg5305S8srq2vrmQ13c2t7Zze7t39tokRzqPNIRvomYAakCKGOAiXcxBqYCiQ0gtvq2G/cgTYiCq9wGENbsX4oeoIztFIne5rPu3GhhXCPWqUIWo1oS4ku/ZGqkhkzOqFuPt/J5ryiNwFdJP6M5MgMtU72s9WNeKIgRD5+pul7MbZTplFwCSO3lRiIGb9lfWhaGjIFpp1OjhrRY6t0aS/StkKkE/X3RMqUMUMV2E7FcGDmvbH4n9dMsFdupyKME4SQTz/qJZJiRMcJ0a7QwFEOLWFcC7sr5QOmGbfpGNeG4M+fvEjqpeJ50b8s5SrlWRoZckiOSIH45IxUyAWpkTrh5IE8kRfy6jw6z86b8z5tXXJmMwfkD5yPbw8JnRo=</latexit><latexit sha1_base64="JZx0YEemw6ewmrbA6CkNQaZn6J8=">AAACE3icbVA9SwNBEN3zM55fUUubxUSICOEujbELpLGMYIyQhLC3mSSLu3fH7pwYjvwIG/+KjYWKrY2d/8bNh6CJDwYe783szrwglsKg5305S8srq2vrmQ13c2t7Zze7t39tokRzqPNIRvomYAakCKGOAiXcxBqYCiQ0gtvq2G/cgTYiCq9wGENbsX4oeoIztFIne5rPu3GhhXCPWqUIWo1oS4ku/ZGqkhkzOqFuPt/J5ryiNwFdJP6M5MgMtU72s9WNeKIgRD5+pul7MbZTplFwCSO3lRiIGb9lfWhaGjIFpp1OjhrRY6t0aS/StkKkE/X3RMqUMUMV2E7FcGDmvbH4n9dMsFdupyKME4SQTz/qJZJiRMcJ0a7QwFEOLWFcC7sr5QOmGbfpGNeG4M+fvEjqpeJ50b8s5SrlWRoZckiOSIH45IxUyAWpkTrh5IE8kRfy6jw6z86b8z5tXXJmMwfkD5yPbw8JnRo=</latexit><latexit sha1_base64="JZx0YEemw6ewmrbA6CkNQaZn6J8=">AAACE3icbVA9SwNBEN3zM55fUUubxUSICOEujbELpLGMYIyQhLC3mSSLu3fH7pwYjvwIG/+KjYWKrY2d/8bNh6CJDwYe783szrwglsKg5305S8srq2vrmQ13c2t7Zze7t39tokRzqPNIRvomYAakCKGOAiXcxBqYCiQ0gtvq2G/cgTYiCq9wGENbsX4oeoIztFIne5rPu3GhhXCPWqUIWo1oS4ku/ZGqkhkzOqFuPt/J5ryiNwFdJP6M5MgMtU72s9WNeKIgRD5+pul7MbZTplFwCSO3lRiIGb9lfWhaGjIFpp1OjhrRY6t0aS/StkKkE/X3RMqUMUMV2E7FcGDmvbH4n9dMsFdupyKME4SQTz/qJZJiRMcJ0a7QwFEOLWFcC7sr5QOmGbfpGNeG4M+fvEjqpeJ50b8s5SrlWRoZckiOSIH45IxUyAWpkTrh5IE8kRfy6jw6z86b8z5tXXJmMwfkD5yPbw8JnRo=</latexit>

p(buy | SPAM)<latexit sha1_base64="/gOzo+zGreUZsdT/LM5P3RmPPEw=">AAACEHicbVA9SwNBEN2LX/H8ilraLCZCbMJdGmMXsbERIhoTyB1hb7NJluzeHbtzYjjyF2z8KzYWKraWdv4bNx+CJj4YeLw3w8y8IBZcg+N8WZml5ZXVtey6vbG5tb2T29271VGiKKvTSESqGRDNBA9ZHTgI1owVIzIQrBEMzsd+444pzaPwBoYx8yXphbzLKQEjtXPFQsGOix6we1AyDZLhCHuSd/CPcl07uxwd24VCO5d3Ss4EeJG4M5JHM9TauU+vE9FEshCoIFq3XCcGPyUKOBVsZHuJZjGhA9JjLUNDIpn208lHI3xklA7uRspUCHii/p5IidR6KAPTKQn09bw3Fv/zWgl0K37KwzgBFtLpom4iMER4HA/ucMUoiKEhhCpubsW0TxShYEK0TQju/MuLpF4unZbcq3K+WpmlkUUH6BAVkYtOUBVdoBqqI4oe0BN6Qa/Wo/VsvVnv09aMNZvZR39gfXwDd6Wbrw==</latexit><latexit sha1_base64="/gOzo+zGreUZsdT/LM5P3RmPPEw=">AAACEHicbVA9SwNBEN2LX/H8ilraLCZCbMJdGmMXsbERIhoTyB1hb7NJluzeHbtzYjjyF2z8KzYWKraWdv4bNx+CJj4YeLw3w8y8IBZcg+N8WZml5ZXVtey6vbG5tb2T29271VGiKKvTSESqGRDNBA9ZHTgI1owVIzIQrBEMzsd+444pzaPwBoYx8yXphbzLKQEjtXPFQsGOix6we1AyDZLhCHuSd/CPcl07uxwd24VCO5d3Ss4EeJG4M5JHM9TauU+vE9FEshCoIFq3XCcGPyUKOBVsZHuJZjGhA9JjLUNDIpn208lHI3xklA7uRspUCHii/p5IidR6KAPTKQn09bw3Fv/zWgl0K37KwzgBFtLpom4iMER4HA/ucMUoiKEhhCpubsW0TxShYEK0TQju/MuLpF4unZbcq3K+WpmlkUUH6BAVkYtOUBVdoBqqI4oe0BN6Qa/Wo/VsvVnv09aMNZvZR39gfXwDd6Wbrw==</latexit><latexit sha1_base64="/gOzo+zGreUZsdT/LM5P3RmPPEw=">AAACEHicbVA9SwNBEN2LX/H8ilraLCZCbMJdGmMXsbERIhoTyB1hb7NJluzeHbtzYjjyF2z8KzYWKraWdv4bNx+CJj4YeLw3w8y8IBZcg+N8WZml5ZXVtey6vbG5tb2T29271VGiKKvTSESqGRDNBA9ZHTgI1owVIzIQrBEMzsd+444pzaPwBoYx8yXphbzLKQEjtXPFQsGOix6we1AyDZLhCHuSd/CPcl07uxwd24VCO5d3Ss4EeJG4M5JHM9TauU+vE9FEshCoIFq3XCcGPyUKOBVsZHuJZjGhA9JjLUNDIpn208lHI3xklA7uRspUCHii/p5IidR6KAPTKQn09bw3Fv/zWgl0K37KwzgBFtLpom4iMER4HA/ucMUoiKEhhCpubsW0TxShYEK0TQju/MuLpF4unZbcq3K+WpmlkUUH6BAVkYtOUBVdoBqqI4oe0BN6Qa/Wo/VsvVnv09aMNZvZR39gfXwDd6Wbrw==</latexit><latexit sha1_base64="/gOzo+zGreUZsdT/LM5P3RmPPEw=">AAACEHicbVA9SwNBEN2LX/H8ilraLCZCbMJdGmMXsbERIhoTyB1hb7NJluzeHbtzYjjyF2z8KzYWKraWdv4bNx+CJj4YeLw3w8y8IBZcg+N8WZml5ZXVtey6vbG5tb2T29271VGiKKvTSESqGRDNBA9ZHTgI1owVIzIQrBEMzsd+444pzaPwBoYx8yXphbzLKQEjtXPFQsGOix6we1AyDZLhCHuSd/CPcl07uxwd24VCO5d3Ss4EeJG4M5JHM9TauU+vE9FEshCoIFq3XCcGPyUKOBVsZHuJZjGhA9JjLUNDIpn208lHI3xklA7uRspUCHii/p5IidR6KAPTKQn09bw3Fv/zWgl0K37KwzgBFtLpom4iMER4HA/ucMUoiKEhhCpubsW0TxShYEK0TQju/MuLpF4unZbcq3K+WpmlkUUH6BAVkYtOUBVdoBqqI4oe0BN6Qa/Wo/VsvVnv09aMNZvZR39gfXwDd6Wbrw==</latexit>

HAM SPAM SPAM SPAM HAM

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money viagra nigeria fly nigeria<latexit sha1_base64="Fk40qlV9xSoCXT3K4KQdocdHpiQ=">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</latexit><latexit sha1_base64="Fk40qlV9xSoCXT3K4KQdocdHpiQ=">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</latexit><latexit sha1_base64="Fk40qlV9xSoCXT3K4KQdocdHpiQ=">AAADaXiclVJLj9MwEHYTHktgobtIgOBi0YL2VCV74XFaxGUvSEVQdqWmqhx3klp17MieLEShEr+RG7+ACz8CpxtWfXDZsSJ93zcPjyeTFFJYDMNfHc+/cfPW7b07wd17+/cfdA8Ov1hdGg4jrqU25wmzIIWCEQqUcF4YYHki4SxZvG/8ZxdgrNDqM1YFTHKWKZEKztBJ04POj34/iBPIhKqZMaxa1vx7e5ZBjPANTV6fvvuwpC/pP/ppeD2+So/jIJ43bdKrsl+1WazHKZGBEWxdSmW1TnOtoNrxN6VbnpQbbl0U2mCpBG7IW1FbdK5z2Ci6c+uFYJlh12v9KiKIQc3aYQf9/rTbCwfhyuguiFrQI60Np92f8UzzMgeFXDJrx1FY4MQVRMEluJ9WWigYX7AMxg4qloOd1KtVWdIXTpnRVBv3KaQrdT2jZrm1VZ64yJzh3G77GvF/vnGJ6etJLVRRIih+eVFaSoqaNntHZ8IAR1k5wLgRrlfK58wwjm47AzeEaPvJu2B0PHgziD4e907ettPYI8/Ic3JEIvKKnJBTMiQjwju/vX3vkffY++Mf+k/8p5ehXqfNeUg2zO/9BU15Giw=</latexit><latexit sha1_base64="Fk40qlV9xSoCXT3K4KQdocdHpiQ=">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</latexit>

p(term | Class) = # instances of term in Class

# total words in Class<latexit sha1_base64="bbnQ04sT4KK3Xpp9KaE1bC8h/jM=">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</latexit><latexit sha1_base64="bbnQ04sT4KK3Xpp9KaE1bC8h/jM=">AAACcXicbVDLahRBFK1uX7F9TRQ3CnrJjBARhu5sjAshkI3LCI4JTA9DdfXtTJF6NFW3o0PTH+DvufMr3PgBVs9MYEy8UHA4j6q6p6iV9JSmv6L41u07d+/t3E8ePHz0+Mlg9+lXbxsncCKssu6s4B6VNDghSQrPaodcFwpPi4vjXj+9ROelNV9oWeNM83MjKyk4BWo++DEaJfmCU1t3+znhd3K6JXS6g1zLEq6oY8W9797CR8jLynHRXgn5EKTxxI1AD7aCPhsYWAe6bR9Z4gravKjgm3Wl77Z8yWg0HwzTcboauAmyDRiyzZzMBz/z0opGoyHR3zLN0ppmLXckhcIuyRuPNRcX/BynARqu0c/aVWUdvAlMCZV14RiCFbudaLn2fqmL4NScFv661pP/06YNVYezVpq6ITRi/VDVqLA99P1DKR0KUssAuHAy/BXEgodKQ3M+CSVk11e+CSYH4w/j7PPB8Ohw08YOe8n22D7L2Ht2xD6xEzZhgv2OnkevotfRn/hFDPHe2hpHm8wz9s/E7/4CbQm9kw==</latexit><latexit sha1_base64="bbnQ04sT4KK3Xpp9KaE1bC8h/jM=">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</latexit><latexit sha1_base64="bbnQ04sT4KK3Xpp9KaE1bC8h/jM=">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</latexit>

p (term I span )

