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Measure theory is a tricky subject. Also consider https://twitter.com/ZachWeiner/status/625711339520954368 .
Could you recommend a good source from which to learn measure theory?
There is no such thing as a uniform probability distribution over a countably infinite event space (see Toggle's comment). The distribution you're assuming in your example doesn't exist.
Maybe a better example for your purposes would be picking a random real number between 0 and 1 (this does correspond to a possible distribution, assuming the axiom of choice is true). The probability of the number being rational is 0, the probability of it being greater than 2 is also 0, yet the latter seems "more impossible" than the former.
Of course, this assumes that "probability 0" entails "impossible". I don't think it does. The probability of picking a rational number may be 0, but it doesn't seem impossible. And then there's the issue of whether the experiment itself is possible. You certainly couldn't construct an algorithm to perform it.
I (now) understand the problem with using a uniform probability distribution over a countably infinite event space. However, I'm kind of confused when you say that the example doesn't exist. Surely, its not logically impossible for such an infinite universe to exist. Do you mean that probability theory isn't expressive enough to describe it?
In response to
Open Thread, Jul. 27 - Aug 02, 2015
There are different levels of impossible.
Imagine a universe with an infinite number of identical rooms, each of which contains a single human. Each room is numbered outside: 1, 2, 3, ...
The probability of you being in the first 100 rooms is 0 - if you ever have to make an expected utility calculation, you shouldn't even consider that chance. On the other hand, it is definitely possible in the sense that some people are in those first 100 rooms.
If you consider the probability of you being in room Q, this probability is also 0. However, it (intuitively) feels "more" impossible.
I don't really think this line of thought leads anywhere interesting, but it definitely violated my intuitions.
In response to
Looking to restart Madison LW meetups, in need of regulars
I'm tentatively interested. I live about an hour east of Madison, but as a college student this is really only relevant during the summer. I'll take a look at potential (cheap) transportation.
If you try to do that, you get a paradox where, if A is not creating anyone, B is creating a new person and letting them lead a sad life, and C is creating a new person and letting them lead a happy life, then U(A) = U(B) < U(C) = U(A). You can't say that it's better for someone to be happy than sad, but both are equivalent to nonexistence.
Interesting. Do you have any idea why this results in a paradox, but not the corrigibility problem in general?
In response to
Stupid Questions May 2015
One common way to think about utilitarianism is to say that each person has a utility function and whatever utilitarian theory you subscribe to somehow aggregates these utility functions. My question, more-or-less, is whether an aggregating function exists that says that (assuming no impact on other sentient beings) the birth of a sentient being is neutral. My other question is whether such a function exists where the birth of the being in question is neutral if and only if that sentient being would have positive utility.
EDIT: I do recall that a similar-seeming post: http://lesswrong.com/lw/l53/introducing_corrigibility_an_fai_research_subfield/
In response to
Probability of coming into existence again ?
I think the probability of you popping into existence again is (1) very small and (2) depends on how you define your "self." Would you consider an atom-for-atom copy of you to be "you"? How about an uploaded copy? etc. The simple fact is that physicists have constructed a very simple model for the universe that hasn't been wrong yet and, so, is very likely to be correct in the vast majority of situations - your existence should be one of them. Faith in the accepted model of the universe constructed by modern physicists can be justified by any reasonable prior coupled with Bayes' theorem. Thus, you can be extremely (99.999%+) that you won't pop into existence with infinite suffering (technically 0 and 1 aren't commonly accepted as probabilities on LessWrong).
Moving on, you will almost certainly not live forever (suffering or otherwise), because, quite simply, the universe will experience heat-death at some point. Justification for this belief is, similarly, based on Bayesian updating.
As a side-note. You say
Similar to what happens if there is no free will and thus nothing matters since no change is possible?
I'm not sure free will is a meaningful mental category when used in philosophy. If we lived in a deterministic universe, I, personally, would still believe that life had value. Ultimately, our universe is either deterministic or it isn't, but I fail to see why this would have any important philosophical implications. Why would it be good if our universe contained randomness?
You might consider reading "Possibility and Could-ness" if you haven't done so for an alternative perspective on what free will actually is.
