alt Hacker NewsSay your forum is a big one and has 1000 active users, with 1 joining every day. Most will be a lot smaller/less active.
What are the chances that someone uses this vpn, joins your forum the day after someone was banned, and has an ip in a similar range?
For most small websites this would be strong evidence.
I think you are (informally and correctly) doing Bayes theorem here. The prior is combined with the conditional to give the posterior estimate; the conditional is not itself the estimate.