FCbuy I SPAM )=

.9

÷

Page 21: Discrete Naïve Bayes

Naïve Bayes

21

To estimate we just have to count instances of term in Class in training data

Example: Estimate from the training data

p(term | Class)<latexit sha1_base64="JZx0YEemw6ewmrbA6CkNQaZn6J8=">AAACE3icbVA9SwNBEN3zM55fUUubxUSICOEujbELpLGMYIyQhLC3mSSLu3fH7pwYjvwIG/+KjYWKrY2d/8bNh6CJDwYe783szrwglsKg5305S8srq2vrmQ13c2t7Zze7t39tokRzqPNIRvomYAakCKGOAiXcxBqYCiQ0gtvq2G/cgTYiCq9wGENbsX4oeoIztFIne5rPu3GhhXCPWqUIWo1oS4ku/ZGqkhkzOqFuPt/J5ryiNwFdJP6M5MgMtU72s9WNeKIgRD5+pul7MbZTplFwCSO3lRiIGb9lfWhaGjIFpp1OjhrRY6t0aS/StkKkE/X3RMqUMUMV2E7FcGDmvbH4n9dMsFdupyKME4SQTz/qJZJiRMcJ0a7QwFEOLWFcC7sr5QOmGbfpGNeG4M+fvEjqpeJ50b8s5SrlWRoZckiOSIH45IxUyAWpkTrh5IE8kRfy6jw6z86b8z5tXXJmMwfkD5yPbw8JnRo=</latexit><latexit sha1_base64="JZx0YEemw6ewmrbA6CkNQaZn6J8=">AAACE3icbVA9SwNBEN3zM55fUUubxUSICOEujbELpLGMYIyQhLC3mSSLu3fH7pwYjvwIG/+KjYWKrY2d/8bNh6CJDwYe783szrwglsKg5305S8srq2vrmQ13c2t7Zze7t39tokRzqPNIRvomYAakCKGOAiXcxBqYCiQ0gtvq2G/cgTYiCq9wGENbsX4oeoIztFIne5rPu3GhhXCPWqUIWo1oS4ku/ZGqkhkzOqFuPt/J5ryiNwFdJP6M5MgMtU72s9WNeKIgRD5+pul7MbZTplFwCSO3lRiIGb9lfWhaGjIFpp1OjhrRY6t0aS/StkKkE/X3RMqUMUMV2E7FcGDmvbH4n9dMsFdupyKME4SQTz/qJZJiRMcJ0a7QwFEOLWFcC7sr5QOmGbfpGNeG4M+fvEjqpeJ50b8s5SrlWRoZckiOSIH45IxUyAWpkTrh5IE8kRfy6jw6z86b8z5tXXJmMwfkD5yPbw8JnRo=</latexit><latexit sha1_base64="JZx0YEemw6ewmrbA6CkNQaZn6J8=">AAACE3icbVA9SwNBEN3zM55fUUubxUSICOEujbELpLGMYIyQhLC3mSSLu3fH7pwYjvwIG/+KjYWKrY2d/8bNh6CJDwYe783szrwglsKg5305S8srq2vrmQ13c2t7Zze7t39tokRzqPNIRvomYAakCKGOAiXcxBqYCiQ0gtvq2G/cgTYiCq9wGENbsX4oeoIztFIne5rPu3GhhXCPWqUIWo1oS4ku/ZGqkhkzOqFuPt/J5ryiNwFdJP6M5MgMtU72s9WNeKIgRD5+pul7MbZTplFwCSO3lRiIGb9lfWhaGjIFpp1OjhrRY6t0aS/StkKkE/X3RMqUMUMV2E7FcGDmvbH4n9dMsFdupyKME4SQTz/qJZJiRMcJ0a7QwFEOLWFcC7sr5QOmGbfpGNeG4M+fvEjqpeJ50b8s5SrlWRoZckiOSIH45IxUyAWpkTrh5IE8kRfy6jw6z86b8z5tXXJmMwfkD5yPbw8JnRo=</latexit><latexit sha1_base64="JZx0YEemw6ewmrbA6CkNQaZn6J8=">AAACE3icbVA9SwNBEN3zM55fUUubxUSICOEujbELpLGMYIyQhLC3mSSLu3fH7pwYjvwIG/+KjYWKrY2d/8bNh6CJDwYe783szrwglsKg5305S8srq2vrmQ13c2t7Zze7t39tokRzqPNIRvomYAakCKGOAiXcxBqYCiQ0gtvq2G/cgTYiCq9wGENbsX4oeoIztFIne5rPu3GhhXCPWqUIWo1oS4ku/ZGqkhkzOqFuPt/J5ryiNwFdJP6M5MgMtU72s9WNeKIgRD5+pul7MbZTplFwCSO3lRiIGb9lfWhaGjIFpp1OjhrRY6t0aS/StkKkE/X3RMqUMUMV2E7FcGDmvbH4n9dMsFdupyKME4SQTz/qJZJiRMcJ0a7QwFEOLWFcC7sr5QOmGbfpGNeG4M+fvEjqpeJ50b8s5SrlWRoZckiOSIH45IxUyAWpkTrh5IE8kRfy6jw6z86b8z5tXXJmMwfkD5yPbw8JnRo=</latexit>

HAM SPAM SPAM SPAM HAM

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money viagra nigeria fly nigeria<latexit sha1_base64="Fk40qlV9xSoCXT3K4KQdocdHpiQ=">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</latexit><latexit sha1_base64="Fk40qlV9xSoCXT3K4KQdocdHpiQ=">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</latexit><latexit sha1_base64="Fk40qlV9xSoCXT3K4KQdocdHpiQ=">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</latexit><latexit sha1_base64="Fk40qlV9xSoCXT3K4KQdocdHpiQ=">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</latexit>

p(buy | HAM)<latexit sha1_base64="7V1Kywa3QvRgPIp3raOa0og9KUU=">AAACD3icbVA9SwNBEN3zM55fUUubxUSMTbhLY+wiNmmECJ4J5ELY22ySJbt3x+6cGI78BBv/io2Fiq2tnf/GzYegiQ8GHu/NMDMviAXX4Dhf1tLyyuraembD3tza3tnN7u3f6ihRlHk0EpFqBEQzwUPmAQfBGrFiRAaC1YPB5div3zGleRTewDBmLUl6Ie9ySsBI7exJPm/HBR/YPSiZBslwhH3JO/hHqV5cjU7tfL6dzTlFZwK8SNwZyaEZau3sp9+JaCJZCFQQrZuuE0MrJQo4FWxk+4lmMaED0mNNQ0MimW6lk4dG+NgoHdyNlKkQ8ET9PZESqfVQBqZTEujreW8s/uc1E+iWWykP4wRYSKeLuonAEOFxOrjDFaMghoYQqri5FdM+UYSCydA2IbjzLy8Sr1Q8L7rXpVylPEsjgw7RESogF52hCqqiGvIQRQ/oCb2gV+vRerberPdp65I1mzlAf2B9fAO6ZJtK</latexit><latexit sha1_base64="7V1Kywa3QvRgPIp3raOa0og9KUU=">AAACD3icbVA9SwNBEN3zM55fUUubxUSMTbhLY+wiNmmECJ4J5ELY22ySJbt3x+6cGI78BBv/io2Fiq2tnf/GzYegiQ8GHu/NMDMviAXX4Dhf1tLyyuraembD3tza3tnN7u3f6ihRlHk0EpFqBEQzwUPmAQfBGrFiRAaC1YPB5div3zGleRTewDBmLUl6Ie9ySsBI7exJPm/HBR/YPSiZBslwhH3JO/hHqV5cjU7tfL6dzTlFZwK8SNwZyaEZau3sp9+JaCJZCFQQrZuuE0MrJQo4FWxk+4lmMaED0mNNQ0MimW6lk4dG+NgoHdyNlKkQ8ET9PZESqfVQBqZTEujreW8s/uc1E+iWWykP4wRYSKeLuonAEOFxOrjDFaMghoYQqri5FdM+UYSCydA2IbjzLy8Sr1Q8L7rXpVylPEsjgw7RESogF52hCqqiGvIQRQ/oCb2gV+vRerberPdp65I1mzlAf2B9fAO6ZJtK</latexit><latexit sha1_base64="7V1Kywa3QvRgPIp3raOa0og9KUU=">AAACD3icbVA9SwNBEN3zM55fUUubxUSMTbhLY+wiNmmECJ4J5ELY22ySJbt3x+6cGI78BBv/io2Fiq2tnf/GzYegiQ8GHu/NMDMviAXX4Dhf1tLyyuraembD3tza3tnN7u3f6ihRlHk0EpFqBEQzwUPmAQfBGrFiRAaC1YPB5div3zGleRTewDBmLUl6Ie9ySsBI7exJPm/HBR/YPSiZBslwhH3JO/hHqV5cjU7tfL6dzTlFZwK8SNwZyaEZau3sp9+JaCJZCFQQrZuuE0MrJQo4FWxk+4lmMaED0mNNQ0MimW6lk4dG+NgoHdyNlKkQ8ET9PZESqfVQBqZTEujreW8s/uc1E+iWWykP4wRYSKeLuonAEOFxOrjDFaMghoYQqri5FdM+UYSCydA2IbjzLy8Sr1Q8L7rXpVylPEsjgw7RESogF52hCqqiGvIQRQ/oCb2gV+vRerberPdp65I1mzlAf2B9fAO6ZJtK</latexit><latexit sha1_base64="7V1Kywa3QvRgPIp3raOa0og9KUU=">AAACD3icbVA9SwNBEN3zM55fUUubxUSMTbhLY+wiNmmECJ4J5ELY22ySJbt3x+6cGI78BBv/io2Fiq2tnf/GzYegiQ8GHu/NMDMviAXX4Dhf1tLyyuraembD3tza3tnN7u3f6ihRlHk0EpFqBEQzwUPmAQfBGrFiRAaC1YPB5div3zGleRTewDBmLUl6Ie9ySsBI7exJPm/HBR/YPSiZBslwhH3JO/hHqV5cjU7tfL6dzTlFZwK8SNwZyaEZau3sp9+JaCJZCFQQrZuuE0MrJQo4FWxk+4lmMaED0mNNQ0MimW6lk4dG+NgoHdyNlKkQ8ET9PZESqfVQBqZTEujreW8s/uc1E+iWWykP4wRYSKeLuonAEOFxOrjDFaMghoYQqri5FdM+UYSCydA2IbjzLy8Sr1Q8L7rXpVylPEsjgw7RESogF52hCqqiGvIQRQ/oCb2gV+vRerberPdp65I1mzlAf2B9fAO6ZJtK</latexit>

p(term | Class) = # instances of term in Class

# total words in Class<latexit sha1_base64="bbnQ04sT4KK3Xpp9KaE1bC8h/jM=">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</latexit><latexit sha1_base64="bbnQ04sT4KK3Xpp9KaE1bC8h/jM=">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</latexit><latexit sha1_base64="bbnQ04sT4KK3Xpp9KaE1bC8h/jM=">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</latexit><latexit sha1_base64="bbnQ04sT4KK3Xpp9KaE1bC8h/jM=">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</latexit>

prelbvyl Ham)

= 1

.

.

Page 22: Discrete Naïve Bayes

Naïve Bayes

22

OK, so we can estimate by computing term probabilities and multiplying them together

We still need to estimate the class priors

Two ways to do this:

o Ask an expert what the priors should be (e.g. experts say 80% of all email is SPAM)

o Estimate class priors from the data

p(x | y)<latexit sha1_base64="e63oY/Eqq9SPDAQt2nlP2DaFOyU=">AAAB/3icbVC7TsNAEDyHVzAvAwUFzYkYKTSRnYZHFYmGMkiYRIqt6Hw+J6ecH7o7IyzLDb9CQwGIlt+g42+4JC4gYaSVRjO72t3xU0aFtKxvrbayura+Ud/Ut7Z3dveM/YN7kWQcEwcnLOF9HwnCaEwcSSUj/ZQTFPmM9PzJ9dTvPRAuaBLfyTwlXoRGMQ0pRlJJQ+PINPW0Wbh+CB9L6EY0gPmZbppDo2G1rBngMrEr0gAVukPjyw0SnEUklpghIQa2lUqvQFxSzEipu5kgKcITNCIDRWMUEeEVswdKeKqUAIYJVxVLOFN/TxQoEiKPfNUZITkWi95U/M8bZDK88Aoap5kkMZ4vCjMGZQKnacCAcoIlyxVBmFN1K8RjxBGWKjNdhWAvvrxMnHbrsmXfthudqyqNOjgGJ6AJbHAOOuAGdIEDMCjBM3gFb9qT9qK9ax/z1ppWzRyCP9A+fwA6KpPD</latexit><latexit sha1_base64="e63oY/Eqq9SPDAQt2nlP2DaFOyU=">AAAB/3icbVC7TsNAEDyHVzAvAwUFzYkYKTSRnYZHFYmGMkiYRIqt6Hw+J6ecH7o7IyzLDb9CQwGIlt+g42+4JC4gYaSVRjO72t3xU0aFtKxvrbayura+Ud/Ut7Z3dveM/YN7kWQcEwcnLOF9HwnCaEwcSSUj/ZQTFPmM9PzJ9dTvPRAuaBLfyTwlXoRGMQ0pRlJJQ+PINPW0Wbh+CB9L6EY0gPmZbppDo2G1rBngMrEr0gAVukPjyw0SnEUklpghIQa2lUqvQFxSzEipu5kgKcITNCIDRWMUEeEVswdKeKqUAIYJVxVLOFN/TxQoEiKPfNUZITkWi95U/M8bZDK88Aoap5kkMZ4vCjMGZQKnacCAcoIlyxVBmFN1K8RjxBGWKjNdhWAvvrxMnHbrsmXfthudqyqNOjgGJ6AJbHAOOuAGdIEDMCjBM3gFb9qT9qK9ax/z1ppWzRyCP9A+fwA6KpPD</latexit><latexit sha1_base64="e63oY/Eqq9SPDAQt2nlP2DaFOyU=">AAAB/3icbVC7TsNAEDyHVzAvAwUFzYkYKTSRnYZHFYmGMkiYRIqt6Hw+J6ecH7o7IyzLDb9CQwGIlt+g42+4JC4gYaSVRjO72t3xU0aFtKxvrbayura+Ud/Ut7Z3dveM/YN7kWQcEwcnLOF9HwnCaEwcSSUj/ZQTFPmM9PzJ9dTvPRAuaBLfyTwlXoRGMQ0pRlJJQ+PINPW0Wbh+CB9L6EY0gPmZbppDo2G1rBngMrEr0gAVukPjyw0SnEUklpghIQa2lUqvQFxSzEipu5kgKcITNCIDRWMUEeEVswdKeKqUAIYJVxVLOFN/TxQoEiKPfNUZITkWi95U/M8bZDK88Aoap5kkMZ4vCjMGZQKnacCAcoIlyxVBmFN1K8RjxBGWKjNdhWAvvrxMnHbrsmXfthudqyqNOjgGJ6AJbHAOOuAGdIEDMCjBM3gFb9qT9qK9ax/z1ppWzRyCP9A+fwA6KpPD</latexit><latexit sha1_base64="e63oY/Eqq9SPDAQt2nlP2DaFOyU=">AAAB/3icbVC7TsNAEDyHVzAvAwUFzYkYKTSRnYZHFYmGMkiYRIqt6Hw+J6ecH7o7IyzLDb9CQwGIlt+g42+4JC4gYaSVRjO72t3xU0aFtKxvrbayura+Ud/Ut7Z3dveM/YN7kWQcEwcnLOF9HwnCaEwcSSUj/ZQTFPmM9PzJ9dTvPRAuaBLfyTwlXoRGMQ0pRlJJQ+PINPW0Wbh+CB9L6EY0gPmZbppDo2G1rBngMrEr0gAVukPjyw0SnEUklpghIQa2lUqvQFxSzEipu5kgKcITNCIDRWMUEeEVswdKeKqUAIYJVxVLOFN/TxQoEiKPfNUZITkWi95U/M8bZDK88Aoap5kkMZ4vCjMGZQKnacCAcoIlyxVBmFN1K8RjxBGWKjNdhWAvvrxMnHbrsmXfthudqyqNOjgGJ6AJbHAOOuAGdIEDMCjBM3gFb9qT9qK9ax/z1ppWzRyCP9A+fwA6KpPD</latexit>

