On the expected value argument, are you referring to this?
The answer I think lies in an oft-overlooked fact about expected values: that while probabilities are random variables, expectations are not. Therefore there are no uncertainties associated with predictions made in expectation. Adding the magic words “in expectation” allows longtermists to make predictions about the future confidently and with absolute certainty.
Based on the link to the wiki page for random variables, I think Vaden didn’t mean that the probabilities themselves follow some distributions, but was rather just identifying probability distributions with the random variables they represent, i.e., given any probability distribution, there’s a random variable distributed according to it.
However, I do think his point does lead us to want to entertain multiple probability distributions.
If you did have probabilities over your outcome probabilities or aggregate utilities, I’d think you could just take iterated expectations. If U is the aggregate utility, U∼p and p∼q, then you’d just take the expected value of p with respect to q first, and calculate:
EV∼Eq[p][V]]
If the dependence is more complicated (you talk about correlations), you might use (something similar to) the law of total expectation.
And you’d use Gilboa and Schmeidler’s maxmin expected value approach if you don’t even have a joint probability distribution over all of the probabilities.
A more recent alternative to maxmin is the maximality rule, which is to rule out any choices whose expected utilities are weakly dominated by the expected utilities of another specific choice.
Mogensen comes out against this rule in the end for being too permissive, though. However, I’m not convinced that’s true, since that depends on your particular probabilities. I think you can get further with hedging.
Yeah, that’s the part I’m referring to. I take his comment that expectations are not random variables to be criticizing taking expectations over expected utility with respect to uncertain probabilities.
I think the critical review of ambiguity aversion I linked to us sufficiently general that any alternatives to taking expectations with respect to uncertain probabilities will have seriously undesirable features.
On the expected value argument, are you referring to this?
Based on the link to the wiki page for random variables, I think Vaden didn’t mean that the probabilities themselves follow some distributions, but was rather just identifying probability distributions with the random variables they represent, i.e., given any probability distribution, there’s a random variable distributed according to it.
However, I do think his point does lead us to want to entertain multiple probability distributions.
If you did have probabilities over your outcome probabilities or aggregate utilities, I’d think you could just take iterated expectations. If U is the aggregate utility, U∼p and p∼q, then you’d just take the expected value of p with respect to q first, and calculate:
EV∼Eq[p][V]]If the dependence is more complicated (you talk about correlations), you might use (something similar to) the law of total expectation.
And you’d use Gilboa and Schmeidler’s maxmin expected value approach if you don’t even have a joint probability distribution over all of the probabilities.
A more recent alternative to maxmin is the maximality rule, which is to rule out any choices whose expected utilities are weakly dominated by the expected utilities of another specific choice.
https://academic.oup.com/pq/article-abstract/71/1/141/5828678
https://globalprioritiesinstitute.org/andreas-mogensen-maximal-cluelessness/
https://forum.effectivealtruism.org/posts/WSytm4XG9DrxCYEwg/andreas-mogensen-s-maximal-cluelessness
Mogensen comes out against this rule in the end for being too permissive, though. However, I’m not convinced that’s true, since that depends on your particular probabilities. I think you can get further with hedging.
Yeah, that’s the part I’m referring to. I take his comment that expectations are not random variables to be criticizing taking expectations over expected utility with respect to uncertain probabilities.
I think the critical review of ambiguity aversion I linked to us sufficiently general that any alternatives to taking expectations with respect to uncertain probabilities will have seriously undesirable features.