Is Probability Real?

https://www.arameb.com/blog/2020/11/22/probability

by EbTech • 6 years ago

220 218 6 years ago

EbTech11• 6 years ago

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!

opheliate• 6 years ago

Such an interesting post, thank you for sharing! I'm in my second year of a maths degree currently, and we obviously studied frequentist probability/stats in the first year, but I'm not taking any probability modules this year. I found the tone & accessibility was just right for me :)

enriquto4• 6 years ago

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 position and speed exactly, and then you say "ok, forget about the coin. You have 5 white and 5 black balls inside this opaque cylinder. What's the probability that the top ball is white? This does not talk about the balls (the color of the top one is already determined) but about your partial knowledge of them".

(EDIT: formatting)

alisonkisk7• 6 years ago

How does the ball situation help? It has the same problems of physical impossibility of measuring however the balls were ordered. (Modelling someone's brain?) I guess the argument works on someone with an unscientific model of the human brain, but that's one step forward and two steps back.

st1x72• 6 years ago

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.

But it does talk about the coin - a weighted coin would have a different probability. Same in the example with the white/black balls - if they weren't 5 white and 5 black but 6 white and 4 black, the probability you would assign to the top one would be different. Again, the probability is a way to describe the balls themselves, not just our knowledge.

I get the general idea of representing probability as uncertainty and partial knowledge but your statements strike me as just straight up incorrect.

Sharlin• 6 years ago

This is basically the Bayesian interpretation of probability.

nimbleal• 6 years ago

It may be deterministic, but are you sure it would be computable? One does not necessarily imply the other.

kps• 6 years ago

Forget the coin and the balls — does the nucleus decay? You're not missing any knowledge; there isn't any.

mannykannot4• 6 years ago

Wigner's friend might have something to say about this...

I don't have a specific argument to make here, only the feeling that if it were all just a matter of what a given observer knows, no-one would be talking about there being a QM measurement problem.

MereInterest• 6 years ago

As a caveat, while this intuition works for classical mechanics, it does not work for quantum mechanics. All observations are consistent with wave function collapse being fundamentally random. Any hidden variables would need to be transmitted many times faster than the speed of light (~10000x, last time I checked the experiments), and are therefore inconsistent with our understanding of special relativity.

JamisonM2• 6 years ago

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.

alisonkisk7• 6 years ago

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.

EbTech11• 6 years ago

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.

TheOtherHobbes• 6 years ago

OK - so apply Kolmogorov complexity to election polling.

How does that work out?

I think you're confusing various possible maps with the territory in a less than useful way.

Given that frequentist interpretations are approximations - and understood as such - and Kolmogorov complexity isn't computable at all, what problem have you solved here?

EbTech11• 6 years ago

Hm I admit it's hard to talk convincingly about election prediction, since we don't have practical algorithms to do this; a lot of it comes down to human judgment.

The philosophical point (which might be approximated algorithmically someday, or by intelligent minds today) is that your election probabilities should come out of an overall highly compressed model of the world. In theory, a Bayesian who uses the prior 2^-K(x) over all strings x should, with sufficient life experience, come up with good estimates, in a certain sense.

I'll have to think about this example more carefully when fulfilling my promise of writing about how this theory relates to everyday decision-making. Thanks for pointing out a potential weakness :)

qsort2• 6 years ago

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 done on an entire model instead. Do you have any objections to that point of view?

2) I don't find the case about Kolmogorov complexity to be actually convincing, at least not as per the requirements the rest of the article sets. "3141592..." could pass as either "random digits" or "first digits of pi". The fact that it's highly unlikely a true RNG would have generated exactly those, we are back to a frequentist argument there. It's likely I'm missing something, could you elaborate more or give me a pointer?

SmooL• 6 years ago

Isn't that what the occam's razor argument was for? Sure, both an RNG and the "40 digits of pie" program can produce that output, but the "40 digits of pie" program is shorter, therefore having less kolmogorov complexity

EbTech11• 6 years ago

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.

JamisonM2• 6 years ago

Some thoughts on the composition:

* "I wrote the case arguing that frequentist interpretations don't work, but algorithmic information theory does": if that's what you are up to here then I think it would be for readers if you stated that up front in some way. And hit me with some kind of summary at the end that makes the concise version of your argument at the end, it's a long article.