p(y)<latexit sha1_base64="QRmTWXEjsPAIAeIgaYzD6yXBXNo=">AAAB8HicbVBNT8JAEJ3iF9Yv1KOXjdQEL6Tl4seJxItHTKwQoSHbZQsbtttmd2tCGv6FFw9qvPpzvPlvXKAHBV8yyct7M5mZF6acKe2631ZpbX1jc6u8be/s7u0fVA6PHlSSSUJ9kvBEdkKsKGeC+pppTjuppDgOOW2H45uZ336iUrFE3OtJSoMYDwWLGMHaSI+OY6e1ybntOP1K1a27c6BV4hWkCgVa/cpXb5CQLKZCE46V6npuqoMcS80Ip1O7lymaYjLGQ9o1VOCYqiCfXzxFZ0YZoCiRpoRGc/X3RI5jpSZxaDpjrEdq2ZuJ/3ndTEeXQc5EmmkqyGJRlHGkEzR7Hw2YpETziSGYSGZuRWSEJSbahGSbELzll1eJ36hf1b27RrV5XaRRhhM4hRp4cAFNuIUW+EBAwDO8wpulrBfr3fpYtJasYuYY/sD6/AHD4o6G</latexit><latexit sha1_base64="QRmTWXEjsPAIAeIgaYzD6yXBXNo=">AAAB8HicbVBNT8JAEJ3iF9Yv1KOXjdQEL6Tl4seJxItHTKwQoSHbZQsbtttmd2tCGv6FFw9qvPpzvPlvXKAHBV8yyct7M5mZF6acKe2631ZpbX1jc6u8be/s7u0fVA6PHlSSSUJ9kvBEdkKsKGeC+pppTjuppDgOOW2H45uZ336iUrFE3OtJSoMYDwWLGMHaSI+OY6e1ybntOP1K1a27c6BV4hWkCgVa/cpXb5CQLKZCE46V6npuqoMcS80Ip1O7lymaYjLGQ9o1VOCYqiCfXzxFZ0YZoCiRpoRGc/X3RI5jpSZxaDpjrEdq2ZuJ/3ndTEeXQc5EmmkqyGJRlHGkEzR7Hw2YpETziSGYSGZuRWSEJSbahGSbELzll1eJ36hf1b27RrV5XaRRhhM4hRp4cAFNuIUW+EBAwDO8wpulrBfr3fpYtJasYuYY/sD6/AHD4o6G</latexit><latexit sha1_base64="QRmTWXEjsPAIAeIgaYzD6yXBXNo=">AAAB8HicbVBNT8JAEJ3iF9Yv1KOXjdQEL6Tl4seJxItHTKwQoSHbZQsbtttmd2tCGv6FFw9qvPpzvPlvXKAHBV8yyct7M5mZF6acKe2631ZpbX1jc6u8be/s7u0fVA6PHlSSSUJ9kvBEdkKsKGeC+pppTjuppDgOOW2H45uZ336iUrFE3OtJSoMYDwWLGMHaSI+OY6e1ybntOP1K1a27c6BV4hWkCgVa/cpXb5CQLKZCE46V6npuqoMcS80Ip1O7lymaYjLGQ9o1VOCYqiCfXzxFZ0YZoCiRpoRGc/X3RI5jpSZxaDpjrEdq2ZuJ/3ndTEeXQc5EmmkqyGJRlHGkEzR7Hw2YpETziSGYSGZuRWSEJSbahGSbELzll1eJ36hf1b27RrV5XaRRhhM4hRp4cAFNuIUW+EBAwDO8wpulrBfr3fpYtJasYuYY/sD6/AHD4o6G</latexit><latexit sha1_base64="QRmTWXEjsPAIAeIgaYzD6yXBXNo=">AAAB8HicbVBNT8JAEJ3iF9Yv1KOXjdQEL6Tl4seJxItHTKwQoSHbZQsbtttmd2tCGv6FFw9qvPpzvPlvXKAHBV8yyct7M5mZF6acKe2631ZpbX1jc6u8be/s7u0fVA6PHlSSSUJ9kvBEdkKsKGeC+pppTjuppDgOOW2H45uZ336iUrFE3OtJSoMYDwWLGMHaSI+OY6e1ybntOP1K1a27c6BV4hWkCgVa/cpXb5CQLKZCE46V6npuqoMcS80Ip1O7lymaYjLGQ9o1VOCYqiCfXzxFZ0YZoCiRpoRGc/X3RI5jpSZxaDpjrEdq2ZuJ/3ndTEeXQc5EmmkqyGJRlHGkEzR7Hw2YpETziSGYSGZuRWSEJSbahGSbELzll1eJ36hf1b27RrV5XaRRhhM4hRp4cAFNuIUW+EBAwDO8wpulrBfr3fpYtJasYuYY/sD6/AHD4o6G</latexit>

p(Class) =# emails from Class

# total emails in training data<latexit sha1_base64="Z1r8YPJ8UJPhpP0yhyUjhAEr738=">AAACWHicbVFNSxxBFOwZ49ck6mqOuTTZDZjLMuNFPQiCF48GslHYWZY3PW92G7t7hu434jLMn/QgiH/FQ3o/BKMpaCiq6vG6q7NKSUdx/BSEa5/WNza3tqPPX3Z29zr7B39cWVuBA1Gq0t5k4FBJgwOSpPCmsgg6U3id3V7M/es7tE6W5jfNKhxpmBhZSAHkpXGn6vWidArUVO1hSnhPVjcXCpxrf/IznuaFBdG8GmmXowapHC9sqfky1761qSRQryFpOFmQRpoJz4GgbaNeb9zpxv14Af6RJCvSZStcjTsPaV6KWqMhMV84TOKKRg1YkkJhG6W1wwrELUxw6KkBjW7ULJpp+Q+v5LworT+G+EJ9O9GAdm6mM5/UQFP33puL//OGNRUno0aaqiY0YrmoqJUvgM9r5rm0KEjNPAFhpb8rF1PwZZL/jMiXkLx/8kcyOOqf9pNfR93zk1UbW+wb+84OWcKO2Tm7ZFdswAR7ZC/BerARPIdBuBluL6NhsJr5yv5BePAXw0C0BQ==</latexit><latexit sha1_base64="Z1r8YPJ8UJPhpP0yhyUjhAEr738=">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</latexit><latexit sha1_base64="Z1r8YPJ8UJPhpP0yhyUjhAEr738=">AAACWHicbVFNSxxBFOwZ49ck6mqOuTTZDZjLMuNFPQiCF48GslHYWZY3PW92G7t7hu434jLMn/QgiH/FQ3o/BKMpaCiq6vG6q7NKSUdx/BSEa5/WNza3tqPPX3Z29zr7B39cWVuBA1Gq0t5k4FBJgwOSpPCmsgg6U3id3V7M/es7tE6W5jfNKhxpmBhZSAHkpXGn6vWidArUVO1hSnhPVjcXCpxrf/IznuaFBdG8GmmXowapHC9sqfky1761qSRQryFpOFmQRpoJz4GgbaNeb9zpxv14Af6RJCvSZStcjTsPaV6KWqMhMV84TOKKRg1YkkJhG6W1wwrELUxw6KkBjW7ULJpp+Q+v5LworT+G+EJ9O9GAdm6mM5/UQFP33puL//OGNRUno0aaqiY0YrmoqJUvgM9r5rm0KEjNPAFhpb8rF1PwZZL/jMiXkLx/8kcyOOqf9pNfR93zk1UbW+wb+84OWcKO2Tm7ZFdswAR7ZC/BerARPIdBuBluL6NhsJr5yv5BePAXw0C0BQ==</latexit><latexit sha1_base64="Z1r8YPJ8UJPhpP0yhyUjhAEr738=">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</latexit> =

Page 23: Discrete Naïve Bayes

Naïve Bayes

23

To estimate we just have to count emails in training data

Example: Estimate and from the training data

HAM SPAM SPAM SPAM HAM

work nigeria fly money fly

buy opportunity buy buy home

money viagra nigeria fly nigeria<latexit sha1_base64="Fk40qlV9xSoCXT3K4KQdocdHpiQ=">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</latexit><latexit sha1_base64="Fk40qlV9xSoCXT3K4KQdocdHpiQ=">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</latexit><latexit sha1_base64="Fk40qlV9xSoCXT3K4KQdocdHpiQ=">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</latexit><latexit sha1_base64="Fk40qlV9xSoCXT3K4KQdocdHpiQ=">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</latexit>

p(Class)<latexit sha1_base64="F2nmYhtpiA5FcPOeLbYXcse6EKU=">AAAB/3icbVC7TsNAEDzzDOZloKCgsYiRQhPZaQhdpDSUQcIkUmJF58s5OeX80N0aEVlu+BUaCkC0/AYdf8M5cQEJI600mtm92x0/4UyCbX9ra+sbm1vblR19d2//4NA4Or6XcSoIdUnMY9HzsaScRdQFBpz2EkFx6HPa9aftwu8+UCFZHN3BLKFeiMcRCxjBoKShcWpZelIbAH0EEWZtjqXML3XLGhpVu27PYa4SpyRVVKIzNL4Go5ikIY2AFK/0HTsBL8MCGOE01weppAkmUzymfUUjHFLpZfMDcvNCKSMziIWqCMy5+nsiw6GUs9BXnSGGiVz2CvE/r59C0PQyFiUp0IgsPgpSbkJsFmmYIyYoAT5TBBPB1K4mmWCBCajMdBWCs3zyKnEb9eu6c9uotpplGhV0hs5RDTnoCrXQDeogFxGUo2f0it60J+1Fe9c+Fq1rWjlzgv5A+/wBpTyUrA==</latexit><latexit sha1_base64="F2nmYhtpiA5FcPOeLbYXcse6EKU=">AAAB/3icbVC7TsNAEDzzDOZloKCgsYiRQhPZaQhdpDSUQcIkUmJF58s5OeX80N0aEVlu+BUaCkC0/AYdf8M5cQEJI600mtm92x0/4UyCbX9ra+sbm1vblR19d2//4NA4Or6XcSoIdUnMY9HzsaScRdQFBpz2EkFx6HPa9aftwu8+UCFZHN3BLKFeiMcRCxjBoKShcWpZelIbAH0EEWZtjqXML3XLGhpVu27PYa4SpyRVVKIzNL4Go5ikIY2AFK/0HTsBL8MCGOE01weppAkmUzymfUUjHFLpZfMDcvNCKSMziIWqCMy5+nsiw6GUs9BXnSGGiVz2CvE/r59C0PQyFiUp0IgsPgpSbkJsFmmYIyYoAT5TBBPB1K4mmWCBCajMdBWCs3zyKnEb9eu6c9uotpplGhV0hs5RDTnoCrXQDeogFxGUo2f0it60J+1Fe9c+Fq1rWjlzgv5A+/wBpTyUrA==</latexit><latexit sha1_base64="F2nmYhtpiA5FcPOeLbYXcse6EKU=">AAAB/3icbVC7TsNAEDzzDOZloKCgsYiRQhPZaQhdpDSUQcIkUmJF58s5OeX80N0aEVlu+BUaCkC0/AYdf8M5cQEJI600mtm92x0/4UyCbX9ra+sbm1vblR19d2//4NA4Or6XcSoIdUnMY9HzsaScRdQFBpz2EkFx6HPa9aftwu8+UCFZHN3BLKFeiMcRCxjBoKShcWpZelIbAH0EEWZtjqXML3XLGhpVu27PYa4SpyRVVKIzNL4Go5ikIY2AFK/0HTsBL8MCGOE01weppAkmUzymfUUjHFLpZfMDcvNCKSMziIWqCMy5+nsiw6GUs9BXnSGGiVz2CvE/r59C0PQyFiUp0IgsPgpSbkJsFmmYIyYoAT5TBBPB1K4mmWCBCajMdBWCs3zyKnEb9eu6c9uotpplGhV0hs5RDTnoCrXQDeogFxGUo2f0it60J+1Fe9c+Fq1rWjlzgv5A+/wBpTyUrA==</latexit><latexit sha1_base64="F2nmYhtpiA5FcPOeLbYXcse6EKU=">AAAB/3icbVC7TsNAEDzzDOZloKCgsYiRQhPZaQhdpDSUQcIkUmJF58s5OeX80N0aEVlu+BUaCkC0/AYdf8M5cQEJI600mtm92x0/4UyCbX9ra+sbm1vblR19d2//4NA4Or6XcSoIdUnMY9HzsaScRdQFBpz2EkFx6HPa9aftwu8+UCFZHN3BLKFeiMcRCxjBoKShcWpZelIbAH0EEWZtjqXML3XLGhpVu27PYa4SpyRVVKIzNL4Go5ikIY2AFK/0HTsBL8MCGOE01weppAkmUzymfUUjHFLpZfMDcvNCKSMziIWqCMy5+nsiw6GUs9BXnSGGiVz2CvE/r59C0PQyFiUp0IgsPgpSbkJsFmmYIyYoAT5TBBPB1K4mmWCBCajMdBWCs3zyKnEb9eu6c9uotpplGhV0hs5RDTnoCrXQDeogFxGUo2f0it60J+1Fe9c+Fq1rWjlzgv5A+/wBpTyUrA==</latexit>

