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)
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.
Now, you see ten white balls get added, then are blinded while other balls are added (maybe). You estimate the jar can't hold more than about a hundred balls. What odds would you take?
Now, you see ten white and ten black get put in, and saw it was empty before. What odds?
Now, you see ten white and fifty black, but the whites are larger, and you get to draw a ball. What odds?
The difference between the second-to-last and the last is the missing information we usually think of when we talk about randomness being missing information.
And you'll see that the previous scenarios don't change anything about that.
[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.
No, if it's weighted it's not 50%. Your prior probability is 50%, but neither a Bayesian nor a frequentist would claim the true probability is known before testing.
Notice that it is possible to build a robot which flips a coin in such a way that it's always heads - sure you might need to build a different robot if the coin is "biased"(you probably mean its weight distribution is uneven) but it's still possible.
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.
Interesting expression.
After testing, it turned out that you flipped it and it landed on heads.
Does that mean that you've discovered that the "true probability" for that flip should have been 100% heads?
What have you learn about the coin's true flip probability?
So after an "infinite" number of tests, we continue to get 50.00% frequency of heads, but with an 0.99 correlation with the orientation when loaded into the machine.
Now I load a coin into the machine and ask you to name the true probability that the result is heads. I don't tell you the initial orientation, but I know it privately.
What's the true probability of heads? Our testing found precisely 50.00% frequency of heads. But are you still sure the probability is an intrinsic property of the system, rather than a property of your state of knowledge of the system?
We can continue the pattern; maybe the 1% error itself correlates to 0.99 with someone running the microwave in the kitchen. This drops the line voltage and causes the servo to impart a little less momentum to the coin, causing it to flip one fewer times on average. Neither of us have currently checked that the microwave is running... And so on...
Say you conduct a thousands trials and calculate the error bar based on the results. If you conduct a hundred such experiments (each consisting of a thousand trials) and one of the experiments violates the error bar, does that invalidate it?
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?
If we observed such a system in nature, what would its "probability" mean?
And there's another interesting point - you could view "there's five white and five black balls" as your model. If in reality, there's one white ball and nine black - then your math is still right, but your model is wrong.
If you do experiments with the wrong model (assuming 50/50, getting samples from 1/10) - your best conclusion would be the model is wrong.
But for many settings, you'd end up declaring your container or the hand used to pull out balls has magical powers. (and to borrow from Douglas Adams go on to prove that black is white, and get killed in the next zebra crossing).
Nobody would a priori assume it is a weighted coin, in this example. A coin is chosen because it's been a standard weight and measure-backed object for centuries. It is about the observer's knowledge, which includes assumptions from every day experience.
You have to base a prior on assumptions. If you assume nothing and flip it 1000x, and calculate the probability, than you base that on assumptions of your own flipping ability, and hand wave it away with count divided by trials.
Probability is about the fact that you don't know everything about the balls themselves. If you could make 100% predictions then your knowledge of the balls would be equivalent to the balls' description.
Why do you even use "describe the balls themselves" to describe this situation? From the perspective of probability you just set an upper bound to how much knowledge you could possibly have about an object, it's still knowledge.
It's your best bet for a scenario where you can be sure that nobody else has more information than you do, though.
Even Einstein struggled with quantum mechanics, famously saying "[God] does not play dice with the universe".
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.
In fact, some people do argue that there is no measurement problem in the Copenhagen formulation of quantum mechanics to begin with – at least if you take it seriously and strictly go by the rule that the laws laid down by Bohr et al. only concern you as the observer and your knowledge about the system, and not the system itself. Following this train of thought, there is nothing "real" about the wavefunction and it is just a tool to come up with predictions. The same goes for the collapse of the wave function (which just describes a change in your ability to predict future measurements, and not a change of the object) and the term "measurement" (which we might as well replace with "enlightenment", i.e. the moment in which we obtain knowledge about the system).