* Shorter might be better: There's a lot of stuff in here that I think you can pare out in the probability discussion that maybe isn't adding that much to your argument. I think there is a lot to be gained by assuming a generous reader.

* Betting might be a distraction to your point: This might be confusing the imperfect knowledge of participants in a market with the imperfect knowledge of all the physical forces involved in a physical phenomenon and how that related to the seeming "randomness" of a coin flip for your reader. (The liquidity and stuff.. this is just not related to your point.)

* Don't undermine your point with unrelated assumptions: "I imagine they wouldn’t consider their world unlikely at all: they would just add a new law to their description of physics: all dice, as if by divine intervention, are deemed to exhibit this strange behaviour" this lead me to think that you were just sort of shooing away the whole last X decades of high vs. low energy physics, we collectively certainly don't think that we have the rules correct precisely because of this complication, we find the idea that we need 2 sets of rules improbable and believe that there must be a way to explain everything with a single set of rules. So your mythical dice society probably would consider their dice exception a very unlikely world.. they would be confident they have the world wrong!

A dubious assertion (or at least one that would need a whole lot of explanation) can be an off-ramp for a subset of readers.

EbTech11• 6 years ago

Thanks for the detailed critique! I'll take some time to think about how to better make the points that I wanted to convey with those sections.

richard_todd• 6 years ago

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.

EbTech11• 6 years ago

Thanks! I hope to better address your concerns in Part 2.

PeterisP• 6 years ago

One aspect of Kolmogorov approach is that implicitly models things like biased probabilities through compressibility.

The shortest representation of rolls of fair dice or coins is their exact results, but if there's "less randomness" in some way (biased coin/die, sum of two dice which means non-uniform probabilities, combination of some predictable pattern with random noise) then there are more compact representations of that information, and all of that gets captured by the Kolmogorov approach without any explicit handling of the various possibilities.

analog312• 6 years ago

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...

topsycatt• 6 years ago

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!

EbTech11• 6 years ago

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]

carapace• 6 years ago

This is awesome! Good work!

Are you going to touch on Chaitin's Omega?

EbTech11• 6 years ago

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?

danabo• 6 years ago

It sounds like you are arguing that i.i.d. frequentism doesn't work. I view AIT as generalizing frequentism to non-i.i.d. timeseries. This is formalized as Martin-Lof tests for randomness, and Solomonoff induction.

REALiSTiC• 6 years ago

Solid write up. If anyone's interesting in knowing more about how probability _came to be_ 'regarded' as real, i highly recommend Ian Hacking's "The Taming of Chance". Thought provoking in so many ways.

MauranKilom3• 6 years ago

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!

willis9362• 6 years ago

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

jhardy54• 6 years ago

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.

karmakaze3• 6 years ago

The patterns in our observations appear simple, the rules are deeper: e.g. newtonian motion vs relativistic.

ordu3• 6 years ago

It is a funny statement, I like it. But due to a different reason: all we know about reality is our theories. How we could state, that rules of the universe are simple? As I see it, we could state, that our theories are full of simple rules. But the universe have no theories nor rules outside of human's mind. It is completely our inventions, our dreams, our hopes that the universe have some rules.

We could state that our simple rules works, but what does it mean "to work"? For example, a spider sees reality not like us, it feels vibrations of it's web, runs to a source of vibrations and start to bite, to wrap intruding object with web. It would do it to a tuning fork, if you pressed it to spider's web. His simple rules of reality works though. Despite the fact that sometimes spider bites steel of a tuning fork without any benefits for the spider.

How could we know that our theories not just extended version of spider's? With the same issues, like they make us to do something absolutely pointless. How could we evaluate this fact? To ask our theories? But our theories already predicted that this pointless thing we do would be a good thing. We might ask our theories again and we'd get the same answer.

This statement seems as a tautology for me. Our rules are simple, because they are simple. Our theories work because they tell us, that they work.

MauranKilom3• 6 years ago

I think there is a disconnect between the "rules of the universe" that the quote talks about and the "our simple rules" that your comment focuses on. Without getting too philosophical, I think we can agree that there are "rules" how the universe works (which is what the quote talks about), which are not necessarily the same as the rules that our theories postulate (what you talk about), although we of course try to get closer and closer.