p(SPAM)<latexit sha1_base64="bQm9B7iPIynqeiCW6wq5W2BgrKI=">AAAB/nicbVBNS8NAEN3Urxq/ooIXL8FGqJeS9GK9Vbx4ESoaW2hD2Ww37dLNB7sTscQe/CtePKh49Xd489+4bXPQ1gcDj/dmmJnnJ5xJsO1vrbC0vLK6VlzXNza3tneM3b07GaeCUJfEPBYtH0vKWURdYMBpKxEUhz6nTX94MfGb91RIFke3MEqoF+J+xAJGMCipaxxYlp6UO0AfQITZTeP8anyiW1bXKNkVewpzkTg5KaEcja7x1enFJA1pBIRjKduOnYCXYQGMcDrWO6mkCSZD3KdtRSMcUull0/vH5rFSemYQC1URmFP190SGQylHoa86QwwDOe9NxP+8dgpBzctYlKRAIzJbFKTchNichGH2mKAE+EgRTARTt5pkgAUmoCLTVQjO/MuLxK1WzirOdbVUr+VpFNEhOkJl5KBTVEeXqIFcRNAjekav6E170l60d+1j1lrQ8pl99Afa5w9MSZPd</latexit><latexit sha1_base64="bQm9B7iPIynqeiCW6wq5W2BgrKI=">AAAB/nicbVBNS8NAEN3Urxq/ooIXL8FGqJeS9GK9Vbx4ESoaW2hD2Ww37dLNB7sTscQe/CtePKh49Xd489+4bXPQ1gcDj/dmmJnnJ5xJsO1vrbC0vLK6VlzXNza3tneM3b07GaeCUJfEPBYtH0vKWURdYMBpKxEUhz6nTX94MfGb91RIFke3MEqoF+J+xAJGMCipaxxYlp6UO0AfQITZTeP8anyiW1bXKNkVewpzkTg5KaEcja7x1enFJA1pBIRjKduOnYCXYQGMcDrWO6mkCSZD3KdtRSMcUull0/vH5rFSemYQC1URmFP190SGQylHoa86QwwDOe9NxP+8dgpBzctYlKRAIzJbFKTchNichGH2mKAE+EgRTARTt5pkgAUmoCLTVQjO/MuLxK1WzirOdbVUr+VpFNEhOkJl5KBTVEeXqIFcRNAjekav6E170l60d+1j1lrQ8pl99Afa5w9MSZPd</latexit><latexit sha1_base64="bQm9B7iPIynqeiCW6wq5W2BgrKI=">AAAB/nicbVBNS8NAEN3Urxq/ooIXL8FGqJeS9GK9Vbx4ESoaW2hD2Ww37dLNB7sTscQe/CtePKh49Xd489+4bXPQ1gcDj/dmmJnnJ5xJsO1vrbC0vLK6VlzXNza3tneM3b07GaeCUJfEPBYtH0vKWURdYMBpKxEUhz6nTX94MfGb91RIFke3MEqoF+J+xAJGMCipaxxYlp6UO0AfQITZTeP8anyiW1bXKNkVewpzkTg5KaEcja7x1enFJA1pBIRjKduOnYCXYQGMcDrWO6mkCSZD3KdtRSMcUull0/vH5rFSemYQC1URmFP190SGQylHoa86QwwDOe9NxP+8dgpBzctYlKRAIzJbFKTchNichGH2mKAE+EgRTARTt5pkgAUmoCLTVQjO/MuLxK1WzirOdbVUr+VpFNEhOkJl5KBTVEeXqIFcRNAjekav6E170l60d+1j1lrQ8pl99Afa5w9MSZPd</latexit><latexit sha1_base64="bQm9B7iPIynqeiCW6wq5W2BgrKI=">AAAB/nicbVBNS8NAEN3Urxq/ooIXL8FGqJeS9GK9Vbx4ESoaW2hD2Ww37dLNB7sTscQe/CtePKh49Xd489+4bXPQ1gcDj/dmmJnnJ5xJsO1vrbC0vLK6VlzXNza3tneM3b07GaeCUJfEPBYtH0vKWURdYMBpKxEUhz6nTX94MfGb91RIFke3MEqoF+J+xAJGMCipaxxYlp6UO0AfQITZTeP8anyiW1bXKNkVewpzkTg5KaEcja7x1enFJA1pBIRjKduOnYCXYQGMcDrWO6mkCSZD3KdtRSMcUull0/vH5rFSemYQC1URmFP190SGQylHoa86QwwDOe9NxP+8dgpBzctYlKRAIzJbFKTchNichGH2mKAE+EgRTARTt5pkgAUmoCLTVQjO/MuLxK1WzirOdbVUr+VpFNEhOkJl5KBTVEeXqIFcRNAjekav6E170l60d+1j1lrQ8pl99Afa5w9MSZPd</latexit>

p(HAM)<latexit sha1_base64="d6eLcIwNrFEk9U5QtNGGgNhYkIg=">AAAB/XicbVBNS8NAEN34WeNXVDx5CbZCvZSkF+ut4qUXoYKxhTaUzXbTLt1Nwu5ELKHgX/HiQcWr/8Ob/8Ztm4O2Phh4vDfDzLwg4UyB43wbK6tr6xubhS1ze2d3b986OLxXcSoJ9UjMY9kOsKKcRdQDBpy2E0mxCDhtBaPrqd96oFKxOLqDcUJ9gQcRCxnBoKWedVwqmUm5C/QRpMgaVzeTc7NU6llFp+LMYC8TNydFlKPZs766/ZikgkZAOFaq4zoJ+BmWwAinE7ObKppgMsID2tE0woIqP5udP7HPtNK3w1jqisCeqb8nMiyUGotAdwoMQ7XoTcX/vE4KYc3PWJSkQCMyXxSm3IbYnmZh95mkBPhYE0wk07faZIglJqATM3UI7uLLy8SrVi4r7m21WK/laRTQCTpFZeSiC1RHDdREHiIoQ8/oFb0ZT8aL8W58zFtXjHzmCP2B8fkDlkqTeA==</latexit><latexit sha1_base64="d6eLcIwNrFEk9U5QtNGGgNhYkIg=">AAAB/XicbVBNS8NAEN34WeNXVDx5CbZCvZSkF+ut4qUXoYKxhTaUzXbTLt1Nwu5ELKHgX/HiQcWr/8Ob/8Ztm4O2Phh4vDfDzLwg4UyB43wbK6tr6xubhS1ze2d3b986OLxXcSoJ9UjMY9kOsKKcRdQDBpy2E0mxCDhtBaPrqd96oFKxOLqDcUJ9gQcRCxnBoKWedVwqmUm5C/QRpMgaVzeTc7NU6llFp+LMYC8TNydFlKPZs766/ZikgkZAOFaq4zoJ+BmWwAinE7ObKppgMsID2tE0woIqP5udP7HPtNK3w1jqisCeqb8nMiyUGotAdwoMQ7XoTcX/vE4KYc3PWJSkQCMyXxSm3IbYnmZh95mkBPhYE0wk07faZIglJqATM3UI7uLLy8SrVi4r7m21WK/laRTQCTpFZeSiC1RHDdREHiIoQ8/oFb0ZT8aL8W58zFtXjHzmCP2B8fkDlkqTeA==</latexit><latexit sha1_base64="d6eLcIwNrFEk9U5QtNGGgNhYkIg=">AAAB/XicbVBNS8NAEN34WeNXVDx5CbZCvZSkF+ut4qUXoYKxhTaUzXbTLt1Nwu5ELKHgX/HiQcWr/8Ob/8Ztm4O2Phh4vDfDzLwg4UyB43wbK6tr6xubhS1ze2d3b986OLxXcSoJ9UjMY9kOsKKcRdQDBpy2E0mxCDhtBaPrqd96oFKxOLqDcUJ9gQcRCxnBoKWedVwqmUm5C/QRpMgaVzeTc7NU6llFp+LMYC8TNydFlKPZs766/ZikgkZAOFaq4zoJ+BmWwAinE7ObKppgMsID2tE0woIqP5udP7HPtNK3w1jqisCeqb8nMiyUGotAdwoMQ7XoTcX/vE4KYc3PWJSkQCMyXxSm3IbYnmZh95mkBPhYE0wk07faZIglJqATM3UI7uLLy8SrVi4r7m21WK/laRTQCTpFZeSiC1RHDdREHiIoQ8/oFb0ZT8aL8W58zFtXjHzmCP2B8fkDlkqTeA==</latexit><latexit sha1_base64="d6eLcIwNrFEk9U5QtNGGgNhYkIg=">AAAB/XicbVBNS8NAEN34WeNXVDx5CbZCvZSkF+ut4qUXoYKxhTaUzXbTLt1Nwu5ELKHgX/HiQcWr/8Ob/8Ztm4O2Phh4vDfDzLwg4UyB43wbK6tr6xubhS1ze2d3b986OLxXcSoJ9UjMY9kOsKKcRdQDBpy2E0mxCDhtBaPrqd96oFKxOLqDcUJ9gQcRCxnBoKWedVwqmUm5C/QRpMgaVzeTc7NU6llFp+LMYC8TNydFlKPZs766/ZikgkZAOFaq4zoJ+BmWwAinE7ObKppgMsID2tE0woIqP5udP7HPtNK3w1jqisCeqb8nMiyUGotAdwoMQ7XoTcX/vE4KYc3PWJSkQCMyXxSm3IbYnmZh95mkBPhYE0wk07faZIglJqATM3UI7uLLy8SrVi4r7m21WK/laRTQCTpFZeSiC1RHDdREHiIoQ8/oFb0ZT8aL8W58zFtXjHzmCP2B8fkDlkqTeA==</latexit>

p(Class) =# emails from Class

# total emails in training data<latexit sha1_base64="Z1r8YPJ8UJPhpP0yhyUjhAEr738=">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</latexit><latexit sha1_base64="Z1r8YPJ8UJPhpP0yhyUjhAEr738=">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</latexit><latexit sha1_base64="Z1r8YPJ8UJPhpP0yhyUjhAEr738=">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</latexit><latexit sha1_base64="Z1r8YPJ8UJPhpP0yhyUjhAEr738=">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</latexit>

pratam) = 2/5

p ( spam ) = %

Page 24: Discrete Naïve Bayes

Naïve Bayes

24

Putting it all together:

Example: Compute the HAM score for

HAM SPAM SPAM SPAM HAM

work nigeria fly money fly

buy opportunity buy buy home

money viagra nigeria fly nigeria<latexit sha1_base64="Fk40qlV9xSoCXT3K4KQdocdHpiQ=">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</latexit><latexit sha1_base64="Fk40qlV9xSoCXT3K4KQdocdHpiQ=">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</latexit><latexit sha1_base64="Fk40qlV9xSoCXT3K4KQdocdHpiQ=">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</latexit><latexit sha1_base64="Fk40qlV9xSoCXT3K4KQdocdHpiQ=">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</latexit>

