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Since this topic isn't so well-known, I wrote the case arguing that frequentist interpretations don't work, but algorithmic information theory (Kolmogorov complexity) does. I want to make this accessible and persuasive, so thoughts, questions, and arguments would be appreciated!
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Ok, so let's imagine you build a servo-driven flipping machine to carry out an infinite series of tests and notice, after a thousand of them, that 99% of the time, the coin flip matches the orientation of the coin when it's loaded into the machine. What have you learn about the coin's true flip probability?
>What have you learn about the coin's true flip probability? Nothing because you only tested the coin flip machine in aggregate. If you have a different throwing mechanism the results could be completely different.
If nothing is flipping the coin, then what coin flips are we making predictions about?
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It's clear that 37.5% sure is 0.5% more sure than 37% sure. The problem remains how to interpret these numbers.
You're asking about the interpretation of a statement such as "I assign the same probability to events A and B"? That would mean that both are equally likely as far as that person knows.
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You're asking about the interpretation of a statement such as "I assign the same probability to events A and B"? That would mean that both are equally likely as far as that person knows.
No, I'm not asking that.
When someone says they're "37% sure" tomorrow will rain they mean that they assign the same probability to "tomorrow will rain" that they do to "if you throw three dice you'll get 9 or less" or "when you threw three dice you got 9 or less". In the second case the event is either true or false already and there is no uncertainty for you, their probability assignment is their best guess with the information they have.
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If quantum is really random, then our universe has infinite Kolmogorov complexity.
Not quite. Under the many-world interpretation, when you send a photon through a half sieved mirror, the universe splits in one version where the photon goes through, and one universe where the photon doesn't. This is all very deterministic. What the researcher subjectively observe however is another matter. If the universe splits, so does the researcher. The problem of observing outcomes turns into an anthropy probl…
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But that’s the thing: the “true“ probability is unknowable, and may even be an ill-defined concept. It is a deterministic process, so “probability“ is just a simplifying concept to describe our best guess belief about how the coin behaves in the aggregate.
The true probability requires an infinite sequence of tests, so it's by definition unknowable. But it's what any sort of statistics attempts to approximate.
Because not only is the true probability unknownable, it is also unencodeable, but, if we are to accept a limitation on encoding, then we CAN give a true probability subject to that.
Like.... if we are to determine a coins probability to 1 decimal place, then we can do that.
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Check out the first chapter of Jayne's probability theory. It's the clearest take down of the frequentist interpretations that you have and introduction to the Bayesian interpretation. http://www.med.mcgill.ca/epidemiology/hanley/bios601/Gaussia...
Thanks, I had a quick skim through it and it seems really helpful. If I understand correctly, the claim is that the laws of probability as they are known comprise the only possible interpretation that satisfies the 3 desiderata set forth in chapter 1.
Chapter 5 is also a doozy in terms of explaining some of the things going on currently.
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No, the distinction is clear. If your assigned probability is something that someone else can reproduce using the same steps given the same information, then it is objective (yet contextual). Your examples each have a clear reasoning behind the assigned probabilities, they're not just opinion-based assertions.
Bob has a very simple algorithm to output probabilities. He just answers 50/50 for any yes or no question. This is reproducible. Is this objective?
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"The probability of that coin toss being 50% does not talk about the coin." I think there is a subtle equivocation going on here.
Well, it's getting philosophical now. We do not have a way to experience the true nature of the coin. We can only experience an "image" [0] of the coin and we summarize all "images" of that object into knowledge. [0] by image I do include your vision, but also hearing, feeling and other methods of perception.
I am intrigued by the idea of the coin having a "true nature" that we have no way to experience; I would like to know what this elusive "true nature" is, but if we cannot experience it, I don't suppose you can tell me. Instead, I will settle for an explanation of how you know it has such a true nature.
How much some data means depends completely on everything else you know about the world. You could imagine under different priors of knowledge, different strings would have differing kolmogorov complexity. not technically true, because kolmogorov complexity Is fixed, but that assumes you have the absolutely omniscient model for everything.