bayes’ theorem. an insurance company divides its clients into two categories: those who are...
TRANSCRIPT
Bayes’ Theorem
Bayes’ Theorem An insurance company divides its clients into two
categories: those who are accident prone and those who are not. Statistics show there is a 40% chance an accident prone person will have an accident within 1 year whereas there is a 20% chance non-accident prone people will have an accident within the first year.
If 30% of the population is accident prone, what is the probability that a new policyholder has an accident within 1 year?
Bayes’ Theorem
Let A be the event a person is accident prone Let F be the event a person has an accident
within 1 year
A AC
F
FAP FAP C
CC FAP CFAP
Bayes’ Theorem
Notice we’ve divided up or partitioned the sample space along accident prone and non-accident prone
A AC
F
FAP FAP C
CC FAP CFAP
Bayes’ Theorem Notice that and are mutually
exclusive events and that
Therefore
We need to find and
How?
FA FAC
FFAFA C
FAPFAPFP C
FAP FAP C
Bayes’ Theorem
Recall from conditional probability
FEPFPFEPFP
FEPFEP
|
|
EFPFEPEPEFPEP
EFPEFP
|
|
Bayes’ Theorem
Thus:
APAFPFAP |
CCC APAFPFAP |
P(A) = 0.30 since 30% of population is accident prone
P(F|A) = 0.40 since if a person is accident prone, then his chance of having an accident within 1 year is 40%
P(F|AC) = 0.2 since non-accident prone people have a 20% chance of having an accident within 1 year
P(AC) = 1- P(A) = 0.70
Bayes’ Theorem Updating our Venn Diagram
Notice again that
A AC
F
APAFPFAP | CCC APAFPFAP |
CC FAP
CFAP
FAPFAPFP C
Bayes’ Theorem
So the probability of having an accident within 1 year is:
26.070.020.030.040.0||
CC
C
APAFPAPAFP
FAPFAPFP
Bayes’ Theorem
Using Tree Diagrams:
Accident Prone P(A) = 0.30
Not Accident Prone P(AC) = 0.70
Accident w/in 1 year P(F|A)=0.40
No Accident w/in 1 year P(FC|A)=0.60
Accident w/in 1 year P(F|AC)=0.20
No Accident w/in 1 year P(FC|AC)=0.80
FAP
FAP C
CC FAP
CFAP
Bayes’ Theorem
Notice you can have an accident within 1 year by following branch A until F is reached The probability that F is reached via branch A is
given by In other words, the probability of being accident
prone and having one within 1 year is
APAFP |
APAFPFAP |
Bayes’ Theorem
You can also have an accident within 1 year by following branch AC until F is reached The probability that F is reached via branch AC is
given by In other words, the probability of NOT being
accident prone and having one within 1 year is
CC APAFP |
CCC APAFPFAP |
Bayes’ Theorem
What would happen if we had partitioned our sample space over more events, say , all them mutually exclusive?
Venn Diagram
nAAA ,,, 21
A1 A2 An-1 An . . . . . .
(etc.)
F
FAP 1 FAP 2 FAP n 1 FAP n
Bayes’ Theorem
For each
FAPFAPFAPFP n 21
iii APAFPFAP |
n
iii
nn
n
APAFP
APAFPAPAFPAPAFPFAPFAPFAPFP
1
2211
21
|
|||
Bayes’ Theorem
Tree Diagram
1AP
2AP
1nAP
nAP
1|AFP
1|AFP C
2|AFP
2|AFP C
1| nAFP
1| nC AFP
nAFP |
nC AFP |
Bayes’ Theorem
Notice that F can be reached via branches
Multiplying across each branch tells us the probability of the intersection
Adding up all these products gives:
nAAA ,,, 21
n
iii APAFPFP
1
|
Bayes’ Theorem
Ex: 2 (text tractor example) Suppose there are 3 assembly lines: Red, White, and Blue. Chances of a tractor not starting for each line are 6%, 11%, and 8%. We know 48% are red and 31% are blue. The rest are white. What % don’t start?
Bayes’ Theorem
Soln. R: red P(R) = 0.48W: white P(W) = 0.21B: blue P(B) = 0.31N: not starting
P(N | R) = 0.06P(N | W) = 0.11P(N | B) = 0.08
Bayes’ Theorem
Soln.
0767.031.008.021.011.048.006.0
|||
BPBNPWPWNPRPRNPNP
Bayes’ Theorem
Main theorem:Suppose we know . We would like to use this information to find if possible.
Discovered by Reverend Thomas Bayes
FEP | EFP |
Bayes’ Theorem
Main theorem:
Ex. Suppose and partition a space and A is some event.
Use and to determine .
