Data-driven Modeling: Lecture 03

Data-driven modeling
APAM E4990

Jake Hofman

Columbia University

February 6, 2012

Jake Hofman (Columbia University) Data-driven modeling February 6, 2012 1 / 13

Diagnoses a la Bayes1

• You’re testing for a rare disease:
• 1% of the population is infected
• You have a highly sensitive and speciﬁc test:
• 99% of sick patients test positive
• 99% of healthy patients test negative
• Given that a patient tests positive, what is probability the
patient is sick?

1
Wiggins, SciAm 2006

Diagnoses a la Bayes

Population
10,000 ppl

1% Sick 99% Healthy
100 ppl 9900 ppl

99% Test + 1% Test - 1% Test + 99% Test -
99 ppl 1 per 99 ppl 9801 ppl



Population
10,000 ppl

1% Sick 99% Healthy
100 ppl 9900 ppl


So given that a patient tests positive (198 ppl), there is a 50%
chance the patient is sick (99 ppl)!



Population
10,000 ppl

1% Sick 99% Healthy
100 ppl 9900 ppl

The small error rate on the large healthy population produces
many false positives.


Inverting conditional probabilities

Bayes’ Theorem
Equate the far right- and left-hand sides of product rule

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

and divide to get the probability of y given x from the probability
of x given y :
p (x|y ) p (y )
p (y |x) =
p (x)

where p (x) = y ∈ΩY p (x|y ) p (y ) is the normalization constant.



Given that a patient tests positive, what is probability the patient
is sick?
99/100 1/100

p (+|sick) p (sick) 99 1
p (sick|+) = = =
p (+) 198 2
99/1002 +99/1002 =198/1002

where p (+) = p (+|sick) p (sick) + p (+|healthy ) p (healthy ).


(Super) Naive Bayes
We can use Bayes’ rule to build a one-word spam classiﬁer:

p (word|spam) p (spam)
p (spam|word) =
p (word)

where we estimate these probabilities with ratios of counts:
# spam docs containing word
p (word|spam) =
ˆ
# spam docs
# ham docs containing word
p (word|ham)
ˆ =
# ham docs
# spam docs
p (spam)
ˆ =
# docs
# ham docs
p (ham)
ˆ =
# docs


(Super) Naive Bayes

$ ./enron_naive_bayes.sh meeting
1500 spam examples
3672 ham examples
16 spam examples containing meeting
153 ham examples containing meeting

estimated P(spam) = .2900
estimated P(ham) = .7100
estimated P(meeting|spam) = .0106
estimated P(meeting|ham) = .0416

P(spam|meeting) = .0923


(Super) Naive Bayes

$ ./enron_naive_bayes.sh money
1500 spam examples
3672 ham examples
194 spam examples containing money
50 ham examples containing money

estimated P(money|spam) = .1293
estimated P(money|ham) = .0136

P(spam|money) = .7957


(Super) Naive Bayes

$ ./enron_naive_bayes.sh enron
1500 spam examples
3672 ham examples
0 spam examples containing enron
1478 ham examples containing enron

estimated P(enron|spam) = 0
estimated P(enron|ham) = .4025

P(spam|enron) = 0


Naive Bayes

Represent each document by a binary vector x where xj = 1 if the
j-th word appears in the document (xj = 0 otherwise).

Modeling each word as an independent Bernoulli random variable,
the probability of observing a document x of class c is:
x
p (x|c) = θjcj (1 − θjc )1−xj
j

where θjc denotes the probability that the j-th word occurs in a
document of class c.


Naive Bayes

Using this likelihood in Bayes’ rule and taking a logarithm, we have:

p (x|c) p (c)
log p (c|x) = log
p (x)
θjc θc
= xj log + log(1 − θjc ) + log
1 − θjc p (x)
j j

where θc is the probability of observing a document of class c.


Naive Bayes

We can eliminate p (x) by calculating the log-odds:

p (1|x) θj1 (1 − θj0 ) 1 − θj1 θ1
log = xj log + log + log
p (0|x) θj0 (1 − θj1 ) 1 − θj0 θ0
j j
wj
w0

which gives a linear classiﬁer of the form w · x + w0


Naive Bayes
We train by counting words and documents within classes to
estimate θjc and θc :

ˆ njc
θjc =
nc
ˆ nc
θc =
n
and use these to calculate the weights wj and bias w0 :
ˆ ˆ

ˆ ˆ
θj1 (1 − θj0 )
ˆ
wj = log
ˆ ˆ
θj0 (1 − θj1 )
ˆ
1 − θj1 ˆ
θ1
w0 =
ˆ log + log .
ˆ
1 − θj0 ˆ
θ0
j

We we predict by simply adding the weights of the words that
appear in the document to the bias term.

Naive Bayes

In practice, this works better than one might expect given its
simplicity2

2
http://www.jstor.org/pss/1403452

Naive Bayes

Training is computationally cheap and scalable, and the model is
easy to update given new observations2

2
http://www.springerlink.com/content/wu3g458834583125/

Naive Bayes

Performance varies with document representations and
corresponding likelihood models2

2
http://ceas.cc/2006/15.pdf

Naive Bayes

It’s often important to smooth parameter estimates (e.g., by
adding pseudocounts) to avoid overﬁtting


Data-driven Modeling: Lecture 03

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