There are definitely well-defined measures on any set (e.g. pick one atomic outcome to have probability 1 and the rest 0); there’s just not only one, and picking exactly one would be arbitrary. But the same is true for any set of outcomes with at least two outcomes, including finite ones (or it’s at least often arbitrary when there’s not enough symmetry for equiprobability).
For the question of how many people will exist in the future, you could use a Poisson distribution. That’s well-defined, whether or not it’s a reasonable distribution to use.
Of course, trying to make your space more and more specific will run into feasibility issues.
There are non-measurable sets (unless you discard the axiom of choice, but then you’ll run into some significant problems.) Indeed, the existence of non-measurable sets is the reason for so much of the measure-theoretic formalism.
If you’re not taking a measure theoretic approach, and instead using propositions (which I guess, it should be assumed that you are, because this approach grounds Bayesianism), then using infinite sets (which clearly one would have to do if reasoning about all possible futures) leads to paradoxes. As E.T. Jaynes writes in Probability Theory and the Logic of Science:
It is very important to note that our consistency theorems have been established only for probabilities assigned on finite sets of propositions … In laying down this rule of conduct, we are only following the policy that mathematicians from Archimedes to Gauss have considered clearly necessary for nonsense avoidance in all of mathematics. (pg. 43-44).
There are non-measurable sets (unless you discard the axiom of choice, but then you’ll run into some significant problems.) Indeed, the existence of non-measurable sets is the reason for so much of the measure-theoretic formalism.
This depends on the space.
It’s at least true for real-valued intervals with continuous measures, of course, but I think you’re never going to ask for the measure of a non-measurable set in real-world applications, precisely because they require the axiom of choice to construct (at least for the real numbers, and I’d assume, by extension, any subset of any Rn), and no natural set you’ll be interested in that comes up in an application will require the axiom of choice (more than dependant choice) to construct. I don’t think the existence of non-measurable sets is viewed as a serious issue for applications.
It is not true in a countable measure space (or, at least, you could always extend the measure to get this to hold), since assuming each singleton (like {x},x∈X) is measurable, every union of countably many singletons is measurable, and hence every subset is measurable (A=∪x∈A{x} is a countable union of singletons, A⊆X, X countable) . In particular, if you’re just interested in the number of future people, assuming there are at most countably infinitely many (so setting aside the many-worlds interpretation of quantum mechanics for now), then your space is just the set of non-negative integers, which is countable.
using infinite sets (which clearly one would have to do if reasoning about all possible futures)
You could group outcomes to represent them with finite sets. Bayesians get to choose the measure spaces/propositions they’re interested in. But again, I don’t think dealing with infinite sets is so bad in applications.
Another way to look at this. What do you think is the probability that everyone will go extinct tomorrow? If you are agnostic about that, then you must also be agnostic about the value of GiveWell-type stuff.
If you are agnostic about that, then you must also be agnostic about the value of GiveWell-type stuff
Why? GiveWell charities have developed theories about the effects of various interventions. The theories have been tested and, typically, found to be relatively robust. Of course, there is always more to know, and always ways we could improve the theories.
I don’t see how this relates to not being able to develop a statistical estimate of the probability we go extinct tomorrow. (Of course, I can just give you a number and call it “my belief that we’ll go extinct tomorrow,” but this doesn’t get us anywhere. The question is whether it’s accurate—and what accuracy means in this case.) What would be the parameters of such a model? There are uncountably many things—most of them unknowable—which could affect such an outcome.
The benefits of GiveWell’s charities are worked out as health or economic benefits which are realised in the future. e.g. AMF is meant to be good because it allows people who would have otherwise died to live for a few more years. If you are agnostic about whether everyone will go extinct tomorrow, then you must be agnostic about whether people will actually get these extra years of life.
What is your probability distribution across the size of the future population, provided there is not an existential catastrophe?
Do you for example think there is a more than 50% chance that it is greater than 10 billion?
