bayes' formula comprehend

Ideology

Old day people study probability forcus on sample distribute. [fix parameters]
Bayes probability study forcus on parameters.[fix samples]

1	prior distribution + sample info -> Posterior distribution

the new experiment result will refresh people cognition on old things.
It’s fit people cognition law on nation explor method.

formula

$P(B|A) = \frac{P(B)P(A|B)}{P(A)}$

First:

$P(AB) = P(B)P(A|B) = P(A)P(B|A)$

the probability event A, event B happen at the same time
, equals the probability A happen multiply with the probability
B happen when A happen.

So we have:

$P(B|A) = \frac{P(B)P(A|B)}{P(A)}$

$P(law|phenomenon) = P(law)\frac{P(phenomenon|law)}{P(phenomenon)}$

a new knowledge = a priori knowledge * a new factor affect knowledge / a new factor

We use sample experiment’s phenomenon verify rules.
Use adjust factor to adust parameters, make rules more accurate.

Example

We need to check a rule:
watermelon sugar content > 10, good probability is 70%

prior condition is good watermelon probability is 20%
exp:bad watermelon has probability 25%, sugar content > 10
exp:good watermelon has probability 99%, sugar content > 10

so we have :

$P(B|A) = P(B) \frac{P(A|B)}{P(A)} = P(B) \frac{P(A|B)}{P(A|B)P(B) + P(A|\overline{B})P(\overline{B})}$ $P(B|A) = 0.2 * ( \frac{0.99}{0.99 * 0.2 + 0.25 * 0.8} ) \approx 0.497$

Now we found the threshold is not right, base on adjust factor, we change rule:
watermelon sugar content > 15, good probability is 70%

prior condition is good watermelon probability is 20%
exp:bad watermelon has probability 15%, sugar content > 15
exp:good watermelon has probability 90%, sugar content > 15

so we have :

$P(B|A) = P(B) \frac{P(A|B)}{P(A)} = P(B) \frac{P(A|B)}{P(A|B)P(B) + P(A|\overline{B})P(\overline{B})}$ $P(B|A) = 0.2 * ( \frac{0.9}{0.9 * 0.2 + 0.15 * 0.8} ) \approx 0.600$

Keep on adjust:
watermelon sugar content > 20, good probability is 70%

prior condition is good watermelon probability is 20%
exp:bad watermelon has probability 9%, sugar content > 20
exp:good watermelon has probability 85%, sugar content > 20

$P(B|A) = 0.2 * ( \frac{0.85}{0.85 * 0.2 + 0.09 * 0.8} ) \approx 0.702$

https://www.cnblogs.com/zhoulujun/p/8893393.html