Benja --
I disagree with Tyrrell (see below), but I can give a version of Tyrrell’s “trivial” formalization:
We want to show that:
Averaging over all theories T, P(T makes correct predictions | T passes 10 tests) > P(T makes correct predictions)
By Bayes’ rule,
P(T makes correct predictions | T passes 10 tests) = P(T makes correct predictions)
P(T passes 10 tests | T makes correct predictions) / P(T passes 10 tests)
So our conclusion is equivalent to:
Averaging over all theories T, P(T passes 10 tests | T makes correct predictions) / P(T passes 10 tests)
1
which is equivalent to
Averaging over all theories T, P(T passes 10 tests | T makes correct predictions) > P(T passes 10 tests)
which has to be true for any plausible definition of “makes correct predictions”. The effect is only small if nearly all theories can pass the 10 tests.
I disagree with Tyrrell’s conclusion. I think his fallacy is to work with the undefined concept of “the best theory”, and to assume that:
If a theory consistent with past observations makes incorrect predictions then there was something wrong with the process by which that theory was formed. (Not true; making predictions is inherently an unreliable process.)
Therefore we can assume that that process produces bad theories with a fixed frequency. (Not meaningful; the observations made so far are a varying input to the process of forming theories.)
In the math above, the fallacy shows up because the set of theories that are consistent with the first 10 observations is different from the set of theories that are consistent with the first 20 observations, so the initial statement isn’t really what we wanted to show. (If that fallacy is a problem with my understanding of Tyrrell’s post, he should have done the “trivial” formalization himself.)
There are lots of ways to apply Bayes’ Rule, and this wasn’t the first one I tried, so I also disagree with Tyrrell’s claim that this is trivial.
>And now the philosopher comes and presents their “thought experiment”—setting up a scenario in which, by
>stipulation, the only possible way to save five innocent lives is to murder one innocent person, and this murder is
>certain to save the five lives. “There’s a train heading to run over five innocent people, who you can’t possibly
>warn to jump out of the way, but you can push one innocent person into the path of the train, which will stop the
>train. These are your only options; what do you do?”
If you are looking out for yourself, it’s an easy decision, at least in the United States. There is no legal requirement to save lives, but dealing with the legal consequences of putting the innocent guy in front of the train is likely to be a real pain in the ass. Therefore, do nothing.
I agree that this isn’t the thought experiment that was originally proposed. If we take inventory of the questions available, we have:
If I’m a real person with real human desires, sit there and let the 5 guys get run over, as I suggest above.
If I’m an AI that is uniformly compassionate and immune from social consequences to my actions, and there’s no compelling reason to value the one above the five, then I’d probably kill one to save five.
* If I’m a person with human desires who is pretending to be perfectly compassionate, then there’s a problem to solve. In this case I prefer to unask the question by stopping the pretense.