Text classification Naive Bayes NB theory Evaluation of TC
PV211: Introduction to Information Retrieval
https://www.fi.muni.cz/~sojka/PV211
IIR 13: Text Classification & Naive Bayes
Handout version
Petr Sojka, Hinrich Schütze et al.
Faculty of Informatics, Masaryk University, Brno
Center for Information and Language Processing, University of Munich
2024-04-25
(compiled on 2024-04-24 10:06:55)
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Text classification Naive Bayes NB theory Evaluation of TC
Overview
1 Text classification
2 Naive Bayes
3 NB theory
4 Evaluation of TC
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Text classification Naive Bayes NB theory Evaluation of TC
Take-away today
Text classification: definition & relevance to information
retrieval
Naive Bayes: simple baseline text classifier
Theory: derivation of Naive Bayes classification rule & analysis
Evaluation of text classification: how do we know it worked /
didn’t work?
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Text classification Naive Bayes NB theory Evaluation of TC
A text classification task: Email spam filtering
From: ‘‘’’ <takworlld@hotmail.com>
Subject: real estate is the only way... gem oalvgkay
Anyone can buy real estate with no money down
Stop paying rent TODAY !
There is no need to spend hundreds or even thousands for similar courses
I am 22 years old and I have already purchased 6 properties using the
methods outlined in this truly INCREDIBLE ebook.
Change your life NOW !
=================================================
Click Below to order:
http://www.wholesaledaily.com/sales/nmd.htm
=================================================
How would you write a program that would automatically detect and delete this
type of message?
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Text classification Naive Bayes NB theory Evaluation of TC
Formal definition of TC: Training
Given:
A document space X
Documents are represented in this space – typically some type
of high-dimensional space.
A fixed set of classes C = {c1, c2, . . . , cJ }
The classes are human-defined for the needs of an application
(e.g., spam vs. nonspam).
A training set D of labeled documents. Each labeled
document d, c ∈ X × C
Using a learning method or learning algorithm, we then wish to
learn a classifier γ that maps documents to classes:
γ : X → C
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Text classification Naive Bayes NB theory Evaluation of TC
Formal definition of TC: Application/Testing
Given: a description d ∈ X of a document
Determine: γ(d) ∈ C, that is, the class that is most appropriate
for d
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Text classification Naive Bayes NB theory Evaluation of TC
Topic classification
classes:
training
set:
test
set:
regions industries subject areas
γ(d′) =China
first
private
Chinese
airline
UK China poultry coffee elections sports
London
congestion
Big Ben
Parliament
the Queen
Windsor
Beijing
Olympics
Great Wall
tourism
communist
Mao
chicken
feed
ducks
pate
turkey
bird flu
beans
roasting
robusta
arabica
harvest
Kenya
votes
recount
run-off
seat
campaign
TV ads
baseball
diamond
soccer
forward
captain
team
d′
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Text classification Naive Bayes NB theory Evaluation of TC
Exercise
Find examples of uses of text classification in information
retrieval
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Text classification Naive Bayes NB theory Evaluation of TC
Examples of how search engines use classification
Language identification (classes: English vs. French, etc.)
The automatic detection of spam pages (spam vs. nonspam)
Sentiment detection: is a movie or product review positive or
negative (positive vs. negative)
Topic-specific or vertical search – restrict search to a
“vertical” like “related to health” (relevant to vertical vs. not)
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Text classification Naive Bayes NB theory Evaluation of TC
Classification methods: 1. Manual
Manual classification was used by Yahoo in the beginning of
the web. Also: MathSciNet (MSC), DMOZ – the Open
Directory Project, PubMed
Very accurate if job is done by experts.
Consistent when the problem size and team is small.
Scaling manual classification is difficult and expensive.
→ We need automatic methods for classification.
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Text classification Naive Bayes NB theory Evaluation of TC
Classification methods: 2. Rule-based
E.g., Google Alerts is rule-based classification.
There are IDE-type development environments for writing very
complex rules efficiently. (e.g., Verity)
Often: Boolean combinations (as in Google Alerts).
Accuracy is very high if a rule has been carefully refined over
time by a subject expert.
Building and maintaining rule-based classification systems is
cumbersome and expensive.
