Comment by mitthrowaway2
3 years ago
Sure, so you add up those terms in the usual way, which you can do by straightforward factorization and summation, until your sum has diminished to probabilities small enough to round off.
This mathematical procedure is usually called "applied common sense", and if you are attempting to argue that it's impossible, you would in the process be proving that nobody can ever learn anything about anything.
You're still assuming that we know all the terms. We use randomized trials with controls improving out medicines because we almost never have enough information to compare two situationally different groups. In a controlled randomized trial, we can declare what we're testing and what we're measuring up front. The data is just too messy, otherwise. Too many hidden variables and too many possible alternate explanations that weren't measured.