In that sense, the only difference between classical and quantum mechanics is that our knowledge (viewed as a mathematical quantity) behaves differently in both theories: In classical physics, when we conduct multiple measurements of a given system in a row, our knowledge about that system will increase – to the point that, once we have measured all system properties to sufficient accuracy, we'll able to predict what any future measurement of any of those properties will yield (again, with some predictable uncertainty). So the knowledge of all our measurements has added up, it is an additive quantity.
In QM, this is fundamentally different: We can only know anything about the object the very moment we look at it. The rules of quantum mechanics (again, in the very strict interpretation laid out above) dictate that the second we conduct a measurement, we can forget about any knowledge obtained through previous measurements of other (conjugate) observables: Future measurements of those observables are inherently unpredictable. In that sense, our knowledge about quantum-mechanical objects never "adds up" to anything. (To see that this is really the the distinguishing feature between classical and quantum mechanics, recall that the existence of conjugate observables really is the only thing setting apart the quantum from the classical world: Without conjugate observables it would be impossible to distinguish, say, 100 electrons in a superposition of spin up and down from an ensemble of 100 electrons of which 50 are in a spin up state and the other 50 are in a spin down state.)
Of course, this whole interpretation is very unsatisfactory to lots of people (myself included) for a whole bunch of reasons. I assume that, to a large degree, this is due to the fact that laws of nature that put human observers in their very center seem rather undesirable. (At least since the time we switched from a geocentric to a heliocentric view of the world.)
But my impression is that there's another reason: Our intuition from classical mechanics & statistics has taught us that objects exist independently of us as observers and behave in a deterministic fashion, at least provided we as observers know enough about them. (Meaning that the more we know about the coin's initial position and velocity, the more likely we are to predict the outcome of the coin toss. If we don't know anything about the coin, though, the outcome is as unpredictable as measuring spin up/down in quantum mechanics.) Unfortunately, this whole line of argument is circular: The reason we believe that the existence of physical objects is independent of us, is precisely because knowledge in classical mechanics is an additive quantity and we can get to the point where we know "enough" to come up with deterministic predictions. That is, we never have to discard knowledge when running new measurements and so our knowledge takes on a independent "role" – which we call reality.
That would mean that both are equally likely as far as that person knows.
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.
- you put a number on it p(A)
- which is between 0 and 1
- and allows you to compare how sure you are about different things p(A) and p(B)
This number can be used to compute how sure you are about composite things:
p(A or B) = p(A) + p(B) - p(A and B)
p(A and B) = p(A given B) p(B) = p(B given A) p(A)
That number p happens to correspond to the notion of probability, but it has not been defined using a pre-existing notion of probability: https://en.wikipedia.org/wiki/Cox%27s_theorem
These numbers have to obey some rules if you require that a set of beliefs is consistent.
The number you assign to the plausability of A and the number you assign to the plausability of not-A have to sum 1.
If you think A and B are equally plausible, you have to put the same number on them.
If you think that A and not-A are equally plausible, you have to assign the number 0.5 to both.
If you put the number p(head)=p(tails)=0.5 as your degree of plausability that the coin I just flipped (I actually did it!) is showing head or tails it's not an "arbitrary" number. It means that you think both (exhaustive) outcomes are equally plausible. Why do you say it cannot mean anything?
In the example of the degree of belief (between 0 and 1) that you have that the coin on my desk is showing one face or the other, don't you agree that the right numbers that represent your indifference are 0.5 and 0.5? The quantity itself is not meaningless.
0) I tossed a coin, it lies flat on my desk
1) You have some degree of belief about the statements H:“the coin shows heads” and T:“the coin shows tails”
2) You want to quantify that degree of belief
3) You postulate that you can put a number on your degree of belief about some statement A with the following properties:
3a) p(A) it is between 0 (false) and 1 (true)
3b) p(A or B) = p(A) + p(B) - p(A and B)
3c) p(A and B) = p(A given B) p(B) = p(B given A) p(A)
4) p(H) + p(T) = 1
5) Unless your degree of belief about H is higher than your degree of belief about T
or your degree of belief about T is higher than your degree of belief about H ...
6) ... it follows that p(H) = p(T) = 0.5
The argument is flawless, the problem is with the interpretation.