That we can only observe, talk about and know reality in our subjective ways does not mean that there is no underlying mechanism by which reality "works". The quote points out that this "mechanism" is apparently sufficiently simple that we can effectively form and test hypotheses about it.

For example, we have no all-encompassing explanation of the universe that concludes R = U/I for electric circuits. Yet we can observe it to be accurate independently from the infinity of conceivable influences - there appears to be no influence on this observation from your lunch, the day of the week, whether your car is green or somebody was just born in Taiwan. We can't explain why these have no influence. We could imagine a reality where all of these (and infinitely many others) are confounders, in which case we could not effectively form theories about these rules. Yet the actual number of things influencing R = U/I observations in our reality are evidently very finite, allowing us to identify them and build our theories.

DiogenesKynikos• 6 years ago

Science has an empirical aspect that you're ignoring here. Our theories are tied to nature through experiment. They are more than just inventions of the human mind. People come up with new ideas about nature all the time. Most of them are wrong, and we find that out by doing experiments.

6gvONxR4sf7o2• 6 years ago

I disagree that the rules just happen to be simple. Thinking in dynamical systems or evolutionary terms, a language and mental model of the universe just won’t stick around unless it’s useful. Imagining that the universe happens to be horribly complex, but has some emergent phenomena that happen to be simple, we’re almost inevitably going to think in those simpler terms and go “wow the universe is simple!”

Then if the foundation underlying those simple emergent concepts turns out to be horrible complex, maybe we get stuck with the foundations at some point. Or maybe simplicity is relative and a visitor from a different hypothetical universe would be astounded at all the shit we have to simulate because the answer isn’t just obvious.

Like how macroscopic objects are an emergent phenomenon, but they’re simple. Maybe the universe at a fundamental level follows some ultra convoluted string theory. Doesn’t matter for us, the same way that flipping a coin is 50-50 regardless of questions about foundational physics.

It’s taken a long time to develop the hierarchy of physical and mathematical concepts in which we can usefully describe much of the universe as simple. And those concepts, the best we have so far, still don’t tell the whole story.

lisper4• 6 years ago

if the foundation underlying those simple emergent concepts turns out to be horrible complex

That is extremely unlikely. The simplicity of the universe is not just an artifact of our best theories, it appears to be baked into the very structure of those theories. If our current theories are even in the ballpark, then there are very few places that hidden complexity could possibly hide.

Take quantum randomness, for example. It used to be thought that the apparent randomness could just be papering over our ignorance of some hidden underlying mechanism, but it turns out that we can prove that this is not the case. We can eliminate entire classes of theories based on finite observations, and one of the classes that we can eliminate (with very high probability) is theories with high Kolmogorov complexity.

dkersten5• 6 years ago

The scientific method only works because the rules of the universe happen to be simple

Are they simple though? That's not the impression I get from physicists, eg, the famous Richard Feynman quote: If you think you understand quantum mechanics, you don't understand quantum mechanics.

We haven't even managed to find a unifying theory yet, but the current contenders, like Loop Quantum Gravity are anything but simple.

BalinKing• 6 years ago

As I understand it: The rules are simple, but not necessarily intuitive.

jhayward• 6 years ago

I've always taken the Feynman quote to mean: just because you understand the rules doesn't mean you understand the implications, which are varied, vast, and counter-intuitive.

Igelau• 6 years ago

The scientific method only works because the rules of the universe happen to be simple

It's simple in the sense that the universe seems to be the kind of place where inductive reasoning mostly works. For the most part we seem to be able to expect things like predictability and causality at the scale of the objects and processes in our everyday life. If it wasn't like that, things would be a lot more complicated.

edna3142• 6 years ago

Because others already questioned the first part, I’ll question this:

the set of observations it offers is vast

Actually, in an experiment there is always only one observation at a time. That we group multiple observations together is kind of arbitrary and relies on the hope that the experimental conditions are the same and therefore one experiment is analogous to the next.

MauranKilom3• 6 years ago

Yes, that's exactly the heart of the quote to me: There is sufficiently little complexity in the causality of observable behavior that with sufficiently controlled and repeated experiments/sampling, we can uncover regularities and patterns in it to fuel our predictions (and hypothesize about these causalities).

