CS 486 - Assignment solutions

CS 486/686 Fall 2012
Assignment 2 Solutions
March 1, 2014

Question 2a

1

Question 2b
Part I - Prior Probability
P r(f raud) = P r(f raud|trav)P r(trav) + P r(f raud|¬trav)P r(¬trav)
= 0.01 ∗ 0.05 + 0.004 ∗ 0.95
= 0.0043

Part II - Posterior Probability
OC
t f f1 (OC) = P r(OC)
0.7
0.3

T rav t f t f

f2 (F raud, T rav) = P r(F raud|T rav)
0.01
0.004
0.99
0.996

T rav t f

f3 (T rav) = P r(T rav)
0.05
0.95

OC t f

f4 (OC) = P r(crp|OC)
0.1
0.001

OC t t f f

F raud t f t f

f5 (OC, F raud) = P r(¬ip|OC, F raud)
0.98
0.99
0.989
0.999

T rav t t f f

F raud t f t f

f6 (T rav, F raud) = P r(f p|T rav, F raud)
0.9
0.9
0.1
0.01

F raud t t f f

Eliminate variable T rav: f7 (F raud) = sumoutT rav [f2 (F raud, T rav) ∗ f3 (trav) ∗ f6 (T rav, F raud)]

2

F raud t f

f7 (F raud)
0.0008
0.0540

Eliminate variable OC: f8 (F raud) = sumoutOC [f1 (OC) ∗ f4 (OC) ∗ f5 (OC, F raud)]
F raud t f

f8 (F raud)
0.0689
0.0696

P r(f raud|f p, ¬ip, crp) = k ∗ f7 (f raud) ∗ f8 (f raud) = 0.0150 where k is a normalizing constant: k= 1 f7 (f raud) ∗ f8 (f raud) + f7 (¬f raud) ∗ f8 (¬f raud)

Question 2c
OC
t f F raud t f

f2 (F raud) = P r(F raud|trav)
0.01
0.99

T rav

f3 () = P r(trav)
0.05

OC t f
OC
t t f f f1 (OC) = P r(OC)
0.65
0.35

f4 (OC) = P r(crp|OC)
0.1
0.001

F raud t f t f

f5 (OC, F raud) = P r(¬ip|OC, F raud)
0.98
0.99
0.989
0.999

F raud t f

f6 (F raud) = P r(f p|trav, F raud)
0.9
0.9

3

Eliminate variable OC: f7 (F raud) = sumoutOC [f1 (OC) ∗ f4 (OC) ∗ f5 (OC, F raud)]
F raud t f

f7 (F raud)
0.0696
0.0689

P r(f raud|f p, ¬ip, crp, trav) = k∗f2 (f raud)∗f3 ()∗f6 (f raud)∗f7 (f raud) = 0.0099 where k is a normalizing constant:
1

k=

f2 (F raud) ∗ f3 () ∗ f6 (F raud) ∗ f7 (F raud)
F raud

Question 2d
When an internet purchase is made, the fraud detection system is