Appendix 2.1: Derivation of Bayes’ Theorem
We know that the following formulas are correct:
P(A|B)=P(AB)P(B)
(1)
P(B|A)=P(AB)P(A)
(2)
[which can be rewritten as P(A'B)=P(B|A')P(A')].
(3)
Furthermore, by definition, we know that
P(B)==P(AB)+P(A′B)P(B|A)P(A)+P(B|A′)P(A′)↖from(2) from(3)↗
(4)
Substituting Equations 2 and into Equation 1, we have
P(A|B)==P(AB)P(B) ↙from (2)P(B|A)P(A)P(B∣∣A)P(A)+P(B∣∣A′)P(A′) from (4)↗
(5)
This is the general form of Bayes’ Theorem, shown as Equation 2- in this chapter.
..................Content has been hidden....................
You can't read the all page of ebook, please click
here login for view all page.