Does the reading of your multimeter depend on the digits of Arnold Schwarzenegger's phone number? Do you have to repeat the experiment if his phone number changes? Indeed, we assume that this is not an "experimental condition" to take into account. There is no way to determine this a-priori, and one could conceive of a universe with an arbitrary amount of such strange influences. But we do not appear to live in such a universe, which is why we get to apply Occam's Razor.

shrubble• 6 years ago

Probably...

rcthompson2• 6 years ago

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!

peterwoerner2• 6 years ago

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...

rcthompson2• 6 years ago

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.

dandanua• 6 years ago

It's complex! (according to quantum mechanics)

xwdv• 6 years ago

Ultimately, probability must be real, because if you zoom in to a low enough level, the movement of particles is basically random, and this randomness is enough to poison any deterministic explanations for which side a coin flip will land.

You could calculate which side a coin will land if you knew all the variables and thus arrive at a higher probability that it will land on a certain side, but it’s still a probability, because the particles that make up the coin could all move in such a way at any point that causes the coin to fall in a way you didn’t expect, even if that probability is very low. A small probability unlikely to happen is still a probability nonetheless.

Try computing how a dice made up of a few particles will land. Good luck...

mellosouls• 6 years ago

I haven't read the article either but skimmed it briefly for a later read; I think it isn't disputing the phenomena that probability is ultimately used to describe (as you seem to claim), but how they are to be interpreted.

walleeee• 6 years ago

Philip McShane expressed this nicely: "...a thing is defined by... systematizations of coincidental aggregates of the properties of lower things". Probability theory "allows for the emergence of the systematic from the non-systematic".

thread_id• 6 years ago

It probably depends on the p-value.

FinanceAnon• 6 years ago

"All models are wrong, but some are useful"

smitty1e• 6 years ago

“Given competing programs whose outputs match our observations, always prefer the shortest.”

I need to know context. Output is necessary, but not sufficient. Is this a stand-alone chat server for some niche voices, or, say, Twitter?

loup-vaillant5• 6 years ago

Tow 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 with its size), the amount of error a perfect Bayesian makes as they observe event and bet on the next one, ad infinitum, is finite. Roughly proportional to the complexity of the simplest theory that correctly predict the whole sequence of events. It's one of the most convincing proofs that Bayesian reasoning works. https://en.wikipedia.org/wiki/Solomonoff's_theory_of_inducti...

There's just a little snag. Perfect Bayesian reasoning is impossible to compute, so us mortals have to resort to approximations. Just as perfect certainty isn't possible, perfect reasoning is not attainable. Oh well.

smallnamespace2• 6 years ago

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.

yters3• 6 years ago

If quantum is really random, then our universe has infinite Kolmogorov complexity.

loup-vaillant5• 6 years ago

Whether the universe (or our observations of the universe) have finite complexity is very much an unresolved philosophical question.

Every time there was a significant advance in physics, it tended to go towards simplification and unification. Geocentrism required epicycles. Then Keppler came up with his ellipses. Then Newton unified celestial and terrestrial laws. Maxwell & Einstein allowed us to view time as less special dimension than we thought it was…

I won't presume about the initial state of the universe, to the extent such a notion is even meaningful. But the fact that it is governed by mathematics, and relatively simple maths at that, sounds likelier and likelier every quarter-century.

And I'm not even talking about everyday life, where we can observe in practice that the simplest theories about who ate the last cookie (little Mike, who lives in the house) are more often true than the more outlandish ones (magical imps, which we never witnessed).

derbOac• 6 years ago

Vitanyi -- one of the authors cited in the linked piece -- has a paper where he and his colleagues show that even though ideal compression isn't knowable/computable, you can show that two encodings approach the ideal differentially with some computable probability (I'm being vague and handwavy because it's been awhile since I've read this literature carefully). That is, if you have two encodings A and B, you can compute some probability that A is closer to ideal than B.

It's kind of a Kolmogorov/algorithmic complexity analogue of a p-value.

I think this literature/inferential paradigm in general is far less known than it should be.

Koshkin3• 6 years ago

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.

loup-vaillant5• 6 years ago

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…

cjkarr• 6 years ago

Maybe?

forrestthewoods• 6 years ago

The 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.

