I know this is a contrived example but I am trying to understand how to use Bayesian statistics and I need your help with my doubts.
Let's say I am visiting an island where I know there are a million people and 99% of these people are "Good" while the remaining 10,000 are "Bad". I try to understand if the people I meet are Good or Bad but Good people don't do anything in particular when I meet them. I hold the belief that if a person I meet tries to punch me, he's probably a Bad person, with a probability of 70%, while if he tries to kill me, he's a Bad person with a 90% probability. The fact is, since it is the first time I visit this island, I am not perfectly sure about these estimates of mine.
So, whenever I meet a person for the first time, I start by considering for my belief about him a Beta Distribution, B(1,99) for the probability that this person is Bad. This distribution is peaked around 0.01 (Mode) and I guess it correctly describes a person I still don't know as probably Good.
After interacting with this newly met person, if he didn't do anything, I have no reason to update my belief about him being good or bad (right?) but unfortunately, this person does try to kill me!
My question is: how do I know how to update the parameters of the Beta Distribution to reflect this new knowledge? I expect the new parameters to be something like B(90,10), so that the Mode is now approximately 0.9. But how do I know it is B(90,10) and not B(9,1) or B(18,2) or B (900,100)?
After escaping from this guy, I try to be more careful, but the day after I meet him again and he tries to punch me! Now I should take into consideration the fact that punching isn't as strong a hint as killing but still it does confirm that this person is probably bad. How do I update my belief now? I would expect to obtain another Beta distribution that is somehow larger and with a lower peak.