x = [work, nigeria]<latexit sha1_base64="/Ej8vUq3Hxea8/xUroSujYyv1ro=">AAACEHicbVA9SwNBEN3zM55fUUubxUSwkHCXxlgIARtLBaNCcoS9zVxcsrd37M5pwpG/YONfsbFQsbW089+4iSk08cHA470ZZuaFqRQGPe/LmZtfWFxaLqy4q2vrG5vFre0rk2SaQ4MnMtE3ITMghYIGCpRwk2pgcSjhOuydjvzrO9BGJOoSBykEMesqEQnO0Ert4kG57OatMKL9IT2hLYQ+6jhv3ie6d0iV6IIWLBi65XK7WPIq3hh0lvgTUiITnLeLn61OwrMYFHLJjGn6XopBzjQKLmHotjIDKeM91oWmpYrFYIJ8/NGQ7lulQ6NE21JIx+rviZzFxgzi0HbGDG/NtDcS//OaGUa1IBcqzRAU/1kUZZJiQkfx0I7QwFEOLGFcC3sr5bdMM442RNeG4E+/PEsa1cpxxb+oluq1SRoFskv2yAHxyRGpkzNyThqEkwfyRF7Iq/PoPDtvzvtP65wzmdkhf+B8fAMRxJt2</latexit><latexit sha1_base64="/Ej8vUq3Hxea8/xUroSujYyv1ro=">AAACEHicbVA9SwNBEN3zM55fUUubxUSwkHCXxlgIARtLBaNCcoS9zVxcsrd37M5pwpG/YONfsbFQsbW089+4iSk08cHA470ZZuaFqRQGPe/LmZtfWFxaLqy4q2vrG5vFre0rk2SaQ4MnMtE3ITMghYIGCpRwk2pgcSjhOuydjvzrO9BGJOoSBykEMesqEQnO0Ert4kG57OatMKL9IT2hLYQ+6jhv3ie6d0iV6IIWLBi65XK7WPIq3hh0lvgTUiITnLeLn61OwrMYFHLJjGn6XopBzjQKLmHotjIDKeM91oWmpYrFYIJ8/NGQ7lulQ6NE21JIx+rviZzFxgzi0HbGDG/NtDcS//OaGUa1IBcqzRAU/1kUZZJiQkfx0I7QwFEOLGFcC3sr5bdMM442RNeG4E+/PEsa1cpxxb+oluq1SRoFskv2yAHxyRGpkzNyThqEkwfyRF7Iq/PoPDtvzvtP65wzmdkhf+B8fAMRxJt2</latexit><latexit sha1_base64="/Ej8vUq3Hxea8/xUroSujYyv1ro=">AAACEHicbVA9SwNBEN3zM55fUUubxUSwkHCXxlgIARtLBaNCcoS9zVxcsrd37M5pwpG/YONfsbFQsbW089+4iSk08cHA470ZZuaFqRQGPe/LmZtfWFxaLqy4q2vrG5vFre0rk2SaQ4MnMtE3ITMghYIGCpRwk2pgcSjhOuydjvzrO9BGJOoSBykEMesqEQnO0Ert4kG57OatMKL9IT2hLYQ+6jhv3ie6d0iV6IIWLBi65XK7WPIq3hh0lvgTUiITnLeLn61OwrMYFHLJjGn6XopBzjQKLmHotjIDKeM91oWmpYrFYIJ8/NGQ7lulQ6NE21JIx+rviZzFxgzi0HbGDG/NtDcS//OaGUa1IBcqzRAU/1kUZZJiQkfx0I7QwFEOLGFcC3sr5bdMM442RNeG4E+/PEsa1cpxxb+oluq1SRoFskv2yAHxyRGpkzNyThqEkwfyRF7Iq/PoPDtvzvtP65wzmdkhf+B8fAMRxJt2</latexit><latexit sha1_base64="/Ej8vUq3Hxea8/xUroSujYyv1ro=">AAACEHicbVA9SwNBEN3zM55fUUubxUSwkHCXxlgIARtLBaNCcoS9zVxcsrd37M5pwpG/YONfsbFQsbW089+4iSk08cHA470ZZuaFqRQGPe/LmZtfWFxaLqy4q2vrG5vFre0rk2SaQ4MnMtE3ITMghYIGCpRwk2pgcSjhOuydjvzrO9BGJOoSBykEMesqEQnO0Ert4kG57OatMKL9IT2hLYQ+6jhv3ie6d0iV6IIWLBi65XK7WPIq3hh0lvgTUiITnLeLn61OwrMYFHLJjGn6XopBzjQKLmHotjIDKeM91oWmpYrFYIJ8/NGQ7lulQ6NE21JIx+rviZzFxgzi0HbGDG/NtDcS//OaGUa1IBcqzRAU/1kUZZJiQkfx0I7QwFEOLGFcC3sr5bdMM442RNeG4E+/PEsa1cpxxb+oluq1SRoFskv2yAHxyRGpkzNyThqEkwfyRF7Iq/PoPDtvzvtP65wzmdkhf+B8fAMRxJt2</latexit>

p(HAM | x) / p(x | HAM)p(HAM) =<latexit sha1_base64="zvvC1tqzjmLw/uYQ82TJZWtMloU=">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</latexit><latexit sha1_base64="zvvC1tqzjmLw/uYQ82TJZWtMloU=">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</latexit><latexit sha1_base64="zvvC1tqzjmLw/uYQ82TJZWtMloU=">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</latexit><latexit sha1_base64="zvvC1tqzjmLw/uYQ82TJZWtMloU=">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</latexit>

p(work | HAM) · p(nigeria | HAM) · p(HAM) =<latexit sha1_base64="gtcvuUAWp8zRUWXgElE1lF7IbV8=">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</latexit><latexit sha1_base64="gtcvuUAWp8zRUWXgElE1lF7IbV8=">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</latexit><latexit sha1_base64="gtcvuUAWp8zRUWXgElE1lF7IbV8=">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</latexit><latexit sha1_base64="gtcvuUAWp8zRUWXgElE1lF7IbV8=">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</latexit> ⇐to . ;

those 's

Page 25: Discrete Naïve Bayes

Naïve Bayes

25

Putting it all together:

Example: Compute the SPAM score for

HAM SPAM SPAM SPAM HAM

work nigeria fly money fly

buy opportunity buy buy home

money viagra nigeria fly nigeria<latexit sha1_base64="Fk40qlV9xSoCXT3K4KQdocdHpiQ=">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</latexit><latexit sha1_base64="Fk40qlV9xSoCXT3K4KQdocdHpiQ=">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</latexit><latexit sha1_base64="Fk40qlV9xSoCXT3K4KQdocdHpiQ=">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</latexit><latexit sha1_base64="Fk40qlV9xSoCXT3K4KQdocdHpiQ=">AAADaXiclVJLj9MwEHYTHktgobtIgOBi0YL2VCV74XFaxGUvSEVQdqWmqhx3klp17MieLEShEr+RG7+ACz8CpxtWfXDZsSJ93zcPjyeTFFJYDMNfHc+/cfPW7b07wd17+/cfdA8Ov1hdGg4jrqU25wmzIIWCEQqUcF4YYHki4SxZvG/8ZxdgrNDqM1YFTHKWKZEKztBJ04POj34/iBPIhKqZMaxa1vx7e5ZBjPANTV6fvvuwpC/pP/ppeD2+So/jIJ43bdKrsl+1WazHKZGBEWxdSmW1TnOtoNrxN6VbnpQbbl0U2mCpBG7IW1FbdK5z2Ci6c+uFYJlh12v9KiKIQc3aYQf9/rTbCwfhyuguiFrQI60Np92f8UzzMgeFXDJrx1FY4MQVRMEluJ9WWigYX7AMxg4qloOd1KtVWdIXTpnRVBv3KaQrdT2jZrm1VZ64yJzh3G77GvF/vnGJ6etJLVRRIih+eVFaSoqaNntHZ8IAR1k5wLgRrlfK58wwjm47AzeEaPvJu2B0PHgziD4e907ettPYI8/Ic3JEIvKKnJBTMiQjwju/vX3vkffY++Mf+k/8p5ehXqfNeUg2zO/9BU15Giw=</latexit>

x = [work, nigeria]<latexit sha1_base64="/Ej8vUq3Hxea8/xUroSujYyv1ro=">AAACEHicbVA9SwNBEN3zM55fUUubxUSwkHCXxlgIARtLBaNCcoS9zVxcsrd37M5pwpG/YONfsbFQsbW089+4iSk08cHA470ZZuaFqRQGPe/LmZtfWFxaLqy4q2vrG5vFre0rk2SaQ4MnMtE3ITMghYIGCpRwk2pgcSjhOuydjvzrO9BGJOoSBykEMesqEQnO0Ert4kG57OatMKL9IT2hLYQ+6jhv3ie6d0iV6IIWLBi65XK7WPIq3hh0lvgTUiITnLeLn61OwrMYFHLJjGn6XopBzjQKLmHotjIDKeM91oWmpYrFYIJ8/NGQ7lulQ6NE21JIx+rviZzFxgzi0HbGDG/NtDcS//OaGUa1IBcqzRAU/1kUZZJiQkfx0I7QwFEOLGFcC3sr5bdMM442RNeG4E+/PEsa1cpxxb+oluq1SRoFskv2yAHxyRGpkzNyThqEkwfyRF7Iq/PoPDtvzvtP65wzmdkhf+B8fAMRxJt2</latexit><latexit sha1_base64="/Ej8vUq3Hxea8/xUroSujYyv1ro=">AAACEHicbVA9SwNBEN3zM55fUUubxUSwkHCXxlgIARtLBaNCcoS9zVxcsrd37M5pwpG/YONfsbFQsbW089+4iSk08cHA470ZZuaFqRQGPe/LmZtfWFxaLqy4q2vrG5vFre0rk2SaQ4MnMtE3ITMghYIGCpRwk2pgcSjhOuydjvzrO9BGJOoSBykEMesqEQnO0Ert4kG57OatMKL9IT2hLYQ+6jhv3ie6d0iV6IIWLBi65XK7WPIq3hh0lvgTUiITnLeLn61OwrMYFHLJjGn6XopBzjQKLmHotjIDKeM91oWmpYrFYIJ8/NGQ7lulQ6NE21JIx+rviZzFxgzi0HbGDG/NtDcS//OaGUa1IBcqzRAU/1kUZZJiQkfx0I7QwFEOLGFcC3sr5bdMM442RNeG4E+/PEsa1cpxxb+oluq1SRoFskv2yAHxyRGpkzNyThqEkwfyRF7Iq/PoPDtvzvtP65wzmdkhf+B8fAMRxJt2</latexit><latexit sha1_base64="/Ej8vUq3Hxea8/xUroSujYyv1ro=">AAACEHicbVA9SwNBEN3zM55fUUubxUSwkHCXxlgIARtLBaNCcoS9zVxcsrd37M5pwpG/YONfsbFQsbW089+4iSk08cHA470ZZuaFqRQGPe/LmZtfWFxaLqy4q2vrG5vFre0rk2SaQ4MnMtE3ITMghYIGCpRwk2pgcSjhOuydjvzrO9BGJOoSBykEMesqEQnO0Ert4kG57OatMKL9IT2hLYQ+6jhv3ie6d0iV6IIWLBi65XK7WPIq3hh0lvgTUiITnLeLn61OwrMYFHLJjGn6XopBzjQKLmHotjIDKeM91oWmpYrFYIJ8/NGQ7lulQ6NE21JIx+rviZzFxgzi0HbGDG/NtDcS//OaGUa1IBcqzRAU/1kUZZJiQkfx0I7QwFEOLGFcC3sr5bdMM442RNeG4E+/PEsa1cpxxb+oluq1SRoFskv2yAHxyRGpkzNyThqEkwfyRF7Iq/PoPDtvzvtP65wzmdkhf+B8fAMRxJt2</latexit><latexit sha1_base64="/Ej8vUq3Hxea8/xUroSujYyv1ro=">AAACEHicbVA9SwNBEN3zM55fUUubxUSwkHCXxlgIARtLBaNCcoS9zVxcsrd37M5pwpG/YONfsbFQsbW089+4iSk08cHA470ZZuaFqRQGPe/LmZtfWFxaLqy4q2vrG5vFre0rk2SaQ4MnMtE3ITMghYIGCpRwk2pgcSjhOuydjvzrO9BGJOoSBykEMesqEQnO0Ert4kG57OatMKL9IT2hLYQ+6jhv3ie6d0iV6IIWLBi65XK7WPIq3hh0lvgTUiITnLeLn61OwrMYFHLJjGn6XopBzjQKLmHotjIDKeM91oWmpYrFYIJ8/NGQ7lulQ6NE21JIx+rviZzFxgzi0HbGDG/NtDcS//OaGUa1IBcqzRAU/1kUZZJiQkfx0I7QwFEOLGFcC3sr5bdMM442RNeG4E+/PEsa1cpxxb+oluq1SRoFskv2yAHxyRGpkzNyThqEkwfyRF7Iq/PoPDtvzvtP65wzmdkhf+B8fAMRxJt2</latexit>

p(work | SPAM) · p(nigeria | SPAM) · p(SPAM) =<latexit sha1_base64="4dPAXFQgOovq5Kgu3fGjeXWsikk=">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</latexit><latexit sha1_base64="4dPAXFQgOovq5Kgu3fGjeXWsikk=">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</latexit><latexit sha1_base64="4dPAXFQgOovq5Kgu3fGjeXWsikk=">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</latexit><latexit sha1_base64="4dPAXFQgOovq5Kgu3fGjeXWsikk=">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</latexit>

p(SPAM | x) / p(x | SPAM)p(SPAM) =<latexit sha1_base64="z1+SUi6fhS2MHjo8Pp0XyqEzAJ0=">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</latexit><latexit sha1_base64="z1+SUi6fhS2MHjo8Pp0XyqEzAJ0=">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</latexit><latexit sha1_base64="z1+SUi6fhS2MHjo8Pp0XyqEzAJ0=">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</latexit><latexit sha1_base64="z1+SUi6fhS2MHjo8Pp0XyqEzAJ0=">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</latexit>

÷ . ÷ . §.