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Text classification Naive Bayes NB theory Evaluation of TC
A Verity topic (a complex classification rule)
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Text classification Naive Bayes NB theory Evaluation of TC
Classification methods: 3. Statistical/Probabilistic
This was our definition of the classification problem – text
classification as a learning problem.
(i) Supervised learning of the classification function γ and
(ii) application of γ to classifying new documents.
We will look at two methods for doing this: Naive Bayes and
SVMs
No free lunch: requires hand-classified training data.
But this manual classification can be done by non-experts.
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Text classification Naive Bayes NB theory Evaluation of TC
The Naive Bayes classifier
The Naive Bayes classifier is a probabilistic classifier.
We compute the probability of a document d being in a class
c as follows:
P(c|d) ∝ P(c)
1≤k≤nd
P(tk |c)
nd is the length of the document. (number of tokens)
P(tk|c) is the conditional probability of term tk occurring in a
document of class c
P(tk|c) as a measure of how much evidence tk contributes
that c is the correct class.
P(c) is the prior probability of c.
If a document’s terms do not provide clear evidence for one
class vs. another, we choose the c with highest P(c).
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Text classification Naive Bayes NB theory Evaluation of TC
Maximum a posteriori class
Our goal in Naive Bayes classification is to find the “best”
class.
The best class is the most likely or maximum a posteriori
(MAP) class cmap:
cmap = arg max
c∈C
ˆP(c|d) = arg max
c∈C
ˆP(c)
1≤k≤nd
ˆP(tk|c)
We write ˆP for P since these values are estimates from the
training set.
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Text classification Naive Bayes NB theory Evaluation of TC
Taking the log
Multiplying lots of small probabilities can result in floating
point underflow.
Since log(xy) = log(x) + log(y), we can sum log probabilities
instead of multiplying probabilities.
Since log is a monotonic function, the class with the highest
score does not change.
So what we usually compute in practice is:
cmap = arg max
c∈C
[log ˆP(c) +
1≤k≤nd
log ˆP(tk|c)]
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Text classification Naive Bayes NB theory Evaluation of TC
Naive Bayes classifier
Classification rule:
cmap = arg max
c∈C
[log ˆP(c) +
1≤k≤nd
log ˆP(tk|c)]
Simple interpretation:
Each conditional parameter log ˆP(tk |c) is a weight that
indicates how good an indicator tk is for c.
The prior log ˆP(c) is a weight that indicates the relative
frequency of c.
The sum of log prior and term weights is then a measure of
how much evidence there is for the document being in the
class.
We select the class with the most evidence.
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Text classification Naive Bayes NB theory Evaluation of TC
Parameter estimation take 1: Maximum likelihood
Estimate parameters ˆP(c) and ˆP(tk|c) from train data: How?
Prior:
ˆP(c) =
Nc
N
Nc: number of docs in class c; N: total number of docs
Conditional probabilities:
ˆP(t|c) =
Tct
t′∈V Tct′
Tct is the number of tokens of t in training documents from
class c (includes multiple occurrences)
We have made a Naive Bayes independence assumption here:
ˆP(Xk1
= t|c) = ˆP(Xk2
= t|c), independent of position
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Text classification Naive Bayes NB theory Evaluation of TC
The problem with maximum likelihood estimates: Zeros
C=China
X1=Beijing X2=and X3=Taipei X4=join X5=WTO
P(China|d) ∝ P(China) · P(Beijing|China) · P(and|China)
· P(Taipei|China) · P(join|China) · P(WTO|China)
If WTO never occurs in class China in the train set:
ˆP(WTO|China) =
TChina,WTO
t′∈V TChina,t′
=
0
t′∈V TChina,t′
= 0
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Text classification Naive Bayes NB theory Evaluation of TC
The problem with maximum likelihood estimates: Zeros
(cont)
If there are no occurrences of WTO in documents in class
China, we get a zero estimate:
ˆP(WTO|China) =