HelloNurse• 6 years ago

Mostly well argued, but there's some very loose language, such as calling "inconsistencies" mere practical relevance issues (such as nonideal markets, nonprobabilistic decisions, ignorance and the ensuing arbitrage, changes in probability values). A bad theory can be logically inconsistent, but it is not the case of probability theory and its competing interpretations.

jmount• 6 years ago

One way to deal with probability is: defer interpretation until after one has axiomatized calculation by measure theory. I have a quick video on the topic here: https://youtu.be/DnTTAd1TDyQ .

alisonkisk7• 6 years ago

Measure theory is a powerful tool, but how does that answer the question of what probability is?

Akin to physics, studying the wave equation doesn't tell you what the wave equation is in the real world.

karmakaze3• 6 years ago

I don't think "first forty digits of pi" should be admissible. As it depends on an external definition that's not computed by the representation. I could come up with a mathematical definition for any prefix of digits, give it a name and say the 'first x digits of C'.

alisonkisk7• 6 years ago

Complexity is defined by size over an entire class of objects, not a single one. Pi is reused many many times at constant cost, wile your C becomes C2, C3, C4,...

skybrian• 6 years ago

That's just because it's a loose description in English. It would actually be a short computer program that calculates pi, and instead of the first 40 digits you could ask for, say, a megabyte of data, or take the limit as the amount of data grows.

Transmitting a program to compute pi would be shorter than the data needed by any compression algorithm that isn't somehow based on knowing the trick.

The same trick could be used for any mathematically interesting number. The point is that incompressible random sequences exist that are not like that. You can't do better than transmitting the sequence itself.

karmakaze3• 6 years ago

Yes the point was merely 'referring to an external' isn't a good example of a minimal size description.

emerged• 6 years ago

Since the lowest levels of our understanding of the universe are probabilistic, it seems reasonable to assume probability is the most real thing we currently know. It might be that will change some day, but at the moment everything we take to be non-probabilistic is just a simplification of underlying probabilities.

leephillips• 6 years ago

Nice article, on the whole, and usefully provocative.

But I have misgivings about making these close connections between information theory and scientific theory-making. As everyone knows, information theory leaves out any notion of semantics, as it should. But the important thing about our theories of the world is that they have meaning to us. The scientist searches for something that makes sense of the world, not an algorithm for computing a series of numbers. The theories that we search for may not have the smallest Kolmogorov complexity; the criteria that they satisfy go a lot deeper.

texasbigdata• 6 years ago

That’s such a great phrase: usefully provocative. Thanks for that.

TaurenHunter• 6 years ago

I like to think that probability is the ratio unknown/known, information available divided by all possibilities.

You know that a coin has 2 faces (known=2) and that if you toss it, 1 face will be up (unknown which=1).

The ratio is more about something in the mind than intrinsic to the objects.

That helped me understand why in the Monty Hall problem, when you switch doors, the probability of getting the prize increases.

alexpetralia• 6 years ago

"Somehow, we must narrow down our hypotheses. Maybe you think that’s easy: only a few hypotheses describe plausible dice behavior; the rest are patently absurd! But now you’re relying on intuitive judgment, not a rigorous methodology."

Maybe, as philosopher Robert Pirsig theorized in "Zen and the Art of Motorcycle Maintenance", you must rely on Quality!

jbeam2• 6 years ago

Very interesting article that cuts to the core of many of the issues with pricing insurance products. While the phrase is hardly limited to the actuarial world, "all models are wrong, but some are useful" is definitely an extension of this.

The fluid nature of probability is the center of the insurance universe. Probability is always a moving target in the insurance world. Indeed, if it weren't, there wouldn't be much of a need for actuaries. Much of actuarial training revolves around the idea of credibility -- how credible is your sample set, what alterations should you make to old data to make it relevant to today, and what data should you add to it as a complement in order to relieve the model of the biases inherent in your sample size. This is inherently Bayesian in it's approach.

Where it truly gets interesting is that insurance companies are very cognizant of tail risk -- the 1-in-100, 1-in-250, 1-in-500 events that can cause insurer insolvency if not properly accounted for. You can survive a miscalculated loss trend within reasonable bounds, but if you haven't thought about the potential Cat 5 hurricane that hits Miami-Dade then you are going to have some very unhappy investors. When it comes to these types of events, you mostly need to be in the right ballpark. The order of magnitude matters more than the exact number -- albeit the exact number matters quite a bit for regulatory reasons. This type of calculation for property lines has largely been outsourced to the stochastic models developed by companies such as AIR and RMS. A sudden change in their models, which I think is likely after this record breaking hurricane season, can inflict capital pressure on the industry almost instantly.[1]

There are some actuarial papers from around 50 years ago that discuss information entropy as another way to approach the issue of constructing probability models, but they never really caught on. It seems that is likely due to the lack of widespread computing power. I'm hoping these ideas can gain some steam now that we can construct some of these distributions from Python and R.

