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Is Probability Real?

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Re: Is Probability Real?

#161

Post by ikeboy »

xvedejas wrote: post #160
ikeboy wrote: post #119

Earlier quoted context omitted.

The distinction between objective and subjective collapses under your usage.

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?

Re: Is Probability Real?

#162

Post by zomglings »

The circularity argument for the frequentist interpretation of probabilities seems lazy to me.

The essay argues that the frequentist view of probability is circular because it "reduces probability claims to probability claims".

One can attempt to resolve this apparent circularity by thinking in terms of claims about the mathematical theory of probabilities versus claims about an empirical theory of probability (involving limiting behavior of experiments). Frequentist statistics could possibly be seen as a means of reconciliing these mathematical and empirical theories.

The argument of the essay precludes this kind of interpretation of frequentist statistics without even considering it.

Re: Is Probability Real?

#163

Post by baobabKoodaa »

ikeboy wrote: post #161
xvedejas wrote: post #160

Earlier quoted context omitted.

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?

This doesn't strike me as a good example of reproducibility in this context. Let me offer another one. Let's go back to your example of the 66% weighted coin. Given the physical properties of the coin, different people could independently come to the same conclusion that the probability of heads is 66%. I would describe this as an "objective" probability, as it's a nice representation of the available information, independently reproducible by different people, given the same information. It's different than "Bob arbitrarily decides that any yes/no question has 50/50 probability", which is inherently subjective.

Re: Is Probability Real?

#164

Post by ikeboy »

baobabKoodaa wrote: post #163
ikeboy wrote: post #161

Earlier quoted context omitted.

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?

This doesn't strike me as a good example of reproducibility in this context. Let me offer another one. Let's go back to your example of the 66% weighted coin. Given the physical properties of the coin, different people could independently come to the same conclusion that the probability of heads is 66%. I would describe this as an "objective" probability, as it's a nice representation of the available information, in…

No two people ever have the same set of information. And very few cases are even as clean as the coin case.

A more typical example is using polls to predict elections. 538's model ended with Biden around 90% to win. Andrew Gelman's model at the Economist ended with Biden around 95%. Do either of those represent objective probabilities?

Or take weather predictions. Per https://www.metaculus.com/questions/4617/will-2020-be-the-wa..., Berkeley Earth gives a 16% chance to something that NOAA gives 29.2% to. Is either of those an objective probability?

I would say no, and I think that just because two people happen to agree on a number in a particular case doesn't make it objective. If you want to use the word objective, I don't have any particular objection. I'm not here to fight over words, and none of these words are really well defined enough to be worth fighting over. I don't think it's useful to think of probabilities as "real" in any sense.

Re: Is Probability Real?

#165

Post by Koshkin »

loup-vaillant wrote: post #153
Koshkin wrote: post #97

Earlier quoted context omitted.

> is real: it's a construct in one's mind, and minds are just as real This is a weird thing to say. Not all mental constructs are "real" in a meaningful sense.

No mental construct can exist without a cognition engine to run it. No idea can be thought without a mind to think it. Information has to be stored somewhere , and that somewhere is ultimately physical: a nervous system, a hard drive, a piece of paper…

What does all this have to do with reality of objects themselves?

Re: Is Probability Real?

#166

Post by EbTech »

alisonkisk wrote: post #9
EbTech wrote: post #2

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!

Why is frequentism bad because it only gives certainty for infinite samples, but complexity is good despite being non-computable? It's two sides of the same coin -- computable uncertainty va non-computable certainty.

I don't think frequentism is "bad"; just insufficient as a gold standard interpretation of probabilistic claims. I liked an analogy from the reference by Rathmanner & Hutter: the most "correct" chess-playing program involves a complete search along the tree of possible games. In practice, we try to approximate this ideal.

In the case of Kolmogorov complexity, a reasonable takeaway might be to use the shortest program that we're able to find, even if it's not the shortest overall.

Re: Is Probability Real?

#167

Post by rcthompson »

peterwoerner wrote: post #29
rcthompson wrote: post #16

I'm glad to see a write-up of this. I've been searching recently for a way to reasonably define probability without having to invoke either hypothetical infinitely repeated experiments or placing bets (since the latter is really just implicitly invoking the former). I'm looking forward to part 2!

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.

Re: Is Probability Real?

#168

Post by EbTech »

carapace wrote: post #134
EbTech wrote: post #2

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!

This is awesome! Good work! Are you going to touch on Chaitin's Omega?

Thanks! :)

I wasn't planning to go there! While I enjoy the idea, for now I'm trying to focus on what's needed to make sense of the problem of induction. Is there a nice connection that I missed?

Re: Is Probability Real?

#169

Post by EbTech »

topsycatt wrote: post #59
EbTech wrote: post #2

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!

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…

Thanks. I should clarify that the computer is deterministic, so as to avoid building randomness into the definition of randomness!

I skimmed over an example too quickly, but your intuition is about right. For that sequence, two possible programs are:

- Compute and print the first 40 digits of pi.

- Decompress the following string according to a Shannon code with probabilities (1/36,1/18,1/12,[etc]): [insert code]

Re: Is Probability Real?

#170

Post by EbTech »

qsort wrote: post #21
EbTech wrote: post #2

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!

The article is excellent, congratulations. A couple observations/questions. 1) You didn't comment on the bayesian viewpoint that probability reflects a subjective idea about the state of the world. One might argue, for example, that probability isn't measurable, and that therefore, strictly speaking, a statement about the objective probability of an event isn't meaningful. Experimental evaluation would have to be don…

1) I rather like the subjective view! Bayesians used to struggle to justify a choice of prior, but it turns out that 2^-K(x) is universal in the sense that it never falls below a constant factor of any given (semi-)computable finite (semi)-measure.

2) Sorry, I should have clarified that the programs are deterministic. So if you want to use an RNG, you also have to supply a string of random bits that cause the RNG to output forty digits of pi.

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