= FE . to

Page 26: Discrete Naïve Bayes

A Not-So-Little Hitch

26

We found that

This should bother you …

The problem is that work did not show up in the SPAM training set

From the spammer’s perspective, including an uncommon word not in the training set would lead to and the filter would never catch it

It would be nice if we could ensure that no emails resulted in a zero SPAM or HAM score …

p(SPAM | x = [work, nigeria]) = 0<latexit sha1_base64="++uN2LPgjv1uVMDoudokSSbU39w=">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</latexit><latexit sha1_base64="++uN2LPgjv1uVMDoudokSSbU39w=">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</latexit><latexit sha1_base64="++uN2LPgjv1uVMDoudokSSbU39w=">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</latexit><latexit sha1_base64="++uN2LPgjv1uVMDoudokSSbU39w=">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</latexit>

p(SPAM | x) = 0<latexit sha1_base64="GPAVJ0fQ0gHzk3VILkyiuYznQG8=">AAACEXicbVA9SwNBEN2L3+dX1NJmMSckTbhL4UchKDY2QkSjQi6Evb29ZMnu3bE7JwlHfoONf8XGQsXWzs5/4yamUOODgcd7M8zMC1LBNbjup1WYmZ2bX1hcspdXVtfWixub1zrJFGUNmohE3QZEM8Fj1gAOgt2mihEZCHYT9E5H/s0dU5on8RUMUtaSpBPziFMCRmoXK45jp2UfWB+UzC/rJ+dD7Ese4twPItwfVvARdjG2HaddLLlVdww8TbwJKaEJ6u3ihx8mNJMsBiqI1k3PTaGVEwWcCja0/UyzlNAe6bCmoTGRTLfy8UtDvGuUEEeJMhUDHqs/J3IitR7IwHRKAl391xuJ/3nNDKKDVs7jNAMW0+9FUSYwJHiUDw65YhTEwBBCFTe3YtolilAwKdomBO/vy9OkUaseVr2LWul4b5LGItpGO6iMPLSPjtEZqqMGougePaJn9GI9WE/Wq/X23VqwJjNb6Bes9y8TZ5oQ</latexit><latexit sha1_base64="GPAVJ0fQ0gHzk3VILkyiuYznQG8=">AAACEXicbVA9SwNBEN2L3+dX1NJmMSckTbhL4UchKDY2QkSjQi6Evb29ZMnu3bE7JwlHfoONf8XGQsXWzs5/4yamUOODgcd7M8zMC1LBNbjup1WYmZ2bX1hcspdXVtfWixub1zrJFGUNmohE3QZEM8Fj1gAOgt2mihEZCHYT9E5H/s0dU5on8RUMUtaSpBPziFMCRmoXK45jp2UfWB+UzC/rJ+dD7Ese4twPItwfVvARdjG2HaddLLlVdww8TbwJKaEJ6u3ihx8mNJMsBiqI1k3PTaGVEwWcCja0/UyzlNAe6bCmoTGRTLfy8UtDvGuUEEeJMhUDHqs/J3IitR7IwHRKAl391xuJ/3nNDKKDVs7jNAMW0+9FUSYwJHiUDw65YhTEwBBCFTe3YtolilAwKdomBO/vy9OkUaseVr2LWul4b5LGItpGO6iMPLSPjtEZqqMGougePaJn9GI9WE/Wq/X23VqwJjNb6Bes9y8TZ5oQ</latexit><latexit sha1_base64="GPAVJ0fQ0gHzk3VILkyiuYznQG8=">AAACEXicbVA9SwNBEN2L3+dX1NJmMSckTbhL4UchKDY2QkSjQi6Evb29ZMnu3bE7JwlHfoONf8XGQsXWzs5/4yamUOODgcd7M8zMC1LBNbjup1WYmZ2bX1hcspdXVtfWixub1zrJFGUNmohE3QZEM8Fj1gAOgt2mihEZCHYT9E5H/s0dU5on8RUMUtaSpBPziFMCRmoXK45jp2UfWB+UzC/rJ+dD7Ese4twPItwfVvARdjG2HaddLLlVdww8TbwJKaEJ6u3ihx8mNJMsBiqI1k3PTaGVEwWcCja0/UyzlNAe6bCmoTGRTLfy8UtDvGuUEEeJMhUDHqs/J3IitR7IwHRKAl391xuJ/3nNDKKDVs7jNAMW0+9FUSYwJHiUDw65YhTEwBBCFTe3YtolilAwKdomBO/vy9OkUaseVr2LWul4b5LGItpGO6iMPLSPjtEZqqMGougePaJn9GI9WE/Wq/X23VqwJjNb6Bes9y8TZ5oQ</latexit><latexit sha1_base64="GPAVJ0fQ0gHzk3VILkyiuYznQG8=">AAACEXicbVA9SwNBEN2L3+dX1NJmMSckTbhL4UchKDY2QkSjQi6Evb29ZMnu3bE7JwlHfoONf8XGQsXWzs5/4yamUOODgcd7M8zMC1LBNbjup1WYmZ2bX1hcspdXVtfWixub1zrJFGUNmohE3QZEM8Fj1gAOgt2mihEZCHYT9E5H/s0dU5on8RUMUtaSpBPziFMCRmoXK45jp2UfWB+UzC/rJ+dD7Ese4twPItwfVvARdjG2HaddLLlVdww8TbwJKaEJ6u3ihx8mNJMsBiqI1k3PTaGVEwWcCja0/UyzlNAe6bCmoTGRTLfy8UtDvGuUEEeJMhUDHqs/J3IitR7IwHRKAl391xuJ/3nNDKKDVs7jNAMW0+9FUSYwJHiUDw65YhTEwBBCFTe3YtolilAwKdomBO/vy9OkUaseVr2LWul4b5LGItpGO6iMPLSPjtEZqqMGougePaJn9GI9WE/Wq/X23VqwJjNb6Bes9y8TZ5oQ</latexit>

Page 27: Discrete Naïve Bayes

Additive Laplace Smoothing

27

We want to avoid zero probability scores when a term isn’t in the training set

Silly Idea: Add 1 to all of the term frequency counts

But… it would be nice if behaved like a probability distribution

We’re going to estimate a probability for every term in vocabulary V

We can restore the sum-to-one nature of these estimates by adding |V| to denominator

p(term | Class) ?=

# instances of term in Class + 1

# total words in Class<latexit sha1_base64="04m6LXAOSLavUOSgFUC09H1FY88=">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</latexit><latexit sha1_base64="04m6LXAOSLavUOSgFUC09H1FY88=">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</latexit><latexit sha1_base64="04m6LXAOSLavUOSgFUC09H1FY88=">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</latexit><latexit sha1_base64="04m6LXAOSLavUOSgFUC09H1FY88=">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</latexit>

p(term | Class)<latexit sha1_base64="u0uU99+HiYB86MlhK+RnmbZ8oOY=">AAACGHicbVBNS8NAEN34WeNX1aOXxVbQS0l6sd4KvXisYFVoStlsp3ZxNwm7E7GE/A0v/hUvHlS89ua/cdNW8OvBwOO9md2ZFyZSGPS8D2dhcWl5ZbW05q5vbG5tl3d2L02cag4dHstYX4fMgBQRdFCghOtEA1OhhKvwtlX4V3egjYijCxwn0FPsJhJDwRlaqV/2qlU3GDHMkvwoQLhHrTIErXIaKDGgX1JLMmPyY7da7ZcrXs2bgv4l/pxUyBztfnkSDGKeKoiQF690fS/BXsY0Ci4hd4PUQML4LbuBrqURU2B62fSynB5aZUCHsbYVIZ2q3ycypowZq9B2KoYj89srxP+8borDRi8TUZIiRHz20TCVFGNaxEQHQgNHObaEcS3srpSPmGbchmNcG4L/++S/pFOvndb883ql2ZinUSL75IAcEZ+ckCY5I23SIZw8kCfyQl6dR+fZeXPeZ60Lznxmj/yAM/kE19ufvQ==</latexit><latexit sha1_base64="u0uU99+HiYB86MlhK+RnmbZ8oOY=">AAACGHicbVBNS8NAEN34WeNX1aOXxVbQS0l6sd4KvXisYFVoStlsp3ZxNwm7E7GE/A0v/hUvHlS89ua/cdNW8OvBwOO9md2ZFyZSGPS8D2dhcWl5ZbW05q5vbG5tl3d2L02cag4dHstYX4fMgBQRdFCghOtEA1OhhKvwtlX4V3egjYijCxwn0FPsJhJDwRlaqV/2qlU3GDHMkvwoQLhHrTIErXIaKDGgX1JLMmPyY7da7ZcrXs2bgv4l/pxUyBztfnkSDGKeKoiQF690fS/BXsY0Ci4hd4PUQML4LbuBrqURU2B62fSynB5aZUCHsbYVIZ2q3ycypowZq9B2KoYj89srxP+8borDRi8TUZIiRHz20TCVFGNaxEQHQgNHObaEcS3srpSPmGbchmNcG4L/++S/pFOvndb883ql2ZinUSL75IAcEZ+ckCY5I23SIZw8kCfyQl6dR+fZeXPeZ60Lznxmj/yAM/kE19ufvQ==</latexit><latexit sha1_base64="u0uU99+HiYB86MlhK+RnmbZ8oOY=">AAACGHicbVBNS8NAEN34WeNX1aOXxVbQS0l6sd4KvXisYFVoStlsp3ZxNwm7E7GE/A0v/hUvHlS89ua/cdNW8OvBwOO9md2ZFyZSGPS8D2dhcWl5ZbW05q5vbG5tl3d2L02cag4dHstYX4fMgBQRdFCghOtEA1OhhKvwtlX4V3egjYijCxwn0FPsJhJDwRlaqV/2qlU3GDHMkvwoQLhHrTIErXIaKDGgX1JLMmPyY7da7ZcrXs2bgv4l/pxUyBztfnkSDGKeKoiQF690fS/BXsY0Ci4hd4PUQML4LbuBrqURU2B62fSynB5aZUCHsbYVIZ2q3ycypowZq9B2KoYj89srxP+8borDRi8TUZIiRHz20TCVFGNaxEQHQgNHObaEcS3srpSPmGbchmNcG4L/++S/pFOvndb883ql2ZinUSL75IAcEZ+ckCY5I23SIZw8kCfyQl6dR+fZeXPeZ60Lznxmj/yAM/kE19ufvQ==</latexit><latexit sha1_base64="u0uU99+HiYB86MlhK+RnmbZ8oOY=">AAACGHicbVBNS8NAEN34WeNX1aOXxVbQS0l6sd4KvXisYFVoStlsp3ZxNwm7E7GE/A0v/hUvHlS89ua/cdNW8OvBwOO9md2ZFyZSGPS8D2dhcWl5ZbW05q5vbG5tl3d2L02cag4dHstYX4fMgBQRdFCghOtEA1OhhKvwtlX4V3egjYijCxwn0FPsJhJDwRlaqV/2qlU3GDHMkvwoQLhHrTIErXIaKDGgX1JLMmPyY7da7ZcrXs2bgv4l/pxUyBztfnkSDGKeKoiQF690fS/BXsY0Ci4hd4PUQML4LbuBrqURU2B62fSynB5aZUCHsbYVIZ2q3ycypowZq9B2KoYj89srxP+8borDRi8TUZIiRHz20TCVFGNaxEQHQgNHObaEcS3srpSPmGbchmNcG4L/++S/pFOvndb883ql2ZinUSL75IAcEZ+ckCY5I23SIZw8kCfyQl6dR+fZeXPeZ60Lznxmj/yAM/kE19ufvQ==</latexit>

11

Page 28: Discrete Naïve Bayes

Additive Laplace Smoothing

28

We want to avoid zero probability scores when a term isn’t in the training set

What about the ridiculous case when we have no SPAM or no HAM documents in training set?

We can do additive smoothing for the class prior estimates as well

where |C| is the number of classes (2 for SPAM vs HAM)

p(term | Class) = # instances of term in Class + 1

# total words in Class + |V |<latexit sha1_base64="UlKOL6lMZOwaJA0cPViHg3O768c=">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</latexit><latexit sha1_base64="UlKOL6lMZOwaJA0cPViHg3O768c=">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</latexit><latexit sha1_base64="UlKOL6lMZOwaJA0cPViHg3O768c=">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</latexit><latexit sha1_base64="UlKOL6lMZOwaJA0cPViHg3O768c=">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</latexit>

p(Class) =# emails from Class + 1

# total emails in training data + |C|<latexit sha1_base64="oy4aLWTPGqzGRSuG4pbOD2CywzY=">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</latexit><latexit sha1_base64="oy4aLWTPGqzGRSuG4pbOD2CywzY=">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</latexit><latexit sha1_base64="oy4aLWTPGqzGRSuG4pbOD2CywzY=">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</latexit><latexit sha1_base64="oy4aLWTPGqzGRSuG4pbOD2CywzY=">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</latexit>

:I

Page 29: Discrete Naïve Bayes

Naïve Bayes

29

Example: Re-compute the SPAM score for with Laplace smoothing

HAM SPAM SPAM SPAM HAM

work nigeria fly money fly

buy opportunity buy buy home

money viagra nigeria fly nigeria<latexit sha1_base64="Fk40qlV9xSoCXT3K4KQdocdHpiQ=">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</latexit><latexit sha1_base64="Fk40qlV9xSoCXT3K4KQdocdHpiQ=">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</latexit><latexit sha1_base64="Fk40qlV9xSoCXT3K4KQdocdHpiQ=">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</latexit><latexit sha1_base64="Fk40qlV9xSoCXT3K4KQdocdHpiQ=">AAADaXiclVJLj9MwEHYTHktgobtIgOBi0YL2VCV74XFaxGUvSEVQdqWmqhx3klp17MieLEShEr+RG7+ACz8CpxtWfXDZsSJ93zcPjyeTFFJYDMNfHc+/cfPW7b07wd17+/cfdA8Ov1hdGg4jrqU25wmzIIWCEQqUcF4YYHki4SxZvG/8ZxdgrNDqM1YFTHKWKZEKztBJ04POj34/iBPIhKqZMaxa1vx7e5ZBjPANTV6fvvuwpC/pP/ppeD2+So/jIJ43bdKrsl+1WazHKZGBEWxdSmW1TnOtoNrxN6VbnpQbbl0U2mCpBG7IW1FbdK5z2Ci6c+uFYJlh12v9KiKIQc3aYQf9/rTbCwfhyuguiFrQI60Np92f8UzzMgeFXDJrx1FY4MQVRMEluJ9WWigYX7AMxg4qloOd1KtVWdIXTpnRVBv3KaQrdT2jZrm1VZ64yJzh3G77GvF/vnGJ6etJLVRRIih+eVFaSoqaNntHZ8IAR1k5wLgRrlfK58wwjm47AzeEaPvJu2B0PHgziD4e907ettPYI8/Ic3JEIvKKnJBTMiQjwju/vX3vkffY++Mf+k/8p5ehXqfNeUg2zO/9BU15Giw=</latexit>