[1] There is a fantastic article by Michael Lewis that describes this issue at great detail: https://www.nytimes.com/2007/08/26/magazine/26neworleans-t.h...

sxzxs2• 6 years ago

For me, it comes down to what kinds of these risky models are most interesting. Some can be interesting because of potential profit or minimizing loss (financial or actuarial) and others are inherently (theoretical physics).

To add to your tail risk point - I wonder how many people foresaw the Venezuelan oil crisis way back when, or even less likely, the Saudi Arabian oil complex attack in 2019. And of course, the current situation we're in with CoVID that an entire university of forward thoughtful looking people didn't call until it was a week away. As an aside, do insurance companies significantly alter their policies when such a cat-5 hurricane is imminent? What preparations would they make in the face of that sort of event?

Are you talking about chaos theory in the last paragraph? I'll read that article you linked in a bit and see what more I have to say, from skimming through it looks as though my question from the previous paragraph may be answered.

jbeam2• 6 years ago

as an aside, do insurance companies significantly alter their policies when such a cat-5 hurricane is imminent? What preparations would they make in the face of that sort of event?

You cannot retroactively alter a policy, for obviously good reasons. The main preparation policy-wise is that insurance companies do not knowingly write new policies in the affected area when disaster is ongoing or imminent. Reinsurers will also avoid writing new treaties (which is what a standard reinsurance policy is called) -- for these reasons the Florida cat reinsurance market is typically dominated by policies that incept on June 1st and run to May 31st of the next year.

Internally, the companies will start modeling what they think their potential losses will be almost immediately, as investors expect a fairly quick turnaround on getting initial loss estimates out the door.

There's a fairly new paper here, albeit not yet peer reviewed, on the promise of maximum entropy models in an actuarial setting. The appendix has references to the earlier papers: https://www.casact.org/pubs/forum/20wforum/07_Evans.pdf

nieve• 6 years ago

Isn't declaring "the raison d’être of probability theory is to explain the decision-making of individuals facing uncertainty" basically a claim that the sole purpose of probability theory is an economist's approach? It's entirely possible that much more work in probability is done without a bet or payoff in sight than is done for the sake of decision-making models. It seems like basing epistemological arguments about all probability theory on a framing friendly to very specific groups of non-mathematicians.

glial• 6 years ago

What is a probability? Nearly always, a probability isn’t a statement about the world, it’s a statement about your knowledge and the information you have. Even for frequentist statisticians, equating probabilities with proportions of outcomes is an admission that you only have partial knowledge about the outcomes.

Of course it also has mathematical structure and properties that may be interesting to people for their own sake. And there may be interesting things to say about quantum physics using probability, but I think a historian of mathematics would not claim that quantum physics was the driving force for the development of probability theory.

Anyway, the “raison d’être” != every conceivable use.

alisonkisk7• 6 years ago

Can you give an alternate definition for probability than "decision making under uncertainty?" I've never heard one. Sure there are mathematical real-analytic "theories of probability" but that's abstract analysis of measure, until it's applied to answer how... probable something is

kolbe• 6 years ago

I work with probably as it relates to financial markets, and my gripe with probably is at an even lower level. Even with dice, the fact of the matter is that you will roll dice with almost entirely Newtonian forces acting on it, and it will settle on a number. This is not unknown to physics. It’s just unknown to us. And because we don’t have the data and computational power to know with any certainty where the dice will land, we call it random.

Everything I do is centered around creating probability distributions of where a stock will be in in the future. I don’t do this because it is fundamentally unknowable, but because I cannot access all of the data necessary to know.

So, I’ve come to regard probably measure as an interesting and useful tool, but one that has no connection to the reality of existence.

