We can meaningfully talk about the map and whether it’s correctly represents the territory
In confusing anthropic situations, we shouldn’t. Correctness implies one-dimensional measure and objectivity and then people start arguing what is “correct” probability in Sleeping Beauty. You can invent some theory of subjective correctness, or label some mathematically isomorphic reasoning as incorrect but useful. Or you can use existing general framework for subjective problems that works every time—utility. Even if you want to know what would maximize correctness, you can just make you utility function care only about being correct—that still makes the necessity of the answer to “correct when?” obvious.
The technical justification for all of this is that the meaning of correctness for probability is not checked, but defined from it being useful—the law of large numbers is a value-laden bridge law. The need for any approximation is derived from it being useful.
Which is of course doesn’t mean that in practice we never can factor out and usefully talk only about correctness. But that’s a shortcut and if it leads to confusion, it can be solved by remembering what was the point of using probability from the start.
I’d say the opposite. The more confusing the case the more important is to make it as simple as possible in order not to multiply possible sources of consufion.
Correctness implies one-dimensional measure and objectivity and then people start arguing what is “correct” probability in Sleeping Beauty.
Well, yes. Sleeping Beauty is actually a great example why you should be more careful with invoking betting, while trying to solve probability theory problems, as you may stumble into a case that doesn’t satisfy Kolmogorov Axioms without noticing it. I’ll talk more about it after I’ll have finished all the prerequisite posts. For now, it’s suffice to say that we can easily talk about different probabilities for Heads: on average awakening and on average experiment, just as we can talk about different betting schemes and the addition of betting scheme doesn’t make the problem clear in any way.
the law of large numbers is a value-laden bridge law.
I’m not sure what you mean by it. Law of large numbers is just a fact from probability theory, it doesn’t require utility functions or betting.
I’m not sure what you mean by it. Law of large numbers is just a fact from probability theory, it doesn’t require utility functions or betting.
I meant that the law is just a statement about probability, not about simulations confirming it. To conclude anything from simulations or any observations you need something more than just probability theory.
For now, it’s suffice to say that we can easily talk about different probabilities for Heads: on average awakening and on average experiment
Or on average odd awakening, if you only value half your days. Or on whatever awakening you need to define to minimize product of squared errors. I feel like the question confused people want answered is more like “can you get new knowledge about coin by awakening?”. But ok, looking forward to your next posts.
In confusing anthropic situations, we shouldn’t. Correctness implies one-dimensional measure and objectivity and then people start arguing what is “correct” probability in Sleeping Beauty. You can invent some theory of subjective correctness, or label some mathematically isomorphic reasoning as incorrect but useful. Or you can use existing general framework for subjective problems that works every time—utility. Even if you want to know what would maximize correctness, you can just make you utility function care only about being correct—that still makes the necessity of the answer to “correct when?” obvious.
The technical justification for all of this is that the meaning of correctness for probability is not checked, but defined from it being useful—the law of large numbers is a value-laden bridge law. The need for any approximation is derived from it being useful.
Which is of course doesn’t mean that in practice we never can factor out and usefully talk only about correctness. But that’s a shortcut and if it leads to confusion, it can be solved by remembering what was the point of using probability from the start.
I’d say the opposite. The more confusing the case the more important is to make it as simple as possible in order not to multiply possible sources of consufion.
Well, yes. Sleeping Beauty is actually a great example why you should be more careful with invoking betting, while trying to solve probability theory problems, as you may stumble into a case that doesn’t satisfy Kolmogorov Axioms without noticing it. I’ll talk more about it after I’ll have finished all the prerequisite posts. For now, it’s suffice to say that we can easily talk about different probabilities for Heads: on average awakening and on average experiment, just as we can talk about different betting schemes and the addition of betting scheme doesn’t make the problem clear in any way.
I’m not sure what you mean by it. Law of large numbers is just a fact from probability theory, it doesn’t require utility functions or betting.
I meant that the law is just a statement about probability, not about simulations confirming it. To conclude anything from simulations or any observations you need something more than just probability theory.
Or on average odd awakening, if you only value half your days. Or on whatever awakening you need to define to minimize product of squared errors. I feel like the question confused people want answered is more like “can you get new knowledge about coin by awakening?”. But ok, looking forward to your next posts.