x = [work, nigeria]<latexit sha1_base64="/Ej8vUq3Hxea8/xUroSujYyv1ro=">AAACEHicbVA9SwNBEN3zM55fUUubxUSwkHCXxlgIARtLBaNCcoS9zVxcsrd37M5pwpG/YONfsbFQsbW089+4iSk08cHA470ZZuaFqRQGPe/LmZtfWFxaLqy4q2vrG5vFre0rk2SaQ4MnMtE3ITMghYIGCpRwk2pgcSjhOuydjvzrO9BGJOoSBykEMesqEQnO0Ert4kG57OatMKL9IT2hLYQ+6jhv3ie6d0iV6IIWLBi65XK7WPIq3hh0lvgTUiITnLeLn61OwrMYFHLJjGn6XopBzjQKLmHotjIDKeM91oWmpYrFYIJ8/NGQ7lulQ6NE21JIx+rviZzFxgzi0HbGDG/NtDcS//OaGUa1IBcqzRAU/1kUZZJiQkfx0I7QwFEOLGFcC3sr5bdMM442RNeG4E+/PEsa1cpxxb+oluq1SRoFskv2yAHxyRGpkzNyThqEkwfyRF7Iq/PoPDtvzvtP65wzmdkhf+B8fAMRxJt2</latexit><latexit sha1_base64="/Ej8vUq3Hxea8/xUroSujYyv1ro=">AAACEHicbVA9SwNBEN3zM55fUUubxUSwkHCXxlgIARtLBaNCcoS9zVxcsrd37M5pwpG/YONfsbFQsbW089+4iSk08cHA470ZZuaFqRQGPe/LmZtfWFxaLqy4q2vrG5vFre0rk2SaQ4MnMtE3ITMghYIGCpRwk2pgcSjhOuydjvzrO9BGJOoSBykEMesqEQnO0Ert4kG57OatMKL9IT2hLYQ+6jhv3ie6d0iV6IIWLBi65XK7WPIq3hh0lvgTUiITnLeLn61OwrMYFHLJjGn6XopBzjQKLmHotjIDKeM91oWmpYrFYIJ8/NGQ7lulQ6NE21JIx+rviZzFxgzi0HbGDG/NtDcS//OaGUa1IBcqzRAU/1kUZZJiQkfx0I7QwFEOLGFcC3sr5bdMM442RNeG4E+/PEsa1cpxxb+oluq1SRoFskv2yAHxyRGpkzNyThqEkwfyRF7Iq/PoPDtvzvtP65wzmdkhf+B8fAMRxJt2</latexit><latexit sha1_base64="/Ej8vUq3Hxea8/xUroSujYyv1ro=">AAACEHicbVA9SwNBEN3zM55fUUubxUSwkHCXxlgIARtLBaNCcoS9zVxcsrd37M5pwpG/YONfsbFQsbW089+4iSk08cHA470ZZuaFqRQGPe/LmZtfWFxaLqy4q2vrG5vFre0rk2SaQ4MnMtE3ITMghYIGCpRwk2pgcSjhOuydjvzrO9BGJOoSBykEMesqEQnO0Ert4kG57OatMKL9IT2hLYQ+6jhv3ie6d0iV6IIWLBi65XK7WPIq3hh0lvgTUiITnLeLn61OwrMYFHLJjGn6XopBzjQKLmHotjIDKeM91oWmpYrFYIJ8/NGQ7lulQ6NE21JIx+rviZzFxgzi0HbGDG/NtDcS//OaGUa1IBcqzRAU/1kUZZJiQkfx0I7QwFEOLGFcC3sr5bdMM442RNeG4E+/PEsa1cpxxb+oluq1SRoFskv2yAHxyRGpkzNyThqEkwfyRF7Iq/PoPDtvzvtP65wzmdkhf+B8fAMRxJt2</latexit><latexit sha1_base64="/Ej8vUq3Hxea8/xUroSujYyv1ro=">AAACEHicbVA9SwNBEN3zM55fUUubxUSwkHCXxlgIARtLBaNCcoS9zVxcsrd37M5pwpG/YONfsbFQsbW089+4iSk08cHA470ZZuaFqRQGPe/LmZtfWFxaLqy4q2vrG5vFre0rk2SaQ4MnMtE3ITMghYIGCpRwk2pgcSjhOuydjvzrO9BGJOoSBykEMesqEQnO0Ert4kG57OatMKL9IT2hLYQ+6jhv3ie6d0iV6IIWLBi65XK7WPIq3hh0lvgTUiITnLeLn61OwrMYFHLJjGn6XopBzjQKLmHotjIDKeM91oWmpYrFYIJ8/NGQ7lulQ6NE21JIx+rviZzFxgzi0HbGDG/NtDcS//OaGUa1IBcqzRAU/1kUZZJiQkfx0I7QwFEOLGFcC3sr5bdMM442RNeG4E+/PEsa1cpxxb+oluq1SRoFskv2yAHxyRGpkzNyThqEkwfyRF7Iq/PoPDtvzvtP65wzmdkhf+B8fAMRxJt2</latexit>

p(HAM | x) / p(x | HAM)p(HAM) =<latexit sha1_base64="zvvC1tqzjmLw/uYQ82TJZWtMloU=">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</latexit><latexit sha1_base64="zvvC1tqzjmLw/uYQ82TJZWtMloU=">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</latexit><latexit sha1_base64="zvvC1tqzjmLw/uYQ82TJZWtMloU=">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</latexit><latexit sha1_base64="zvvC1tqzjmLw/uYQ82TJZWtMloU=">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</latexit>

p(work | HAM) · p(nigeria | HAM) · p(HAM) =<latexit sha1_base64="gtcvuUAWp8zRUWXgElE1lF7IbV8=">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</latexit><latexit sha1_base64="gtcvuUAWp8zRUWXgElE1lF7IbV8=">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</latexit><latexit sha1_base64="gtcvuUAWp8zRUWXgElE1lF7IbV8=">AAACaXiclVFNSxtBGJ5dW6vbalMFW9rL0ETQS9j1Uj0IihcvgoWmCtkQZmffJEPmY5l5tzUse/AvevMXePFHOPkotMZLXxh4eD54Z57JCikcxvF9EK68er36Zm09evtuY/N948PWT2dKy6HDjTT2OmMOpNDQQYESrgsLTGUSrrLx2VS/+gXWCaN/4KSAnmJDLQaCM/RUv3HbakXpiGFV1Hspwg1aVf02dlzTVImc/qHOTy/qfZry3CBd8msxBCvY/0Tm4jGNWq1+oxm349nQZZAsQJMs5rLfuEtzw0sFGrlkznWTuMBexSwKLqGO0tJBwfiYDaHroWYKXK+aVVXTXc/kdGCsPxrpjP07UTHl3ERl3qkYjtxzbUq+pHVLHBz2KqGLEkHz+aJBKSkaOu2d5sICRznxgHEr/F0pHzHLOPrfiXwJyfMnL4POQfuonXw/aJ4cLtpYI1/IV7JHEvKNnJBzckk6hJOHYCPYCT4Gj+FW+Cn8PLeGwSKzTf6ZsPkE9Su44w==</latexit><latexit sha1_base64="gtcvuUAWp8zRUWXgElE1lF7IbV8=">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</latexit>

In = 1111111=8

÷a .i÷s . :#

'¥÷¥÷-Et6

=• 0087

Page 30: Discrete Naïve Bayes

Naïve Bayes

30

Example: Re-compute the HAM score for with Laplace smoothing

HAM SPAM SPAM SPAM HAM

work nigeria fly money fly

buy opportunity buy buy home

money viagra nigeria fly nigeria<latexit sha1_base64="Fk40qlV9xSoCXT3K4KQdocdHpiQ=">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</latexit><latexit sha1_base64="Fk40qlV9xSoCXT3K4KQdocdHpiQ=">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</latexit><latexit sha1_base64="Fk40qlV9xSoCXT3K4KQdocdHpiQ=">AAADaXiclVJLj9MwEHYTHktgobtIgOBi0YL2VCV74XFaxGUvSEVQdqWmqhx3klp17MieLEShEr+RG7+ACz8CpxtWfXDZsSJ93zcPjyeTFFJYDMNfHc+/cfPW7b07wd17+/cfdA8Ov1hdGg4jrqU25wmzIIWCEQqUcF4YYHki4SxZvG/8ZxdgrNDqM1YFTHKWKZEKztBJ04POj34/iBPIhKqZMaxa1vx7e5ZBjPANTV6fvvuwpC/pP/ppeD2+So/jIJ43bdKrsl+1WazHKZGBEWxdSmW1TnOtoNrxN6VbnpQbbl0U2mCpBG7IW1FbdK5z2Ci6c+uFYJlh12v9KiKIQc3aYQf9/rTbCwfhyuguiFrQI60Np92f8UzzMgeFXDJrx1FY4MQVRMEluJ9WWigYX7AMxg4qloOd1KtVWdIXTpnRVBv3KaQrdT2jZrm1VZ64yJzh3G77GvF/vnGJ6etJLVRRIih+eVFaSoqaNntHZ8IAR1k5wLgRrlfK58wwjm47AzeEaPvJu2B0PHgziD4e907ettPYI8/Ic3JEIvKKnJBTMiQjwju/vX3vkffY++Mf+k/8p5ehXqfNeUg2zO/9BU15Giw=</latexit><latexit sha1_base64="Fk40qlV9xSoCXT3K4KQdocdHpiQ=">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</latexit>

x = [work, nigeria]<latexit sha1_base64="/Ej8vUq3Hxea8/xUroSujYyv1ro=">AAACEHicbVA9SwNBEN3zM55fUUubxUSwkHCXxlgIARtLBaNCcoS9zVxcsrd37M5pwpG/YONfsbFQsbW089+4iSk08cHA470ZZuaFqRQGPe/LmZtfWFxaLqy4q2vrG5vFre0rk2SaQ4MnMtE3ITMghYIGCpRwk2pgcSjhOuydjvzrO9BGJOoSBykEMesqEQnO0Ert4kG57OatMKL9IT2hLYQ+6jhv3ie6d0iV6IIWLBi65XK7WPIq3hh0lvgTUiITnLeLn61OwrMYFHLJjGn6XopBzjQKLmHotjIDKeM91oWmpYrFYIJ8/NGQ7lulQ6NE21JIx+rviZzFxgzi0HbGDG/NtDcS//OaGUa1IBcqzRAU/1kUZZJiQkfx0I7QwFEOLGFcC3sr5bdMM442RNeG4E+/PEsa1cpxxb+oluq1SRoFskv2yAHxyRGpkzNyThqEkwfyRF7Iq/PoPDtvzvtP65wzmdkhf+B8fAMRxJt2</latexit><latexit sha1_base64="/Ej8vUq3Hxea8/xUroSujYyv1ro=">AAACEHicbVA9SwNBEN3zM55fUUubxUSwkHCXxlgIARtLBaNCcoS9zVxcsrd37M5pwpG/YONfsbFQsbW089+4iSk08cHA470ZZuaFqRQGPe/LmZtfWFxaLqy4q2vrG5vFre0rk2SaQ4MnMtE3ITMghYIGCpRwk2pgcSjhOuydjvzrO9BGJOoSBykEMesqEQnO0Ert4kG57OatMKL9IT2hLYQ+6jhv3ie6d0iV6IIWLBi65XK7WPIq3hh0lvgTUiITnLeLn61OwrMYFHLJjGn6XopBzjQKLmHotjIDKeM91oWmpYrFYIJ8/NGQ7lulQ6NE21JIx+rviZzFxgzi0HbGDG/NtDcS//OaGUa1IBcqzRAU/1kUZZJiQkfx0I7QwFEOLGFcC3sr5bdMM442RNeG4E+/PEsa1cpxxb+oluq1SRoFskv2yAHxyRGpkzNyThqEkwfyRF7Iq/PoPDtvzvtP65wzmdkhf+B8fAMRxJt2</latexit><latexit sha1_base64="/Ej8vUq3Hxea8/xUroSujYyv1ro=">AAACEHicbVA9SwNBEN3zM55fUUubxUSwkHCXxlgIARtLBaNCcoS9zVxcsrd37M5pwpG/YONfsbFQsbW089+4iSk08cHA470ZZuaFqRQGPe/LmZtfWFxaLqy4q2vrG5vFre0rk2SaQ4MnMtE3ITMghYIGCpRwk2pgcSjhOuydjvzrO9BGJOoSBykEMesqEQnO0Ert4kG57OatMKL9IT2hLYQ+6jhv3ie6d0iV6IIWLBi65XK7WPIq3hh0lvgTUiITnLeLn61OwrMYFHLJjGn6XopBzjQKLmHotjIDKeM91oWmpYrFYIJ8/NGQ7lulQ6NE21JIx+rviZzFxgzi0HbGDG/NtDcS//OaGUa1IBcqzRAU/1kUZZJiQkfx0I7QwFEOLGFcC3sr5bdMM442RNeG4E+/PEsa1cpxxb+oluq1SRoFskv2yAHxyRGpkzNyThqEkwfyRF7Iq/PoPDtvzvtP65wzmdkhf+B8fAMRxJt2</latexit><latexit sha1_base64="/Ej8vUq3Hxea8/xUroSujYyv1ro=">AAACEHicbVA9SwNBEN3zM55fUUubxUSwkHCXxlgIARtLBaNCcoS9zVxcsrd37M5pwpG/YONfsbFQsbW089+4iSk08cHA470ZZuaFqRQGPe/LmZtfWFxaLqy4q2vrG5vFre0rk2SaQ4MnMtE3ITMghYIGCpRwk2pgcSjhOuydjvzrO9BGJOoSBykEMesqEQnO0Ert4kG57OatMKL9IT2hLYQ+6jhv3ie6d0iV6IIWLBi65XK7WPIq3hh0lvgTUiITnLeLn61OwrMYFHLJjGn6XopBzjQKLmHotjIDKeM91oWmpYrFYIJ8/NGQ7lulQ6NE21JIx+rviZzFxgzi0HbGDG/NtDcS//OaGUa1IBcqzRAU/1kUZZJiQkfx0I7QwFEOLGFcC3sr5bdMM442RNeG4E+/PEsa1cpxxb+oluq1SRoFskv2yAHxyRGpkzNyThqEkwfyRF7Iq/PoPDtvzvtP65wzmdkhf+B8fAMRxJt2</latexit>

p(work | SPAM) · p(nigeria | SPAM) · p(SPAM) =<latexit sha1_base64="4dPAXFQgOovq5Kgu3fGjeXWsikk=">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</latexit><latexit sha1_base64="4dPAXFQgOovq5Kgu3fGjeXWsikk=">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</latexit><latexit sha1_base64="4dPAXFQgOovq5Kgu3fGjeXWsikk=">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</latexit><latexit sha1_base64="4dPAXFQgOovq5Kgu3fGjeXWsikk=">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</latexit>

p(SPAM | x) / p(x | SPAM)p(SPAM) =<latexit sha1_base64="z1+SUi6fhS2MHjo8Pp0XyqEzAJ0=">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</latexit><latexit sha1_base64="z1+SUi6fhS2MHjo8Pp0XyqEzAJ0=">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</latexit><latexit sha1_base64="z1+SUi6fhS2MHjo8Pp0XyqEzAJ0=">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</latexit><latexit sha1_base64="z1+SUi6fhS2MHjo8Pp0XyqEzAJ0=">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</latexit>

•008 vs o 0055

a HATT

#+8 . 3¥ . 3¥z¥'Ft ' ¥ -

¥A. 7 =. Ooss

Page 31: Discrete Naïve Bayes

Naïve Bayes

31

Example: How would Naïve Bayes classify the email ?

