Maybe?
Is Probability Real?
51–60 of 227 posts
Re: Is Probability Real?
#52Earlier quoted context omitted.
Sure, I just can't agree with the parent statement that probability has nothing to do with the object it describes, which is demonstrably false.
They didn't say that it has "nothing to do with the object it describes" though. It has to do with your knowledge, which itself has to do with the object your knowledge describes
"The probability of that coin toss being 50% does not talk about the coin." I think there is a subtle equivocation going on here.
Re: Is Probability Real?
#53The best part of this is that dice are known to not be fair. There’s a whole niche for people who want to buy fair dice.
I can’t find it but I once saw a post that stacked ~20 d20 dice. The difference in height based on what number you picked to stack was shocking. The dice were incredibly non-uniform.
Re: Is Probability Real?
#54Since 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!
I would have liked to know what the Kolmogorov approach has to say about the examples used to deflate frequentism ("which of my friends will start a business?"). I don't see from the article how the "smallest-program" approach could say anything useful about those, either. Maybe that wasn't the point--but after poking holes in the frequentist view, it uses unrelated examples like digits of pi to illustrate the Kolmogorov idea, so I'm left unable to directly compare the kinds of statements the two approaches can make.
Even going back to dice or coins would have helped me compare them. Like, I know frequentists can show how the variance in coin-toss outcomes decreases as the sample size increases. What can the smallest-program approach say about that? Or was the point that those variant outcomes aren't "real" enough to talk about? Does that mean there is a connection to constructivism in mathematics here?
It seems either approach benefits from more data, and there must be a concept related to a "confidence interval" where, as 100, then 200, then 300 digits of pi roll in, your pi-program stays the same size while other programs have to keep growing to accommodate the new data. Like, the ratio of the smallest program to the naive encoding ought to say something about how potentially predictive the small program is.
Thanks for the interesting article. It definitely made me think about the issues, and now I'm curious to know more about the topic.
Re: Is Probability Real?
#55Tow things. First, probability is real: it's a construct in one's mind, and minds are just as real as dice or coins. https://www.lesswrong.com/posts/f6ZLxEWaankRZ2Crv/probabilit... Second, (and the author may be leading up to this), there's Solomonoff's theory of inductive inference, which he has proven complete : when we apply Occam's razor (where the prior probability of each possible theory drops exponentially wit…
> the amount of error a perfect Bayesian makes as they observe event and bet on the next one, ad infinitum, is finite.
There's a little assumption that you're leaving out, namely that the Kolmogorov complexity of the data generating process is finite. From Wikipedia:
> expected cumulative errors made by the predictions based on Solomonoff's induction are upper-bounded by the Kolmogorov complexity of the (stochastic) data generating process
Whether the universe (or our observations of the universe) have finite complexity is very much an unresolved philosophical question.
Re: Is Probability Real?
#56Since 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!
My only thought is that discovering a workable definition of "scientific method" is a reach. Philosophers spent a century searching for such a thing, in vain.
On the other hand, providing something that just works would be beneficial enough, even if falling short of the philosophical holy grail, so it's worth pursuing.
I'm a physicist, and physicists have always wondered why math works so well in physics. There's this famous essay by Eugene Wigner:
https://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.htm...
Re: Is Probability Real?
#57Since 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!
A simple sentence that I've found useful for pedagogy: "the probability of that coin toss being 50% does not talk about the coin; it talks about you , and about your partial knowledge of the universe." You can add: "The coin toss itself is deterministic and the result can be computed if you know the initial position and speed." They will inevitably bother you about the physical impossibility to measure the starting p…
There seems to be a lot of quibbling about the simple sentence here but I find it clarifying. Discussing coin tosses is a thought experiment with a very practical physical analog so spelling out clearly what the thought experiment's actual subject is has valuable properties so you don't get lost in the weeds of the physical execution of flipping coins.
Re: Is Probability Real?
#58> The scientific method only works because the rules of the universe happen to be simple, while the set of observations it offers is vast. Kolmogorov complexity captures this defining characteristic of our reality. I've never seen this spelled out so beautifully!
Not everything that is true is simple or catchy. Biasing towards it is a trap. PBS SpaceTime has a good discussion on this topic. https://youtu.be/xFKgIOX8IRE
Neat video, but it specifically disagrees with your point. He advocates for applying scientific rigor between leaps oc intuition. There's nothing wrong with GP appreciating beauty.
Re: Is Probability Real?
#59Since 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!
There were a few points that I, as someone unfamiliar with many of the ideas presented, got hung up on.
First, the paragraph that begins with "At first blush, the requirement to use..." Seems to be a non sequitur. I don't fully understand how the previous section creates a requirement to use deterministic programs, so I could use more explanation on how that requirement is established.
Second, a very simple concrete example of what one of these programs would look like would be immensely helpful. After re-reading the article a bit I have a mental image of a program that contains a long, compressed string and a decompression algorithm that somehow models the system you're interested in. I can imagine how you might get a useful interpretation of probability from the decompression system, but there are enough open questions there that I'm not sure I have the correct interpretation.
Hope that helps!
Re: Is Probability Real?
#60Earlier quoted context omitted.
They didn't say that it has "nothing to do with the object it describes" though. It has to do with your knowledge, which itself has to do with the object your knowledge describes
"The probability of that coin toss being 50% does not talk about the coin." I think there is a subtle equivocation going on here.
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