I reject the probabilistic argument that any possibility of infinity allows them to dominate all EV calculations. I just don’t think the argument is coherent, at least not in the formulations I’ve seen.
What do you mean by “coherent”? Without inconsistencies/contradictions?
I’m not sure which formulations you’ve seen, but infinite (and undefined) expected values are implied by textbook measure theory with infinities, which uses the extended real numbers, R∪{−∞,+∞}. Basically, we define, for any extended real number a>0,a×(+∞)=+∞, and similarly b+∞=+∞ for any b∈R∪{+∞}, plus some other rules.
Then, for any extended real number-valued random variable X, you can show the following:
If P(X=+∞)>0 and P(X>c)=1 for some c∈R, then E[X]=∞.
If P(X<c)=1 for some c∈R, and P(X=−∞)>0, then E[X]=−∞.
If P(X=+∞)>0 and P(X=−∞)>0, then E[X] is undefined.
If P(X=+∞)>0 or P(X=−∞)>0, then E[X] is not finite (it will be infinite or undefined).
Are you denying that for any standard real number p>0,p×∞ is either not definable coherently, or not ∞? Or something else?
Here’s a proof of 1, where C:=∫X<+∞XdP≥∫X<+∞cdP=cP(X<+∞)>−∞ :
The last equality follows because C>−∞, and the one before because P(X=+∞)>0. All the steps are pretty standard, either following (almost) directly from definitions, or from propositions that are simple to prove from definitions and usually covered early on when integrals are defined.
Infinities make mathematical sense, but they don’t make real world sense. When you’re measuring expected value and the value comes from the actual physical world, it can (empirically) never be infinite.
*You could say black hole singularities are an infinity, but even those are a delta measure with a finite integral.
What do you mean by “coherent”? Without inconsistencies/contradictions?
I’m not sure which formulations you’ve seen, but infinite (and undefined) expected values are implied by textbook measure theory with infinities, which uses the extended real numbers, R∪{−∞,+∞}. Basically, we define, for any extended real number a>0,a×(+∞)=+∞, and similarly b+∞=+∞ for any b∈R∪{+∞}, plus some other rules.
Then, for any extended real number-valued random variable X, you can show the following:
If P(X=+∞)>0 and P(X>c)=1 for some c∈R, then E[X]=∞.
If P(X<c)=1 for some c∈R, and P(X=−∞)>0, then E[X]=−∞.
If P(X=+∞)>0 and P(X=−∞)>0, then E[X] is undefined.
If P(X=+∞)>0 or P(X=−∞)>0, then E[X] is not finite (it will be infinite or undefined).
Are you denying that for any standard real number p>0,p×∞ is either not definable coherently, or not ∞? Or something else?
Here’s a proof of 1, where C:=∫X<+∞X dP≥∫X<+∞c dP=cP(X<+∞)>−∞ :
E[X]=∫X dP=∫X=+∞X dP+∫X<+∞X dP=∫X=+∞X dP+C=∫X=+∞+∞ dP+C
=+∞×P(X=+∞)+C=+∞+C=+∞
The last equality follows because C>−∞, and the one before because P(X=+∞)>0. All the steps are pretty standard, either following (almost) directly from definitions, or from propositions that are simple to prove from definitions and usually covered early on when integrals are defined.
Infinities make mathematical sense, but they don’t make real world sense. When you’re measuring expected value and the value comes from the actual physical world, it can (empirically) never be infinite.
*You could say black hole singularities are an infinity, but even those are a delta measure with a finite integral.
You can still assign actual infinities a decent chance of existing without measuring them directly. Not being able to measure them doesn’t make them incoherent or impossible. I think the universe is probably infinite in spatial extent based on current physics: https://forum.effectivealtruism.org/posts/5nzDZTsi35mDaJny3/punching-utilitarians-in-the-face?commentId=G3DELj34he2bsqszf
I think you’re using “sense” too restrictively.