In response to
Justifying (Improper) Priors
I also think that it would be a positive trait for a function to say that there is a finite positive probability of randomly picking more than one in 3^^^3.
I don't see why prior probabilities specifically should have this property. If it was a good property, I'd expect I'd still have it even after establishing a probability distribution. Since I know I won't, I tend to think the entire expectation that it should work that way is just some sort of misunderstanding of probability. Specifically, it reminds me of the base rate fallacy. You should be worried about the expected value of the variable given your belief, not the expected value of your belief given the variable.
I had typed up an eloquent reply to address these issues, but instead wrote a program that scored uniform priors vs 1/x^2 priors for this problem. (Un)fortunately, my idea does consistently (slightly) worse using the p*log(p) metric. So, you are correct in your skepticism. Thank you for the feeback!
Justifying (Improper) Priors
EDIT: I tested this idea in a simulation and it (un)fortunately failed.
This post is an attempt to combine some parts of frequentism with some parts of Bayesianism. Unfortunately, there is a good deal of probability theory. I assume that "improper priors" are okay to use and that we have no prior knowledge about the population of which we are trying to learn.
THEORY
Let's say we're trying to estimate some parameter, B. If we interpret the likelihood function as a conditional probability function, we get a function for our posterior probability density function after taking n samples:
This yields an expected value of B:
Now, intuitively, it would make a lot of sense if, given B=b, the expected value of our posterior probability density function turned out to be actually be B. In frequentist terms, it would be nice if our posterior distribution was unbiased:
So, the question is: If we're trying to estimate a parameter of some distribution, can we pick a prior distribution such that this property is true?
EXAMPLE
Let's say that we have a uniform distribution from 0 to B. So, we draw a sample of "N" numbers with some maximum value "M". Now, let's examine the hypothesis that B=b, for some "b". If b<M, we know that B=b is impossible. If b>M, we know it is possible. In particular, the likelihood function becomes:
The expected value calculation turns out to be:
Now, I leave solving this equation as an exercise for the reader, but it turns out that if we let we get an unbiased estimator for B if
Now, of course, 1/x^2 is definitely not a proper prior distribution for the simple reason that its integral is infinite. However, if we use it as an improper prior distribution, then, any sample with maximum≠0 will yield a valid probability distribution.
DISCUSSION
As far as I know there is no law of probability that says our posterior probability function "should" have an expected value equal to the parameter. That being said it does seem like a positive trait for the function to have. I also suspect that this idea can be extended to a wide variety of distributions. That being said, we obviously tend to have prior information in real life about our populations, so the practical effects of this idea are probably minimal.
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When I say the probability distribution doesn't exist, I'm not talking about the possibility of the world you described. I'm talking about the coherence of the belief state you described. When you say "The probability of you being in the first 100 rooms is 0", it's a claim about a belief state, not about the mind-independent world. The world just has a bunch of rooms with people in them. A probability distribution isn't an additional piece of ontological furniture.
If you buy the Cox/Jaynes argument that your beliefs must adhere to the probability calculus to be rationally coherent, then assigning probability 0 to being in any particular room is not a coherent set of beliefs. I wouldn't say this is a case of probability theory not being "expressive enough". Maybe you want to argue that the particular belief state you described ("Being in any room is equally likely") is clearly rational, in which case you would be rejecting the idea that adherence to the Kolmogorov axioms is a criterion for rationality. But do you think it is clearly rational? On what grounds?
(Incidentally, I actually do think there are issues with the LW orthodoxy that probability theory limns rationality, but that's a discussion for another day.)
From a decision-theory perspective, I should essentially just ignore the possibility that I'm in the first 100 rooms - right?
Similarly, if I'm born in a universe with infinite such rooms and someone tells me to guess whether my room is a multiple of 10 or not. If I guess correctly, I get a dollar; otherwise I lose a dollar.
Theoretically there are as many multiples of 10 as not (both being equinumerous to the integers), but if we define rationality as the "art of winning", then shouldn't I guess "not in a multiple of 10"? I admit that my intuition may be broken here - maybe it just truly doesn't matter which you guess - after all its not like we can sample a bunch of people born into this world without some sampling function. However, doesn't the question still remain: what would a rational being do?