TChina,WTO
t′∈V TChina,t′
= 0
→ We will get P(China|d) = 0 for any document that
contains WTO
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Text classification Naive Bayes NB theory Evaluation of TC
To avoid zeros: Add-one smoothing
Before:
ˆP(t|c) =
Tct
t′∈V Tct′
Now: Add one to each count to avoid zeros:
ˆP(t|c) =
Tct + 1
t′∈V (Tct′ + 1)
=
Tct + 1
( t′∈V Tct′ ) + B
B is the number of bins – in this case the number of different
words or the size of the vocabulary |V | = M
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Text classification Naive Bayes NB theory Evaluation of TC
Naive Bayes: Summary
Estimate parameters from the training corpus using add-one
smoothing
For a new document, for each class, compute sum of (i) log of
prior and (ii) logs of conditional probabilities of the terms
Assign the document to the class with the largest score
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Text classification Naive Bayes NB theory Evaluation of TC
Naive Bayes: Training
TrainMultinomialNB(C, D)
1 V ← ExtractVocabulary(D)
2 N ← CountDocs(D)
3 for each c ∈ C
4 do Nc ← CountDocsInClass(D, c)
5 prior[c] ← Nc/N
6 textc ← ConcatenateTextOfAllDocsInClass(D, c)
7 for each t ∈ V
8 do Tct ← CountTokensOfTerm(textc, t)
9 for each t ∈ V
10 do condprob[t][c] ← Tct +1
t′ (Tct′ +1)
11 return V , prior, condprob
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Text classification Naive Bayes NB theory Evaluation of TC
Naive Bayes: Testing
ApplyMultinomialNB(C, V , prior, condprob, d)
1 W ← ExtractTokensFromDoc(V , d)
2 for each c ∈ C
3 do score[c] ← log prior[c]
4 for each t ∈ W
5 do score[c]+ = log condprob[t][c]
6 return arg maxc∈C score[c]
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Text classification Naive Bayes NB theory Evaluation of TC
Exercise
docID words in document in c = China?
training set 1 Chinese Beijing Chinese yes
2 Chinese Chinese Shanghai yes
3 Chinese Macao yes
4 Tokyo Japan Chinese no
test set 5 Chinese Chinese Chinese Tokyo Japan ?
Estimate parameters of Naive Bayes classifier
Classify test document
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Text classification Naive Bayes NB theory Evaluation of TC
Example: Parameter estimates
Priors: ˆP(c) = 3/4 and ˆP(c) = 1/4
Conditional probabilities:
ˆP(Chinese|c) = (5 + 1)/(8 + 6) = 6/14 = 3/7
ˆP(Tokyo|c) = ˆP(Japan|c) = (0 + 1)/(8 + 6) = 1/14
ˆP(Chinese|c) = (1 + 1)/(3 + 6) = 2/9
ˆP(Tokyo|c) = ˆP(Japan|c) = (1 + 1)/(3 + 6) = 2/9
The denominators are (8 + 6) and (3 + 6) because the lengths of
textc and textc are 8 and 3, respectively, and because the constant
B is 6 as the vocabulary consists of six terms.
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Text classification Naive Bayes NB theory Evaluation of TC
Example: Classification
ˆP(c|d5) ∝ 3/4 · (3/7)3
· 1/14 · 1/14 ≈ 0.0003
ˆP(c|d5) ∝ 1/4 · (2/9)3
· 2/9 · 2/9 ≈ 0.0001
Thus, the classifier assigns the test document to c = China.
The reason for this classification decision is that the three
occurrences of the positive indicator Chinese in d5 outweigh the
occurrences of the two negative indicators Japan and Tokyo.
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Text classification Naive Bayes NB theory Evaluation of TC
Time complexity of Naive Bayes
mode time complexity
training Θ(|D|Lave + |C||V |)
testing Θ(La + |C|Ma) = Θ(|C|Ma)
Lave: average length of a training doc, La: length of the test
doc, Ma: number of distinct terms in the test doc, D: training
set, V : vocabulary, C: set of classes
Θ(|D|Lave) is the time it takes to compute all counts.
Θ(|C||V |) is the time it takes to compute the parameters
from the counts.
Generally: |C||V | < |D|Lave
Test time is also linear (in the length of the test document).
Thus: Naive Bayes is linear in the size of the training set
(training) and the test document (testing). This is optimal.
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Text classification Naive Bayes NB theory Evaluation of TC
Naive Bayes: Analysis
Now we want to gain a better understanding of the properties
of Naive Bayes.
We will formally derive the classification rule . . .
. . . and make our assumptions explicit.
