d) Repeat part (b) using the conditional probabilities given in part (c). e) Compare the two methods for estimating probabilities. Which method is better and why?

Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
Section: Chapter Questions
Problem 1PE
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Complete questions d and e only
Question 1: Consider the data set shown in the following Table:
Record
A B C
0
0 0
2
0
0
1
3
0
1
1
4
0 1
1
0
1
1
1
1
1
10
1
0 1
a) Estimate the conditional probabilities for P(A+), P(B+), P(CI+), P(A-), P(B|-), and P(C-).
b) Use the estimate of conditional probabilities given in the previous question to predict the class label for a test
sample (A=0, B=1, C=0) using the naive Bayes approach.
c) Estimate the conditional probabilities using the m-estimate approach, with p=1/2 and m=4.
d) Repeat part (b) using the conditional probabilities given in part (c).
e) Compare the two methods for estimating probabilities. Which method is better and why?
Ans a,b,c
a)
P (A = 1|?) = 2/5 = 0.4, P (B = 11?) = 2/5 = 0.4, P (C = 11?) = 1, P (A= 01?) = 3/5 = 0.6,
P (B = 01?) = 3/5 = 0.6, P (C = 0[?) = 0; P (A = 1|+) = 3/5 = 0.6, P (B = 1|+) = 1/5 = 0.2, P (C = 11+) = 4/5 = 0.8,
P (A=0|+) = 2/5 = 0.4, P (B = 0+) = 4/5 = 0.8, P(C=0]+)=1/5=0.2
b)
Let's assume that P (A = 0, B = 1, C = 0) = W.
P (+|A=0, B = 1, C = 0) = (P (A = 0, B = 1, C = 0+) x P (+))/(P (A = 0, B = 1, C = 0))
= (P (A = 0]+)P (B= 1+)P (C = 0+) * P (+))/ W
= .008/W
P (?|A = 0, B = 1, C = 0) = (P (A = 0, B = 1, C = 0[?) x P (?))/(P (A = 0, B = 1, C = 0))
P (A= 01?) x P (B = 1|?) x P (C = 0[?) * P (?) /W
= 0/W
Hence, the class label should be '+'.
c)
P (A=0|+) = (2+2)/(5 + 4) = 4/9,
P (A = 0[?)=(3+2)/(5 + 4) = 5/9,
P (B= 1+) = (1+2)/(5 + 4) = 3/9,
P (B=11?) = (2+2)/(5 + 4) = 4/9,
P (C = 0+) = (1 + 2)/(5 + 4) = 3/9,
P (C = 01?) = (0+2)/(5 + 4) = 2/9.
Class
++
Transcribed Image Text:Complete questions d and e only Question 1: Consider the data set shown in the following Table: Record A B C 0 0 0 2 0 0 1 3 0 1 1 4 0 1 1 0 1 1 1 1 1 10 1 0 1 a) Estimate the conditional probabilities for P(A+), P(B+), P(CI+), P(A-), P(B|-), and P(C-). b) Use the estimate of conditional probabilities given in the previous question to predict the class label for a test sample (A=0, B=1, C=0) using the naive Bayes approach. c) Estimate the conditional probabilities using the m-estimate approach, with p=1/2 and m=4. d) Repeat part (b) using the conditional probabilities given in part (c). e) Compare the two methods for estimating probabilities. Which method is better and why? Ans a,b,c a) P (A = 1|?) = 2/5 = 0.4, P (B = 11?) = 2/5 = 0.4, P (C = 11?) = 1, P (A= 01?) = 3/5 = 0.6, P (B = 01?) = 3/5 = 0.6, P (C = 0[?) = 0; P (A = 1|+) = 3/5 = 0.6, P (B = 1|+) = 1/5 = 0.2, P (C = 11+) = 4/5 = 0.8, P (A=0|+) = 2/5 = 0.4, P (B = 0+) = 4/5 = 0.8, P(C=0]+)=1/5=0.2 b) Let's assume that P (A = 0, B = 1, C = 0) = W. P (+|A=0, B = 1, C = 0) = (P (A = 0, B = 1, C = 0+) x P (+))/(P (A = 0, B = 1, C = 0)) = (P (A = 0]+)P (B= 1+)P (C = 0+) * P (+))/ W = .008/W P (?|A = 0, B = 1, C = 0) = (P (A = 0, B = 1, C = 0[?) x P (?))/(P (A = 0, B = 1, C = 0)) P (A= 01?) x P (B = 1|?) x P (C = 0[?) * P (?) /W = 0/W Hence, the class label should be '+'. c) P (A=0|+) = (2+2)/(5 + 4) = 4/9, P (A = 0[?)=(3+2)/(5 + 4) = 5/9, P (B= 1+) = (1+2)/(5 + 4) = 3/9, P (B=11?) = (2+2)/(5 + 4) = 4/9, P (C = 0+) = (1 + 2)/(5 + 4) = 3/9, P (C = 01?) = (0+2)/(5 + 4) = 2/9. Class ++
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