[inciampati]

Is the posterior really an update of our prior? That doesn't make any sense to me. It's P(A|B). I can't then use it as P(A) in another inference based on different observations. What am I missing about your description of Bayesian inference?

[cgel]

Imagine you have two random variables A and B, which are 0 with prob 0.5 and 1 with prob 0.5. They just have the property that when A=1 then B is always 0 and vice versa. Thus, when you have seen the value of B, that clearly has changed the distribution of A. You should read P(A) as: the distribution of A when I know nothing about the world. And P(A|B=0) as: the distribution of A when I know that B took on value 0.