For clarity, what I think in the case of “scenario starts with one small bag containing 1-10 and one large bag containing 1-10,000, each piece of paper goes to exactly one person, each person gets exactly one piece of paper, you are one of the people” is that, prior to seeing what number is on your piece of paper, you should think there’s a 10,00010,000+10≈0.999 chance that the paper you received came from the large bag—i.e. not a uniform prior of “50/50 small bag or large bag”.
Do you mean a situation where every piece of paper from each of the two bags is given, therefore 10010 people got a piece of paper? If so this is very much not what I’ve been talking about. We are dealing with an either/or case, where only one bag is used to give papers, but you have no idea which one. This should be obvious in the context of doomsday argument, because humanity doesn’t simultaneously has long and short history.
Anyway, as I said, lets forget about anthropics for now and deal with the marble picking example from the previous comment.
the main justification I use is “probabilities are facts about my model of the world, not facts about the world itself, and so the mechanics of my model dictate the probability, and the prior is no exception”.
Oh sure, probabilities are about the model. But we want our models of the world to correspond to the way world actually is, so that our models were useful. You want a model that systematically produces correct answers in reality, not just being self-consistent.
“A uniform prior” seems like a pretty good one in the complete absence of any other information
It sure does. I encorage you to think about why. Anyway, lets, for now, simply accept that if we do not have any information about some alternatives we assume equiprobable prior. Now consider these scenarios. I think They highlight the crux of disagreement very well:
1. There is a bag on your table. You know that it’s either bag 1 or bag 2 and you know nothing else. What should be your credence that it’s bag 1?
Here we have uniform prior between bags. P(Bag 1) = P(Bag 2) = 1⁄2
2. A marble was picked. You know that there are 20 possible marbles that could be picked: 14 red and 6 blue. What is your credence that the marble blue?
Here we have uniform prior between marbles: P(Blue) = 3⁄10
3. There is a bag on your table. You know that it’s either bag 1 which contains 9 red and 1 blue marbles or bag 2 which contains 5 red and 5 blue marbles. You see a person blindly picked a marble from the bag. What’s the probability that the marble the person picked is blue.
Are we supposed to be using uniform prior over bags or are we supposed to be using uniform prior over marbles here? This is an important question because even though In this case both produce the same answer:
If the arrangement of the marbles was different, say, bag 1 one contained 10 red and 2 blue while bag 2 contained 4 red and 4 blue, the situation changes:
So how do we decide what to do? What is the principled position here? Notice that this case has absolutely nothing to do with anthropics and is basic probability theory.
Do you mean a situation where every piece of paper from each of the two bags is given, therefore 10010 people got a piece of paper? If so this is very much not what I’ve been talking about. We are dealing with an either/or case, where only one bag is used to give papers, but you have no idea which one. This should be obvious in the context of doomsday argument, because humanity doesn’t simultaneously has long and short history.
Anyway, as I said, lets forget about anthropics for now and deal with the marble picking example from the previous comment.
Oh sure, probabilities are about the model. But we want our models of the world to correspond to the way world actually is, so that our models were useful. You want a model that systematically produces correct answers in reality, not just being self-consistent.
It sure does. I encorage you to think about why. Anyway, lets, for now, simply accept that if we do not have any information about some alternatives we assume equiprobable prior. Now consider these scenarios. I think They highlight the crux of disagreement very well:
1. There is a bag on your table. You know that it’s either bag 1 or bag 2 and you know nothing else. What should be your credence that it’s bag 1?
Here we have uniform prior between bags. P(Bag 1) = P(Bag 2) = 1⁄2
2. A marble was picked. You know that there are 20 possible marbles that could be picked: 14 red and 6 blue. What is your credence that the marble blue?
Here we have uniform prior between marbles: P(Blue) = 3⁄10
3. There is a bag on your table. You know that it’s either bag 1 which contains 9 red and 1 blue marbles or bag 2 which contains 5 red and 5 blue marbles. You see a person blindly picked a marble from the bag. What’s the probability that the marble the person picked is blue.
Are we supposed to be using uniform prior over bags or are we supposed to be using uniform prior over marbles here? This is an important question because even though In this case both produce the same answer:
Over bags:
P(Bag 1) = P(Bag 2) = 1⁄2
P(Blue) = P(Blue|Bag 1)P(Bag 1) + P(Blue| Bag 2)P(Bag 2) = 1⁄20 + 1⁄4 = 6⁄20 = 3⁄10
Over marbles:
P(Blue) = 3⁄10
If the arrangement of the marbles was different, say, bag 1 one contained 10 red and 2 blue while bag 2 contained 4 red and 4 blue, the situation changes:
Over bags:
P(Blue) = P(Blue|Bag 1)P(Bag 1) + P(Blue| Bag 2)P(Bag 2) = 1⁄22 + 1⁄4 = 6⁄20 = 13⁄44
Over marbles:
P(Blue) = 3⁄10
So how do we decide what to do? What is the principled position here? Notice that this case has absolutely nothing to do with anthropics and is basic probability theory.