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Text classification Naive Bayes NB theory Evaluation of TC
Derivation of Naive Bayes rule
We want to find the class that is most likely given the document:
cmap = arg max
c∈C
P(c|d)
Apply Bayes rule P(A|B) = P(B|A)P(A)
P(B) :
cmap = arg max
c∈C
P(d|c)P(c)
P(d)
Drop denominator since P(d) is the same for all classes:
cmap = arg max
c∈C
P(d|c)P(c)
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Text classification Naive Bayes NB theory Evaluation of TC
Too many parameters / sparseness
cmap = arg max
c∈C
P(d|c)P(c)
= arg max
c∈C
P( t1, . . . , tk, . . . , tnd
|c)P(c)
There are too many parameters P( t1, . . . , tk , . . . , tnd
|c), one
for each unique combination of a class and a sequence of
words.
We would need a very, very large number of training examples
to estimate that many parameters.
This is the problem of data sparseness.
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Text classification Naive Bayes NB theory Evaluation of TC
Naive Bayes conditional independence assumption
To reduce the number of parameters to a manageable size, we
make the Naive Bayes conditional independence assumption:
P(d|c) = P( t1, . . . , tnd
|c) =
1≤k≤nd
P(Xk = tk|c)
We assume that the probability of observing the conjunction of
attributes is equal to the product of the individual probabilities
P(Xk = tk|c).
Recall from earlier the estimates for these conditional probabilities:
ˆP(t|c) = Tct +1
( t′∈V
Tct′ )+B
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Text classification Naive Bayes NB theory Evaluation of TC
Generative model
C=China
X1=Beijing X2=and X3=Taipei X4=join X5=WTO
P(c|d) ∝ P(c) 1≤k≤nd
P(tk|c)
Generate a class with probability P(c)
Generate each of the words (in their respective positions),
conditional on the class, but independent of each other, with
probability P(tk |c)
To classify docs, we “reengineer” this process and find the
class that is most likely to have generated the doc.
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Text classification Naive Bayes NB theory Evaluation of TC
Second independence assumption
ˆP(Xk1
= t|c) = ˆP(Xk2
= t|c)
For example, for a document in the class UK, the probability
of generating queen in the first position of the document is
the same as generating it in the last position.
The two independence assumptions amount to the bag of
words model.
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Text classification Naive Bayes NB theory Evaluation of TC
A different Naive Bayes model: Bernoulli model
UAlaska=0 UBeijing=1 UIndia=0 Ujoin=1 UTaipei=1 UWTO=1
C=China
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Text classification Naive Bayes NB theory Evaluation of TC
Violation of Naive Bayes independence assumptions
Conditional independence:
P( t1, . . . , tnd
|c) =
1≤k≤nd
P(Xk = tk|c)
Positional independence:
ˆP(Xk1
= t|c) = ˆP(Xk2
= t|c)
The independence assumptions do not really hold of
documents written in natural language.
Exercise
Examples for why conditional independence assumption is not
really true?
Examples for why positional independence assumption is not
really true?
How can Naive Bayes work if it makes such inappropriate
assumptions?
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Text classification Naive Bayes NB theory Evaluation of TC
Why does Naive Bayes work?
Naive Bayes can work well even though conditional
independence assumptions are badly violated.
Example:
c1 c2 class selected
true probability P(c|d) 0.6 0.4 c1
ˆP(c) 1≤k≤nd
ˆP(tk|c) 0.00099 0.00001
NB estimate ˆP(c|d) 0.99 0.01 c1
Double counting of evidence causes underestimation (0.01)
and overestimation (0.99).
Classification is about predicting the correct class and not
about accurately estimating probabilities.
Naive Bayes is terrible for correct estimation . . .
. . . but if often performs well at accurate prediction (choosing
the correct class).