EbTech11• 6 years ago

Fair enough! I'd like to point out that the Kolmogorov complexity approach can make sense of subjective probability too. Since you lack precise enough information to predict the dice roll, the most compressed way to write down your observations will involve a Shannon-style code with your subjective probabilities. If you have enough information but not enough computational power, the resource-bounded variants of Kolmogorov complexity may be more applicable.

jtsuken• 6 years ago

Can you please define Shannon-style code? (Obviously, you don't have the shannon code, the compression method, in mind).

Also, isn't Kolmogorov complexity uncomputable and you run into multiple "who shaves the barber" issues, when trying to determine it?

kempbellt• 6 years ago

The article probably has more value than my comment.

nolite• 6 years ago

its pretty complex

abdulhaq• 6 years ago

Surely you have to define reality before approaching this question?

Koshkin3• 6 years ago

I wonder if it is even possible. In math, science, and philosophy some (basic) terms are necessarily left undefined and their meaning is either left open to interpretation or assumed to be evident based on common experience or convention.

ikeboy10• 6 years ago

Probability is not real. Probability is subjective, it depends on what you know, and everyone has a different set of things they know.

I flip a coin and look at it, then ask two other people for their probabilities. One of them knows the coin is biased towards head such that it's twice as likely to land heads than tails on any given flip. The other knows nothing about the coin.

The first person guesses a 66% chance of heads. The second guesses a 50% chance of heads. I, having seen the coin, say it's a 0% chance of heads.

None of these probabilities are wrong. They're all correct given the set of knowledge that person had. Probability is subjective.

xvedejas3• 6 years ago

Your example suggests to me the opposite: Probability is a real and objective way to describe the information you have. "Garbage in, garbage out" still applies when you have no information.

ikeboy10• 6 years ago

The distinction between objective and subjective collapses under your usage.

mojomark• 6 years ago

It sounds like you're argument is actually stating that probability is in fact a real attribute of physical systems, but the accuracy with which you can accurately predict a probabilistic event is dependant on the amount of information you have about the system. Interestingly, someone only observing results of a probabilistic event will eventually learn to predict events with an accuracy approaching ideal (I.e. the true probability distribution of the system - or someone who knows every single detail about the causal system down to the limits of physical reality - subatomic particles and all of the underlying physics we don't understand yet).

There's that phrase again - "true probability" (implying the word "True" == "Real"). You could imagine a system that yields non-linear results, such that no matter how many observations were made, your ability to accuratelt predict outcomes in the system never increases. In this case, you might conclude that you're dealing with one of two special cases: 1.) a non-probablistic event (i.e. a purely random (0% predicatable) event or 2.) on the other end of hyperbole, a purely deterministic event (100% predictable)).

ikeboy10• 6 years ago

Interestingly, someone only observing results of a probabilistic event will eventually learn to predict events with an accuracy approaching ideal

This relies on various unstated assumptions.

I would deny the assumption that an event can be probablistic in nature. There are only ever predictions. The distinction between a probablistic world and a deterministic world is incoherent.

dkersten5• 6 years ago

Probability is relative to the information you have. That doesn't make it not be real.

Probability is something we invented to describe systems, based on what we know about them, when we have imperfect information. If we had perfect information about the coin, its surroundings and how it was flipped, we could tell which side it will land on, but we don't. (Ignoring that there are some quantum things that physicist say can never have perfect information)

ikeboy10• 6 years ago

Generally we use the term real to refer to objective claims and consider subjective claims not to be real. You can certainly use the term real in a broader sense if you'd like. I don't think the concept of reality is particularly useful or coherent, but to the extent others use the term, probability is not real in the sense it's being used in.

grandrew• 6 years ago

I have a question about how Kolmogorov complexity is related to "Human complexity" or complexity of understanding?

E.g. a program may be very complex in Kolmogorov terms, like describing 1000 random numbers - easy to understand: you have a database of numbers, and a simple procedure that would scan through it. You can also imagine some real-world microservices-based program with a good architecture and a lot of code that handles all the exception cases of incoming data, all easily understandable.

And now imagine an optimizing compiler for prolog programs. It may have much less code but the algorithm will be so complex that it might be impossible to fully understand its behaviour. E.g. fast-downward is a great example of such a program.

