I presented a mathematical theory on a medical testing method. This fundamental theory can be taken account of both cases when the resource of the testing is limited or not. One implication is that "negative proof" may not function well, and another implication is that excessively high specificity and accuracy are required for meaningful diagnosis unless the careful usage of the diagnosis is considered.
5. Prob( hᵢ | e ) ∝ Prob( hᵢ ) Prob( e | hᵢ ) (∝ )
Prob( hᵢ | e ) = Prob( hᵢ ) Prob( e | hᵢ ) / Prob ( e )
P( e ∩ hᵢ ) / P ( e )
{ P ( hᵢ ) P ( e ∩ hᵢ ) } / { P ( hᵢ ) P ( e ) }
P ( hᵢ )
6. (1702-1761)
Prob( hᵢ | e ) ∝ Prob( hᵢ ) Prob( e | hᵢ )
•
• ( 0 )
•
• 100%
100%
•
•
•
Wikipedia
7. 1. (hypothesis) :
h₀ h₁
2. (evidence) Prob ( e | h₀ ) : Prob ( e | h₁ )
• : (100% - )
• (100% - ) :
3. : 99% 70%
• ( ) 99% : 30% = 3.3 : 1 ( ) 1% : 70% = 1 : 70
• :
3.3 70
• 100% ( )
a : b 3.3 a : b a : 70 b
9. 1.
1/5000 5000
2. 1, 2, 5 10
3. 1, 2, 5, 10, 20,
50, 100, 20, 500, 1000,
2000, 5000
(1, 1/2, 1/5, 1/10, ..)
4.
5. 70
( ) 1/3.33 ( )
6. 70
5%
79%
Y = L X / ( L X + ( 1 – X ) )
10. Y = L X / ( L X + ( 1 – X ) )
0 ≦ X ≦ 1 , 0 ≦ Y ≦ 1
L=M 10 ⁻ᴱ ; E ∈ { }
(1) :
M=1: -3 ≦ E ≦ 3
(2) :
M=2,5: -4 ≦ E ≦ 3
(3) :
M=3,4,6,7,8,9: -3≦E≦ 2
(4) :
M=1.2, 1.4, 1.6, 1.8,
2.5, 3.5, 4.5 :
-2≦ E ≦ 2
L
X
Y
11. p log ( p / (1-p) )
[0, 1] ( 0% 100%)
[-∞ , +∞]
e ˣ / ( 1 + e ˣ )