What is everyone's favorite function / formula? Mine is n/(n+x) It has a bunch of interesting aspects: As n gets bigger, it goes from 0 to 1. When n equals x, it is 0.5 As n gets bigger, the difference between n and n+1 gets smaller For two sufficiently large n's, the results are equal. Say somebody told you about a new cafe in town and that it is completely awesome. The best cafe ever. What probability do you assign…
Essentially if you have a Beta(a, b) prior then your prior mean is a/(a+b) and after observing n samples from a Bernoulli distribution that are all positive, your posterior is Beta(a+n, b) with posterior mean (a+n)/(a+n+b). So in your example you effectively have a Beta(0, x) prior and x (“suspicious”/“gullible”) is directly interpreted as the strength of your prior!