HAM SPAM SPAM SPAM HAM

work nigeria fly money fly

buy opportunity buy buy home

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x = [work, nigeria]<latexit sha1_base64="/Ej8vUq3Hxea8/xUroSujYyv1ro=">AAACEHicbVA9SwNBEN3zM55fUUubxUSwkHCXxlgIARtLBaNCcoS9zVxcsrd37M5pwpG/YONfsbFQsbW089+4iSk08cHA470ZZuaFqRQGPe/LmZtfWFxaLqy4q2vrG5vFre0rk2SaQ4MnMtE3ITMghYIGCpRwk2pgcSjhOuydjvzrO9BGJOoSBykEMesqEQnO0Ert4kG57OatMKL9IT2hLYQ+6jhv3ie6d0iV6IIWLBi65XK7WPIq3hh0lvgTUiITnLeLn61OwrMYFHLJjGn6XopBzjQKLmHotjIDKeM91oWmpYrFYIJ8/NGQ7lulQ6NE21JIx+rviZzFxgzi0HbGDG/NtDcS//OaGUa1IBcqzRAU/1kUZZJiQkfx0I7QwFEOLGFcC3sr5bdMM442RNeG4E+/PEsa1cpxxb+oluq1SRoFskv2yAHxyRGpkzNyThqEkwfyRF7Iq/PoPDtvzvtP65wzmdkhf+B8fAMRxJt2</latexit><latexit sha1_base64="/Ej8vUq3Hxea8/xUroSujYyv1ro=">AAACEHicbVA9SwNBEN3zM55fUUubxUSwkHCXxlgIARtLBaNCcoS9zVxcsrd37M5pwpG/YONfsbFQsbW089+4iSk08cHA470ZZuaFqRQGPe/LmZtfWFxaLqy4q2vrG5vFre0rk2SaQ4MnMtE3ITMghYIGCpRwk2pgcSjhOuydjvzrO9BGJOoSBykEMesqEQnO0Ert4kG57OatMKL9IT2hLYQ+6jhv3ie6d0iV6IIWLBi65XK7WPIq3hh0lvgTUiITnLeLn61OwrMYFHLJjGn6XopBzjQKLmHotjIDKeM91oWmpYrFYIJ8/NGQ7lulQ6NE21JIx+rviZzFxgzi0HbGDG/NtDcS//OaGUa1IBcqzRAU/1kUZZJiQkfx0I7QwFEOLGFcC3sr5bdMM442RNeG4E+/PEsa1cpxxb+oluq1SRoFskv2yAHxyRGpkzNyThqEkwfyRF7Iq/PoPDtvzvtP65wzmdkhf+B8fAMRxJt2</latexit><latexit sha1_base64="/Ej8vUq3Hxea8/xUroSujYyv1ro=">AAACEHicbVA9SwNBEN3zM55fUUubxUSwkHCXxlgIARtLBaNCcoS9zVxcsrd37M5pwpG/YONfsbFQsbW089+4iSk08cHA470ZZuaFqRQGPe/LmZtfWFxaLqy4q2vrG5vFre0rk2SaQ4MnMtE3ITMghYIGCpRwk2pgcSjhOuydjvzrO9BGJOoSBykEMesqEQnO0Ert4kG57OatMKL9IT2hLYQ+6jhv3ie6d0iV6IIWLBi65XK7WPIq3hh0lvgTUiITnLeLn61OwrMYFHLJjGn6XopBzjQKLmHotjIDKeM91oWmpYrFYIJ8/NGQ7lulQ6NE21JIx+rviZzFxgzi0HbGDG/NtDcS//OaGUa1IBcqzRAU/1kUZZJiQkfx0I7QwFEOLGFcC3sr5bdMM442RNeG4E+/PEsa1cpxxb+oluq1SRoFskv2yAHxyRGpkzNyThqEkwfyRF7Iq/PoPDtvzvtP65wzmdkhf+B8fAMRxJt2</latexit><latexit sha1_base64="/Ej8vUq3Hxea8/xUroSujYyv1ro=">AAACEHicbVA9SwNBEN3zM55fUUubxUSwkHCXxlgIARtLBaNCcoS9zVxcsrd37M5pwpG/YONfsbFQsbW089+4iSk08cHA470ZZuaFqRQGPe/LmZtfWFxaLqy4q2vrG5vFre0rk2SaQ4MnMtE3ITMghYIGCpRwk2pgcSjhOuydjvzrO9BGJOoSBykEMesqEQnO0Ert4kG57OatMKL9IT2hLYQ+6jhv3ie6d0iV6IIWLBi65XK7WPIq3hh0lvgTUiITnLeLn61OwrMYFHLJjGn6XopBzjQKLmHotjIDKeM91oWmpYrFYIJ8/NGQ7lulQ6NE21JIx+rviZzFxgzi0HbGDG/NtDcS//OaGUa1IBcqzRAU/1kUZZJiQkfx0I7QwFEOLGFcC3sr5bdMM442RNeG4E+/PEsa1cpxxb+oluq1SRoFskv2yAHxyRGpkzNyThqEkwfyRF7Iq/PoPDtvzvtP65wzmdkhf+B8fAMRxJt2</latexit>

Page 32: Discrete Naïve Bayes

Naïve Bayes

32

o The Naïve Bayes classifier is a probabilistic classifier

o We compute the posterior score of message belonging to class c as

o Predicted class is the one with the highest posterior score

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y = arg maxc p(y = c | x) = arg maxc p(c)Y

k

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p(y = c | x) / p(c)Y

k

p(xk | c)<latexit sha1_base64="q5Il4z+u6bm+ZTPZ1pxtjTbEtoQ=">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</latexit><latexit sha1_base64="q5Il4z+u6bm+ZTPZ1pxtjTbEtoQ=">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</latexit><latexit sha1_base64="q5Il4z+u6bm+ZTPZ1pxtjTbEtoQ=">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</latexit><latexit sha1_base64="q5Il4z+u6bm+ZTPZ1pxtjTbEtoQ=">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</latexit>

Page 33: Discrete Naïve Bayes

Numerical Hiccup

33

For long messages, have to multiply a lot of probabilities together

Probabilities are numbers less than 1. If you multiply enough of them, you could get underflow

Fix: Compute and store the log of the term-class score estimates

Recall: The log of a product is the sum of the logs:

Predicted class becomes:

log(ab) = log(a) + log(b)<latexit sha1_base64="/x8Avx4VJdGe0ZgAxZLIbHa1mIE=">AAACC3icbZDLSgMxFIYz9VbHW9Wlm2BHaBHKTDdeQCi4cVnBsYV2KJk0bUMzyZBkhDL0Adz4Km5cqLj1Bdz5NqadWWjrD4GP/5zDyfnDmFGlXffbKqysrq1vFDftre2d3b3S/sG9EonExMeCCdkOkSKMcuJrqhlpx5KgKGSkFY6vZ/XWA5GKCn6nJzEJIjTkdEAx0sbqlcqOY3eZGFZQWIVXMMMqPM3IeLbjmC635s4Fl8HLoQxyNXulr25f4CQiXGOGlOp4bqyDFElNMSNTu5soEiM8RkPSMchRRFSQzo+ZwhPj9OFASPO4hnP390SKIqUmUWg6I6RHarE2M/+rdRI9OA9SyuNEE46zRYOEQS3gLBnYp5JgzSYGEJbU/BXiEZIIa5OfbULwFk9eBr9eu6h5t/Vy4zJPowiOwDGoAA+cgQa4AU3gAwwewTN4BW/Wk/VivVsfWWvBymcOwR9Znz866ZbS</latexit><latexit sha1_base64="/x8Avx4VJdGe0ZgAxZLIbHa1mIE=">AAACC3icbZDLSgMxFIYz9VbHW9Wlm2BHaBHKTDdeQCi4cVnBsYV2KJk0bUMzyZBkhDL0Adz4Km5cqLj1Bdz5NqadWWjrD4GP/5zDyfnDmFGlXffbKqysrq1vFDftre2d3b3S/sG9EonExMeCCdkOkSKMcuJrqhlpx5KgKGSkFY6vZ/XWA5GKCn6nJzEJIjTkdEAx0sbqlcqOY3eZGFZQWIVXMMMqPM3IeLbjmC635s4Fl8HLoQxyNXulr25f4CQiXGOGlOp4bqyDFElNMSNTu5soEiM8RkPSMchRRFSQzo+ZwhPj9OFASPO4hnP390SKIqUmUWg6I6RHarE2M/+rdRI9OA9SyuNEE46zRYOEQS3gLBnYp5JgzSYGEJbU/BXiEZIIa5OfbULwFk9eBr9eu6h5t/Vy4zJPowiOwDGoAA+cgQa4AU3gAwwewTN4BW/Wk/VivVsfWWvBymcOwR9Znz866ZbS</latexit><latexit sha1_base64="/x8Avx4VJdGe0ZgAxZLIbHa1mIE=">AAACC3icbZDLSgMxFIYz9VbHW9Wlm2BHaBHKTDdeQCi4cVnBsYV2KJk0bUMzyZBkhDL0Adz4Km5cqLj1Bdz5NqadWWjrD4GP/5zDyfnDmFGlXffbKqysrq1vFDftre2d3b3S/sG9EonExMeCCdkOkSKMcuJrqhlpx5KgKGSkFY6vZ/XWA5GKCn6nJzEJIjTkdEAx0sbqlcqOY3eZGFZQWIVXMMMqPM3IeLbjmC635s4Fl8HLoQxyNXulr25f4CQiXGOGlOp4bqyDFElNMSNTu5soEiM8RkPSMchRRFSQzo+ZwhPj9OFASPO4hnP390SKIqUmUWg6I6RHarE2M/+rdRI9OA9SyuNEE46zRYOEQS3gLBnYp5JgzSYGEJbU/BXiEZIIa5OfbULwFk9eBr9eu6h5t/Vy4zJPowiOwDGoAA+cgQa4AU3gAwwewTN4BW/Wk/VivVsfWWvBymcOwR9Znz866ZbS</latexit><latexit sha1_base64="/x8Avx4VJdGe0ZgAxZLIbHa1mIE=">AAACC3icbZDLSgMxFIYz9VbHW9Wlm2BHaBHKTDdeQCi4cVnBsYV2KJk0bUMzyZBkhDL0Adz4Km5cqLj1Bdz5NqadWWjrD4GP/5zDyfnDmFGlXffbKqysrq1vFDftre2d3b3S/sG9EonExMeCCdkOkSKMcuJrqhlpx5KgKGSkFY6vZ/XWA5GKCn6nJzEJIjTkdEAx0sbqlcqOY3eZGFZQWIVXMMMqPM3IeLbjmC635s4Fl8HLoQxyNXulr25f4CQiXGOGlOp4bqyDFElNMSNTu5soEiM8RkPSMchRRFSQzo+ZwhPj9OFASPO4hnP390SKIqUmUWg6I6RHarE2M/+rdRI9OA9SyuNEE46zRYOEQS3gLBnYp5JgzSYGEJbU/BXiEZIIa5OfbULwFk9eBr9eu6h5t/Vy4zJPowiOwDGoAA+cgQa4AU3gAwwewTN4BW/Wk/VivVsfWWvBymcOwR9Znz866ZbS</latexit>

y = arg maxc log p(c) +X

k

log p(xk | c)<latexit sha1_base64="85GSNIiZN4v3YGJ/bbsWeHhmnto=">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</latexit><latexit sha1_base64="85GSNIiZN4v3YGJ/bbsWeHhmnto=">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</latexit><latexit sha1_base64="85GSNIiZN4v3YGJ/bbsWeHhmnto=">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</latexit><latexit sha1_base64="85GSNIiZN4v3YGJ/bbsWeHhmnto=">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</latexit>

y = arg maxc p(c)Y

k

p(xk | c)<latexit sha1_base64="uTdDh4a239l8Qd/LGMbU/KtqY1M=">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</latexit><latexit sha1_base64="uTdDh4a239l8Qd/LGMbU/KtqY1M=">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</latexit><latexit sha1_base64="uTdDh4a239l8Qd/LGMbU/KtqY1M=">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</latexit><latexit sha1_base64="uTdDh4a239l8Qd/LGMbU/KtqY1M=">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</latexit>

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Training and Prediction Complexity

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How much does Naïve Bayes cost to train?

How much does Naïve Bayes cost to make a prediction?

Page 35: Discrete Naïve Bayes

If-Time: Generative Models

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Why is Naïve Bayes called a generative model?

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