Question

Consider the following data set. You are asked to predict the class label (if Stolen = Yes or No) for a test data point “X” where “X” = (color=Red, Type=SUV, Origin=Domestic) using the Naïve Bayes approach. Show every step and calculation details.

	Color	Type	Origin	Stolen
1	Red	Sports	Domestic	Yes
2	Red	Sports	Domestic	Yes
3	Red	Sports	Domestic	Yes
4	Yellow	Sports	Domestic	No
5	Yellow	Sports	Import	Yes
6	Yellow	SUV	Import	No
7	Yellow	SUV	Import	Yes
8	Yellow	SUV	Domestic	No
9	Red	SUV	Import	Yes
10	Red	Sports	Import	Yes

Answer 1

By Naive Bayes approach we can classify samples which are unseen and here we have to classify (color=Red, Type=SUV, Origin=Domestic) so we have the formula for classification by this approach which is as follows:

P(C|X)=[P(X|C)*P(C)] / P(X)

where 'C' stands for values of the class(Yes and no in this case from stolen class) and 'X' stands values of particular attribute other than class attribute(Red and yellow from the attribute color likewise other values also from their respective attribute)

P(C)=prior probability

P(X)=probability of X

P(X|C)=likelihood

P(C|X)=posterior probability

But for practical classification of data we only use the "P(X|C)*P(C)" from the formula and search for it's maximum value.

P(C) can be found out easily and for P(X|C) we use the the following formula

P(X|C)= P(Xk|C)

II k-1

Now let's get back to your problem

By the class label "stolen" from the dataset we can find the probabilities of "yes"and "no" as follows:

P(Yes)=Number of yes/Total number of tuples = 7/10 P(No)=Number of no/Total number of tuples = 3/10

Now let us find out P(X|C) for all other attributes

1)Color

P(Red|Yes)=5/7	P(Red|No)=0
P(Yellow|Yes)=2/7	P(Yellow|No)=3/3

2)Type

P(SUV|Yes)=2/7	P(SUV|No)=2/3
P(Sports|Yes)=5/7	P(Sports|No)=1/3

3)Origin

P(Domestic|Yes)=3/7	P(Domestic|No)=2/3
P(Import|Yes)=4/7	P(Import|No)=1/3

In order to classify the unseen sample (color=Red, Type=SUV, Origin=Domestic) the final formula that is P(X|C)*P(C) is used and the working is as below:

P(X|Yes)*P(Yes) = P(Red|Yes) * P(Domestic|Yes) * P(SUV|Yes) * P(Yes) =5/7 * 3/7 * 2/7 * 7/10 =0.0612 P(X|No)*P(No) = P(Red|No) * P(Domestic|No) * P(SUV|No) * P(No) =0 * 2/3 * 2/3 * 3/10 =0

After finding the final values we can clearly see 0.0612 > 0 hence our unseen sample (color=Red, Type=SUV, Origin=Domestic) gets classifies as "Yes" that means under class label "stolen" this unseen sample is having value as "Yes".