Q1) At a certain stage of a criminal investigation, ①the inspector in charge is 60% convinced of the guilty of a certain suspect.
Suppose now that a new piece of evidence that shows that ②the criminal has a left-handedness is uncovered.
If ③20% of population possesses this characteristic, ④how certain of the guilt of the suspect should the inspector now be if it turns out that the suspect is among this group?
(how certain of A if B = P(A|B)로 해석)
A) according to Bayesian Theorem'...
<Bayesian Theorem 두 가지 가정 충족 확인!>
1> <Bayesian 가정1> - 표본공간의 분할 -
→ '분할된 원인들(G / Gc)은 상호배반이며 합집합은 전체 표본공간이다'
- P(G) = 'the inspector in charge가 certain suspect가 guilty라 확신할 확률'
- P(Gc) = 'the inspector in charge가 certain suspect가 not guilty라 확신할 확률'
(즉, 해당 용의자가 죄가 있거나 & 있지 않거나 둘 중 하나로 확신해야 한다는 뜻!
가정에 의해, '죄가 있을 것 같기도, 없을 것 같기도 생각하는 건' 없다는 전제를 깔고 들어감)
2> <Bayesian 가정2> - 전확률공식 -
→ '원인(G) & 결과(L; Left-handedness)'가 무엇인지 안다면 결과의 확률을 아래와 같이 표현할 수 있다'
"P(L) = P(L∩G) + P(L∩Gc) = P(G)P(L|G) + P(Gc)P(L|Gc)" (사건 G, Gc는 상호배반이며, G∪Gc = S라고 함)
<Bayesian Theorem - 3종류 확률>
P(A)(조건: guilty 여부) → P(B)(결과: 왼손잡이라는 특징을 가지고 있는 지의 여부)
1> 사전확률 (조건 P(A))
* P(A) = (①) = 0.6
*(베이시안 가정1에 의해 원인 사건들은 서로 상호배반이므로) P(Ac) = (1-①) = 0.4
2> TP(True Positive Rate)
* TP = P(B|A) = (②) = 1 (모든 criminal은 left-handedness라 하였음)
3> FP(False Positive Rate)
* FP = P(B|Ac) = (③) = 0.2 (guilty하지 않은 population 중 left-handed 비율)
<Bayesian 계산>
4> P(A|B) = (④) = (P(A∩B)) / (P(B) = P(A){P(A∩B)/P(A)} / {P(Ac)P(B|Ac) + P(A)P(B|A)}
= (0.6*1) / {(0.6*1) + (0.4*0.2)} = 88.23529411764707(%)
*(베이시안 가정2 전확률 공식 적용)
Q2) After that, ⑤the new evidence(Bayesian UPDATE) is subject to different possible interpretations, and in fact only shows that it is ⑥90% likely that the criminal possess this characteristic.
In this case ⑦how likely would it be that the suspect is guilty?
4> (⑤) the new evidence updated ≫ (4>의 결과 posterior probability = P(A) = 0.8823529411764707)
5> TP update) TP = P(B|A) = (⑥) = 0.9
6> (4> & 5> 반영하면..!) P(A|B) = (⑦) = (P(A∩B)) / (P(B) = P(A){P(A∩B)/P(A)} / {P(Ac)P(B|Ac) + P(A)P(B|A)}
= 97.12230215827338(%)
<결과>
→ after the new evidence has been found, prior Bayesian Probability '88.2%' has been updated to '97.1%'
- this shows that when the inspector finds out the suspect is left-handed, the possibility of making himself feel certain of the guilt of the suspect is about 97.1%, which we can mostly speculate that this suspect is probably guilty
- this case showed us that as new evidence accumulates the suspect will be more likely found to be guilty (but doesn't apply in every experimental cases and in vice versa & as long as we could prove Mathematical Induction in this case. but, no. end now. too far.)
code demonstration (간단)
#prior) 사전 확률
#tpr) True Positive Rate
#fpr) False Positive Rate
def Bayesian(prior, tpr, fpr):
return (prior*tpr) / (prior*tpr + (1-prior)*fpr)
Q1.
Bayesian(0.6,1,0.2)
#ans) 0.8823529411764707
Q2.
Bayesian(Bayesian(0.6,1,0.2),0.9,0.2)
#ans) 0.9712230215827338** Bayesian 이론 관련해서는 추후 포스팅 예정 **
* 출처) Introduction to Probability and Statistics for Engineers and Scientists, 4th Ed.
* 썸네일 출처) http://doingbayesiandataanalysis.blogspot.com/2013/12/icons-for-essence-of-bayesian-and.html
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