I don’t have a probability distribution across the size of the future population. That said, I’m happy to interpret the question in the colloquial, non-formal sense, and just take >50% to mean “likely”. In that case, sure, I think it’s likely that the population will exceed 10 billion. Credences shouldn’t be taken any more seriously than that—epistemologically equivalent to survey questions where the respondent is asked to tick a very unlikely, unlikely, unsure, likely, very likely box.
I don’t think the question makes sense. I agree with Vaden’s argument that there’s no well-defined measure over all possible futures.
There are definitely well-defined measures on any set (e.g. pick one atomic outcome to have probability 1 and the rest 0); there’s just not only one, and picking exactly one would be arbitrary. But the same is true for any set of outcomes with at least two outcomes, including finite ones (or it’s at least often arbitrary when there’s not enough symmetry for equiprobability).
For the question of how many people will exist in the future, you could use a Poisson distribution. That’s well-defined, whether or not it’s a reasonable distribution to use.
Of course, trying to make your space more and more specific will run into feasibility issues.
There are non-measurable sets (unless you discard the axiom of choice, but then you’ll run into some significant problems.) Indeed, the existence of non-measurable sets is the reason for so much of the measure-theoretic formalism.
If you’re not taking a measure theoretic approach, and instead using propositions (which I guess, it should be assumed that you are, because this approach grounds Bayesianism), then using infinite sets (which clearly one would have to do if reasoning about all possible futures) leads to paradoxes. As E.T. Jaynes writes in Probability Theory and the Logic of Science:
(Vaden makes this point in the podcast.)
This depends on the space.
It’s at least true for real-valued intervals with continuous measures, of course, but I think you’re never going to ask for the measure of a non-measurable set in real-world applications, precisely because they require the axiom of choice to construct (at least for the real numbers, and I’d assume, by extension, any subset of any Rn), and no natural set you’ll be interested in that comes up in an application will require the axiom of choice (more than dependant choice) to construct. I don’t think the existence of non-measurable sets is viewed as a serious issue for applications.
It is not true in a countable measure space (or, at least, you could always extend the measure to get this to hold), since assuming each singleton (like {x},x∈X) is measurable, every union of countably many singletons is measurable, and hence every subset is measurable (A=∪x∈A{x} is a countable union of singletons, A⊆X, X countable) . In particular, if you’re just interested in the number of future people, assuming there are at most countably infinitely many (so setting aside the many-worlds interpretation of quantum mechanics for now), then your space is just the set of non-negative integers, which is countable.
You could group outcomes to represent them with finite sets. Bayesians get to choose the measure spaces/propositions they’re interested in. But again, I don’t think dealing with infinite sets is so bad in applications.
Do you for example think there is a more than 50% chance that it is greater than 10 billion?
Another way to look at this. What do you think is the probability that everyone will go extinct tomorrow? If you are agnostic about that, then you must also be agnostic about the value of GiveWell-type stuff.
Why? GiveWell charities have developed theories about the effects of various interventions. The theories have been tested and, typically, found to be relatively robust. Of course, there is always more to know, and always ways we could improve the theories.
I don’t see how this relates to not being able to develop a statistical estimate of the probability we go extinct tomorrow. (Of course, I can just give you a number and call it “my belief that we’ll go extinct tomorrow,” but this doesn’t get us anywhere. The question is whether it’s accurate—and what accuracy means in this case.) What would be the parameters of such a model? There are uncountably many things—most of them unknowable—which could affect such an outcome.
The benefits of GiveWell’s charities are worked out as health or economic benefits which are realised in the future. e.g. AMF is meant to be good because it allows people who would have otherwise died to live for a few more years. If you are agnostic about whether everyone will go extinct tomorrow, then you must be agnostic about whether people will actually get these extra years of life.
I don’t have a probability distribution across the size of the future population. That said, I’m happy to interpret the question in the colloquial, non-formal sense, and just take >50% to mean “likely”. In that case, sure, I think it’s likely that the population will exceed 10 billion. Credences shouldn’t be taken any more seriously than that—epistemologically equivalent to survey questions where the respondent is asked to tick a very unlikely, unlikely, unsure, likely, very likely box.