So I'm wondering what does Kolmogorov complexity actually tell? Or does it tell anything useful in "real"-world?

jdc• 6 years ago

What is easy or useful for one person to understand is often much less so for another.

renewiltord• 6 years ago

Well, that's altering the 'language' that you're measuring the KC against. For instance, in the language that CS majors use the phrase "Kolmogorov Complexity" is sufficient to encode the concept of KC itself. And in the context of that language plus the concept of KC and adding in this entire thread, the letters KC themselves encode Kolmogorov Complexity in entirety.

So the 'real world' version isn't so easy to tell in absolute terms like that because the languages can differ. But if you were thinking of it like a compressor/decompressor pair, then moving things into the language is like moving things into the compressor/decompressor.

Naturally then you conclude that the "human complexity" depends greatly on the humans since any pair of humans creates a new universal description language that we can see as some base language plus the jargon that they are both familiar with.

qayxc2• 6 years ago

in the language that CS majors use the phrase "Kolmogorov Complexity" is sufficient to encode the concept of KC itself

Wait, isn't that just conflating "language" with "knowledge"/"information"? The underlying assumption here is that the CS major has an association of a concept encoded by the letters "Kolmogorov Complexity".

This is not universal, though, i.e. there's no computation that could derive the meaning behind these letters from the encoding alone. It's like claiming "620" is sufficient to encode Mozart's "Die Zauberflöte" ("The Magic Flute"), because in the language of a musician, the Köchel catalogue number along with the context would enable them to decode the full meaning.

But in reality you would still have to look up the number and the score somewhere so it's not really an encoding but more of a pointer or index. I'd see any technical term that way, in that the term itself is not an encoding, but a key/index/identifier of a concept, not a full definition of the concept itself.

roenxi2• 6 years ago

Nice article, there is an even easier interpretation though:

The mathematics of probability defines a bunch of objects that do not exist, anywhere. Random variables, expectations, probabilities, maybe a few others. Then there are situations in the real world that look a lot like those objects from certain perspectives.

It is a bit like Escher's Ascending and Descending - sometimes people make things in the real world that look like the infinite staircase when viewed from the right spot. Similarly, sometimes we find things that look probabilistic (dice rolls, coin flips) when viewed in ignorance of the sum totality of the entire universe. That is why there are a bunch of statistical tests that determine if outcomes are distinguishable from a hypothetical process generating that outcome.

tl; dr; I like the article, but it seems to answer it's own headline with "yes" and it is much easier to answer it with "no". There is, philosophically speaking, nothing that we can guarantee looks random from all perspectives.

analog312• 6 years ago

Perhaps the most useful connection is that statistics lets us study what kind of math we can do, when we know something about a set, but not everything about it. And our experience with the world is like that too.

burrows• 6 years ago

There is, philosophically speaking, nothing that we can guarantee looks random from all perspectives.

Does this imply that the universe is deterministic?

I believe it is false to claim “Humans have knowledge that the universe is deterministic.”

roenxi2• 6 years ago

It means we can't be certain. Whether the universe is random or deterministic is, ironically, a matter of probabilities.

thisiszilff• 6 years ago

I think this is related to the bayesian/frequentist approach to statistics. Bayesian statistics revolves around incorporating knowledge to update beliefs and probabilities express degrees of belief. IE, bayesian probability is the logic of uncertainty and how we should reason with imperfect information.

The most famous example of this is the "draw a ball from an urn" example, and how we frequently day we "shake" the urn after we add the balls to "randomize" it. There is a lot of wordplay going on in stats to effectively forget certain information so that the problem I'd workable

zomglings• 6 years ago

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.

zaptheimpaler• 6 years ago

Kolmogorov complexity is a useful concept in the abstract but since it's not computable I find it hard to see anywhere it can really be applied. It doesn't solve the impossible problem of somehow deciding the true information content of any piece of data, it only locks it away in a slightly neater box of impossible.

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.

EbTech11• 6 years ago

The reference by Rathmanner & Hutter presents a useful analogy. It argues that Kolmogorov complexity (and Solomonoff induction) are best viewed as a conceptual gold standard, like a perfect chess computer that does an exhaustive tree search. Practical methods are approximations.

There are a few results where researchers were able to automatically infer evolutionary trees and such, by using a standard compression algorithm in place of K(x).

danielearwicker• 6 years ago

I think of probability as a summary of the structure/symmetries in a model. A model of a jar of red and blue beads is characterised by the ratio of red to blue, and that beads are only distinguished by colour.

Is Probability Real?

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