> p(A or B) = p(A) + p(B) - p(A and B)
How does one add degrees of belief and what sense do we make out of the result?
> p(H) = p(T) = 0.5
Sure, two equal quantities representing degrees of belief must mean the degrees of belief are of the same magnitude. But what about P(H) = 2P(T)? What does it mean for one degree of belief to be twice as large as the other?
> How does one add degrees of belief and what sense do we make out of the result?
That's how we postulate [1] that the numeric representations of degrees of belief are added. Doesn't that look like a property that you want a numeric representation of degrees of belief to have?
If you have some degree of belief about A, some degree of belief about B, and you believe that A and B are mutually exclusive, wouldn't you want the number representing the degree of belief of "any of them" p(A or B) to be the sum p(A)+p(B)?
>> p(H) = p(T) = 0.5
> Sure, two equal quantities representing degrees of belief must mean the degrees of belief are of the same magnitude. But what about P(H) = 2P(T)? What does it mean for one degree of belief to be twice as large as the other?
Consider p(H or T) = p(H) + p(T) = 2 p(H) = 2 p(T). Isn't it natural to quantify the degree of belief that I got any outcome with a number that is the sum of the numeric representations of the degrees of belief that I got each outcome?
Or say that, instead of flipping a coin, I toss two of them. They're lying flat on my desk right now. The number of heads up is 0, 1, or 2.
How would you describe your degree of belief about the statements "X=0: there are no heads", "X=1: there is one" and "X=2: there are two"?
Wouldn't you say that your degree of belief about "X=1" is of the same magnitude as your degree of belief about "X=0 or X=2"?
Wouldn't you say that your degree of belief about "X=0" is of the same magnitude as your degree of belief about "X=2"?
Wouldn't that make the numerical representation of your degree of belief about "X=1" twice as large as the numerical representations of your degrees of belief about each of "X=0" and "X=2"? (Where you assign numbers to degrees of belief using the representation we're discussing.)
p(X=1) = p(X=0) + p(X=2) = 2 p(X=0) = 2 p(X=2)
[1] in fact I think this is what we get from postulates which are a bit more general, but for the sake of the discussion we may stay in this level
All in all, to me it's clear that these degrees of belief are a theoretical construct, not an empirical reality. I don't think people assess the truth value of a statement on a continuum from truth to false. This is not how the human psyche works. Personally, no, it's not natural for me to have a degree of belief (in the way that you have defined them) about a statement, and I have no idea how to interpret arithmetic operations involving these "things".
Ok, what can you say then regarding your degrees of belief about the statements "X=0: there are no heads", "X=1: there is one", "X=2: there are two" and “X={0,2}: there are none or two”?
Of course, if you don't think you have degrees of belief to start with there is no way you can make sense of assigning numbers to them. But I thought we had progressed to the point where you could accept that they existed and that they could be ordered.
> I don't think people assess the truth value of a statement on a continuum from truth to false.
So they cannot be ordered? Or there are no extremes?
> When interpreting probability as a degree of belief this property is not only not useful, but is particularly troublesome, because adding and subtracting degrees of beliefs doesn't appear to make a lot of sense.
Are the rules I proposed problematic in some specific way? They make a lot of sense as far as I can tell.
Do the rules make sense? The rules don't have to make sense, they're axioms. They're assumed to be true whether they make sense or not. We are not discussing the axioms. We are discussing interpretations of probability. In my opinion a good interpretation of probability must provide a context in which these axioms (kind of) make sense. And that's one of the problems I have with the interpretation of probability as a degrees of belief, the rules don't make sense in the provided context, at least to me, because I don't know how to make sense of arithmetic operations involving degrees of belief. (But that doesn't mean that I think the rules themselves don't make sense.)
Finally, even if you think that the human mind doesn't work in the way degrees of belief are hypothesised to work, you may still find the concept useful as a means of giving an interpretation to probability. Personally, I don't think that the mind works like that, nor that they're useful as an interpretation of probability. This is basically my position.
