Consider an example as follows.
I am running a mobile app that allows users to apply for a loan on the app.
Say a guy signed in to my app to use his phone to apply for a loan.
A = a person has a smartphone
B = default
First of all, I assign
P(B)=0.8 for this guy without any information (just be conservative).
Assume that in my country
P(A) = 0.5, i.e. only 50% of the population do have a smartphone.
P(A|B) = 1, i.e. when I look to my database, all the guys who did not pay back so far do have a smartphone, that is obvious because users need a smartphone to install my app.
So apply Bayes:
P(B|A) = P(A|B) * P (B) / P(A) = 1 * 0.8 / 0.5 = 1.6
Two problems here indeed:
P(B|A) > 1. I know that more than one thread on StackExchange discussed this problem in theory to prove that
P(B|A) <= 1 in all cases but could not find why my inference is wrong.
2) Adding one more bit of information, such as "this guy has a smartphone", according to my Bayesian inference, in fact will increase the probability of default of his case, while in my intuitive inference, it does not bring any information because I know all my customers do have smartphone already. How to explain that?