Bayesian Classifier (naive Boryes)

Abhijeet Singh Rajput

@abhijeetsingh

Created

Apr 17, 2025

Last Modified

3 weeks ago

Bayesian Classifier

Given the training data set, use naive Boryes algorithms to classify a particular species if its features are (slow, rarely, no).

s.no	Swim	Fly	Crowl	Class
1	Fast	No	No	Fish
2	Fast	No	Yes	Animal
3	Slow	No	No	Animal
4	Fast	No	No	Animal
5	No	Short	No	Bird
6	No	Short	No	Bird
7	No	Rarely	No	Animal
8	Slow	No	Yes	Animal
9	Slow	No	No	Fish
10	Slow	No	Yes	Fish
11	No	Large	No	Bird
12	Fast	No	No	Bird

$F_{1} = "swim" F_{2} = "Fly" F_{3} = "Crowl"$

The class Labels are
$C_{1} = "Animal" C_{2} = "Bird" C_{3} = "Fish"$

Construct the frequency table which summaries the data [Not the part of algo]

Class		Swim (F1)			Fly (F2)			Crowl (F3)		Total
	Fast	Swim	No	Long	Short	Rarely	No	Yes	No
Animal	2	2	1	0	0	1	4	2	3	8
Bird	1	0	3	1	2	0	1	0	4	4
Fish	1	2	0	0	0	0	3	1	2	3

Total	4	4	4	1	2	1	8	3	9	12

Step 1: Compute the probability

P (C_{1}) = \frac{no of records with class label "Animal"}{total number of examples} P (C_{1}) = \frac{5}{12}

P (C_{2}) = \frac{no of records with class label "Bird"}{total number of examples} P (C_{2}) = \frac{4}{12}

P (C_{1}) = \frac{no of records with class label "Fish"}{total number of examples} P (C_{1}) = \frac{3}{12}

Step 2: Constructing Table of Conditional Propability

Class	Swim $(F_{1})$ Fast Slow No	Fly $(F_{2})$ Long Short Rarely No	Crowl $(F_{3})$ Yes No	Total
Animal	2/5 2/5 1/5	0/5 0/5 1/5 4/5	2/5 3/5	5
Bird	1/4 0/4 3/4	1/4 2/4 0/4 1/4	0/4 4/4	4
Fish	1/3 2/3 0/3	0/3 0/3 0/3 3/3	1/3 2/3	3

Class

Swim $(F_{1})$

Fast Slow No

Fly $(F_{2})$

Long Short Rarely No

Crowl $(F_{3})$

Yes No

Total

Animal

2/5 2/5 1/5

0/5 0/5 1/5 4/5

2/5 3/5

Bird

1/4 0/4 3/4

1/4 2/4 0/4 1/4

0/4 4/4

Fish

1/3 2/3 0/3

0/3 0/3 0/3 3/3

1/3 2/3

The conditional probability are calculated as
$P (F_{1} = slow / C_{1}) = \frac{Number of records where F _{1} = slow and class label C _{1}}{Number of records with class label C _{1}} = \frac{2}{5}$

Step 3: we now calculate the following numbers

$S w im = S l o w F l y = R a r e l y C r a w l = N o$

$q_{1} = P (swim/animal) . P (fly/animal) . P (crawl/animal) . p (animal)$

$q_{1} = 2/5 \times 1/5 \times 3/5 \times 5/12$

$q_{2} = P (S w im / c_{2}) . P (F l y / c_{2}) . P (C r a w l / c_{2}) . p (c_{2})$

$q_{2} = 0/4 \times 0/4 \times 3/4 \times 4/12 = 0$

$q_{3} = P (S w im / c_{3}) . P (F l y / c_{3}) . P (C r a w l / c_{3}) . p (c_{3})$

$q_{3} = 2/3 \times 0/3 \times 3/3 \times 3/12 = 0$

Step 4: Find Maximum

$max (q_{1}, q_{2}, q_{3}) = 0.0200$

Step 5: The maximum is $q_{1}$ as it corresponds to class $C_{1} = Animal$

so we assign the class Label "Animal" to the test instance $(Slow, Rarely, No)$