Anyway, I think you're getting the direction of the argument wrong. It's not that you have probabilities and force an interpretation of them as degrees of belief.
You start with real numbers representing degrees of belief (with an ordinal meaning only, a larger number means more plausible) and some "common sense" properties they should have to be "rational":
- having identical information should result in the same degree of belief
- the degree of belief in "not A" should be a function of the degree of belief in A
- the degree of belief in "A and B" should be a function of the degree of belief in "A given B" and the degree of belief in B
p(A) + p(not A) = 1
p(A and/or B) = p(A) + p(B) - p(A and B)
p(A and B) = p(A|B)p(B) = p(A)p(B|A)
In pages 5-9 he derives the rules of probability as a "reasonable expectations" extension to symbolic logic.
Of course you're right that this is a description of how rational thought should be and not necessarily a description of how people think. Actual beliefs can be inconsistent in the same way that one can believe things that go against the laws of logic.
It's always fun to discuss the foundations of probability. Thanks.
Probably have to use actual quantum phenomena that behave probabilistically by definition if you want a currently irrefutable physical example.
I'm personally not convinced even this is fundamentally probabilistic and we currently have to rely on probability theory as a crutch for complex behaviors we just quite don't understand yet or don't have the time and resources to compute.
My point exactly. Probability theory is a precise mathematical formalization of the concept of "not enough information".
I'm not sure if it helps, but I suppose a compromise here would be the assumption that you don't really know the starting configuration of yourself, why you draw probabilistic inferences naturally, that the sun will go up tomorrow like every day.
If that has a biologic explanation, then the top comment was not just to the illusive argument of platonic ideals.
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.
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.
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?
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 :)
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?
For all I know this is a false dichotomy. True RNGs don't exist, for one, in a deterministic philosophy.
RNGs are per definition non-deterministic, but algorithms are deterministic--by the definition that I learned. We use floating gates picking up cosmic background radiation or the fallout from radioactive elements to come close to really random. XKCD's butterfly joke applies.
Computable numbers have a computable complexity. For a non-deterministic program the fair comparison to a true RNG would be a programm that outputs all digits of Pi up to infinity, I suppose. Vice-versa, I'm not sure if a pseudo-random number generator couldn't be fit into the equivalent kolmogorov complexity of a 40-digits of PIE.
Are you saying it couldn't be? On the one-hand it is the case that pseudo RNGs simply rely on intractible complexity, which must regularly supercede 40 digits of Pi equivalent to 40 byte key, as opposed to your 4kb RSA key chains and what not.
On the other hand I think your argument is akin to the gamblers fallacy: I have absolutely no clue but I will hope it will not have been a primitive RNG, right?
For reference, here's a simple pseudo RNG taken from TinyPTC example code to produce a TV-noise graphic in a loop
noise = seed;
noise >>= 3;
noise ^= seed;
carry = noise & 1;
noise >>= 1;
seed >>= 1;
seed |= (carry << 30);
noise &= 0xFF;
pixel[index] = (noise<<16) | (noise<<8) | noise;However, you can ask for resource-bounded Kolmogorov complexity. Suppose you ask for the shortest length of the description of a PTIME machine which output the N bits of an RNG versus that which outputs the first N bits of PI. Complexity theorists believe that the first will be longer. Proving that conclusively might possibly have a bearing on P=BPP.
When you go all the way down to finite-state machines, KC becomes something equivalent to finite-state information-lossless compression, like Lempel-Ziv compressibility.
Long story short: by restricting the resources available for the computation, it is possible to discriminate among some such examples as you point out.
In the case of true random numbers, how is that so?
Very few random sequences can be generated by a logarithmic-sized program, since most strings are not significantly compressible [1]. A simple counting argument shows that: there are 2^n strings of n bits, but only 2^(lg n) = n logarithmic-sized strings, a much smaller number!
[1] http://theory.stanford.edu/~trevisan/cs154-12/kolcomplexity-...
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.
It's two sides of the same coin -- computable uncertainty va non-computable certainty.
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.
* "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.
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!
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]
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...
Are you going to touch on Chaitin's Omega?