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Text classification Naive Bayes NB theory Evaluation of TC
Naive Bayes is not so naive
Naive Bayes has won some bakeoffs (e.g., KDD-CUP 97)
More robust to irrelevant features than some more complex
learning methods
More robust to concept drift (changing of definition of class
over time) than some more complex learning methods
Better than methods like decision trees when we have many
equally important features
A good dependable baseline for text classification (but not the
best)
Optimal if independence assumptions hold (never true for
text, but true for some domains)
Very fast
Low storage requirements
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Text classification Naive Bayes NB theory Evaluation of TC
Evaluation on Reuters
classes:
training
set:
test
set:
regions industries subject areas
γ(d′) =China
first
private
Chinese
airline
UK China poultry coffee elections sports
London
congestion
Big Ben
Parliament
the Queen
Windsor
Beijing
Olympics
Great Wall
tourism
communist
Mao
chicken
feed
ducks
pate
turkey
bird flu
beans
roasting
robusta
arabica
harvest
Kenya
votes
recount
run-off
seat
campaign
TV ads
baseball
diamond
soccer
forward
captain
team
d′
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Text classification Naive Bayes NB theory Evaluation of TC
Example: The Reuters collection
symbol statistic value
N documents 800,000
L avg. # word tokens per document 200
M word types 400,000
type of class number examples
region 366 UK, China
industry 870 poultry, coffee
subject area 126 elections, sports
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Text classification Naive Bayes NB theory Evaluation of TC
A Reuters document
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Text classification Naive Bayes NB theory Evaluation of TC
Evaluating classification
Evaluation must be done on test data that are independent of
the training data, i.e., training and test sets are disjoint.
It’s easy to get good performance on a test set that was
available to the learner during training (e.g., just memorize
the test set).
Measures: Precision, recall, F1, classification accuracy
Average measures over multiple training and test sets (splits
of the overall data) for best results.
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Text classification Naive Bayes NB theory Evaluation of TC
Precision P and recall R
in the class not in the class
predicted to be in the class true positives (TP) false positives (FP)
predicted to not be in the class false negatives (FN) true negatives (TN)
TP, FP, FN, TN are counts of documents. The sum of these four
counts is the total number of documents.
precision:P = TP/(TP + FP)
recall:R = TP/(TP + FN)
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Text classification Naive Bayes NB theory Evaluation of TC
A combined measure: F
F1 allows us to trade off precision against recall.
F1 =
1
1
2
1
P + 1
2
1
R
=
2PR
P + R
This is the harmonic mean of P and R: 1
F = 1
2( 1
P + 1
R )
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Text classification Naive Bayes NB theory Evaluation of TC
Averaging: Micro vs. Macro
We now have an evaluation measure (F1) for one class.
But we also want a single number that measures the
aggregate performance over all classes in the collection.
Macroaveraging
Compute F1 for each of the C classes
Average these C numbers
Microaveraging
Compute TP, FP, FN for each of the C classes
Sum these C numbers (e.g., all TP to get aggregate TP)
Compute F1 for aggregate TP, FP, FN
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Text classification Naive Bayes NB theory Evaluation of TC
Naive Bayes vs. other methods
(a) NB Rocchio kNN SVM
micro-avg-L (90 classes) 80 85 86 89
macro-avg (90 classes) 47 59 60 60
(b) NB Rocchio kNN trees SVM
earn 96 93 97 98 98
acq 88 65 92 90 94
money-fx 57 47 78 66 75
grain 79 68 82 85 95
crude 80 70 86 85 89
trade 64 65 77 73 76
interest 65 63 74 67 78
ship 85 49 79 74 86
wheat 70 69 77 93 92
corn 65 48 78 92 90
micro-avg (top 10) 82 65 82 88 92
micro-avg-D (118 classes) 75 62 n/a n/a 87
Evaluation measure: F1
Naive Bayes does pretty well, but some methods beat it consistently (e.g., SVM).
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Text classification Naive Bayes NB theory Evaluation of TC
Take-away today
Text classification: definition & relevance to information
retrieval
Naive Bayes: simple baseline text classifier
Theory: derivation of Naive Bayes classification rule & analysis
Evaluation of text classification: how do we know it worked /
didn’t work?
Sojka, IIR Group: PV211: Text Classification & Naive Bayes 51 / 52
Text classification Naive Bayes NB theory Evaluation of TC
Resources
Chapter 13 of IIR
Resources at https://www.fi.muni.cz/~sojka/PV211/
and http://cislmu.org, materials in MU IS and FI MU
library
Weka: A data mining software package that includes an
implementation of Naive Bayes
Reuters-21578 – text classification evaluation set
Vulgarity classifier fail
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