1B 2B
,|,, 121 BAPBPBP 2|BAP ABP |1
Bayes’ Theorem
Recall the formulas:
So,
2211 || BAPBPBAPBPAP
AP
ABPABP
1
1 |
2211
11
11
|||
|
BAPBPBAPBPBPBAP
APABP
ABP
1111 | BPBAPBAPABP
Bayes’ Theorem
Bayes’ Theorem:
n
iii
kkk
BPBAP
BPBAPABP
1
|
||
Bayes’ Theorem Ex. 4 (text tractor example) 3 assembly lines:
Red, White, and Blue. Some tractors don’t start (see Ex. 2). Find prob. of each line producing a non-starting tractor.
P(R) = 0.48 P(N | R) = 0.06P(W) = 0.21 P(N | W) = 0.11P(B) = 0.31 P(N | B) = 0.08
Bayes’ Theorem
Soln. Find P(R | N), P(W | N), and P(B | N)P(R) = 0.48 P(N | R) = 0.06P(W) = 0.21 P(N | W) = 0.11P(B) = 0.31 P(N | B) = 0.08
Bayes’ Theorem
Soln.
3755.031.008.021.011.048.006.0
48.006.0|||
|
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BPBNPWPWNPRPRNPRPRNP
NPNRP
NRP
Bayes’ Theorem
Soln.
3012.0
31.008.021.011.048.006.021.011.0
||||
|
BPBNPWPWNPRPRNP
WPWNPNWP
3233.0| NBP
Bayes’ Theorem
Focus on the Project:
We want to find the following probabilities: and .
To get these, use Bayes’ Theorem
CTYSP | CTYFP |
Bayes’ Theorem
Focus on the Project:
FPFCTYPSPSCTYP
SPSCTYPCTYSP
||
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FPFCTYPSPSCTYP
FPFCTYPCTYFP
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Bayes’ Theorem
Focus on the Project:
In Excel, we find the probability to be approx. 0.4774
FPFCTYPSPSCTYP
SPSCTYPCTYSP
||
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Bayes’ Theorem
Focus on the Project:
In Excel,we find the probability to be approx. 0.5226
FPFCTYPSPSCTYP
FPFCTYPCTYFP
||
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Bayes’ Theorem
Focus on the Project:
Let Z be the value of a loan work out for a borrower with 7 years, Bachelor’s, Normal…
000,040,2$5226.0000,2504774.0000,000,4
Failure Prob. Failure Success Prob. Success
ZE
Bayes’ Theorem
Focus on the Project:
Since foreclosure value is $2,100,000 and on average we would receive $2,040,000 from a borrower with John Sanders characteristics, we should foreclose.
Bayes’ Theorem
Focus on the Project:
However, there were only 239 records containing 7 years experience.
Look at range of value 6, 7, and 8 (1 year more and less)
Bayes’ Theorem
Focus on the Project:
Use DCOUNT function with an extra “Years in Business” heading
Same for “no”
Added a new column
Former BankYears In Business Education Level
State Of Economy
Loan Paid Back?
Years In Business
BR >=6 yes <=8
Bayes’ Theorem Focus on the Project:
From this you get 349 successful and 323 failed records
Let be a borrower with 6, 7, or 8 years experience
and 1470349| SYP
Y
1779323| FYP
Bayes’ Theorem
Focus on the Project:
0504.05222.05314.01816.0
||||
FCPFTPFYPFCTYP
0733.05823.05301.02374.0
||||
SCPSTPSYPSCTYP
Bayes’ Theorem Focus on the Project:
Use Bayes’ Theorem to get new probabilities
: 6, 7, or 8 years, Bachelor’s, Normal (indicates work out)
5575.0| CTYSP 4425.0| CTYFP
000,341,2$ZE
Z
Bayes’ Theorem
Focus on the Project:
We can look at a large range of years.
Look at range of value 5, 6, 7, 8, and 9 (2 years more and less)
Bayes’ Theorem
Focus on the Project:
Use DCOUNT function with an extra “Years in Business” heading
Same for “no”
Added a new column
Former BankYears In Business Education Level
State Of Economy
Loan Paid Back?
Years In Business
BR >=5 yes <=9
Bayes’ Theorem Focus on the Project:
From this you get 566 successful and 564 failed records
Let be a borrower with 5, 6, 7, 8, or 9 years exper.
and 1470566|" SYP
Y
1779564|" FYP
Bayes’ Theorem
Focus on the Project:
0880.05222.05314.03170.0
||||
FCPFTPFYPFCTYP
1189.05823.05301.03850.0
||||
SCPSTPSYPSCTYP
Bayes’ Theorem Focus on the Project:
Use Bayes’ Theorem to get new probabilities
: 5, 6, 7, 8, or 9 years, Bachelor’s, Normal
(indicates work out)
5392.0| CTYSP
4608.0| CTYFP
000,272,2$ZEZ
Bayes’ Theorem
Focus on the Project:
Since both indicated a work out while only indicated a foreclosure, we will work out a new payment schedule.
Any further extensions…
ZEZE and ZE