Is there a definition of randomness?

[entropy1]

Is there a definition of "random(ness)"? Is it defined?

 

[andrewkirk]

There is no formal definition. The items used in probability theory, such as random variables and stochastic processes, have formal definitions, but these do not help with questions such as 'what does random mean', which are philosophical rather than mathematical or scientific.

There is also a formal definition about infinite sequences of digits, called 'normal', which has some similarities to the folk notion of 'randomness'. But again it does not help in philosophical discussions on topics such as 'Is the world deterministic or random?'

 

[WWGD]

As I understand , a Random process is one that cannot be predicted but can be described probabilistically.

 

[andrewkirk]

As I understand , a Random process is one that cannot be predicted but can be described probabilistically.

That's the common understanding of it. The twist comes in the meaning of 'can be predicted'. A process may be unpredictable in one theory but predictable in a more sophisticated theory. There's no way we can know that there isn't some currently unknown, more sophisticated theory that can predict outcomes that currently seem random to us. So what we can say is that a given process is random with respect to theory T. That is, predictability depends on what theory we are using to predict.

 

[FactChecker]

It's often good to think of "random" in terms of the information available to make a guess at the outcome, rather than what the outcome will be. It's the theory of guessing a result rather than the theory of the process itself. That avoids the complaint about calling an outcome that has already occurred "random" (such as a coin toss that has occurred but not seen). It also makes Bayesian theory, where probabilities are adjusted when more information is obtained, more natural. And it allows us to call something "random" if we know that it is deterministic, but we do not know enough to know the outcome and must guess.

 

[entropy1]

If you can't predict the next outcome, how come you can predict the average outcome?

 

[FactChecker]

If you can't predict the next outcome, how come you can predict the average outcome?

Are you talking about the population average or a sample average?
For population average:
It's just common experience, like predicting a coin toss or the roll of dice. We have seen enough to estimate the probabilities.

For a sample average:
You can not predict exactly. You can calculate the expected mean and the variance of a sample average. The answers will depend on the sample size.

 

[WWGD]

That's the common understanding of it. The twist comes in the meaning of 'can be predicted'. A process may be unpredictable in one theory but predictable in a more sophisticated theory. There's no way we can know that there isn't some currently unknown, more sophisticated theory that can predict outcomes that currently seem random to us. So what we can say is that a given process is random with respect to theory T. That is, predictability depends on what theory we are using to predict.

True. Depending on the tools, information available at a given point, I should have said, and within a theory. A good question is whether there are phenomena that are somehow intrinsically unpredictable, i.e., not predictable within any system.

 

[StoneTemplePython]

True. Depending on the tools, information available at a given point, I should have said, and within a theory. A good question is whether there are phenomena that are somehow intrinsically unpredictable, i.e., not predictable within any system.

Your original post was awfully close to the (Frank) Knight definition that:

risk is where a future outcome is unknown but the distribution is known/knowable. Uncertainty is where neither the future outcome or distribution is known.

Economists are fond of bringing this up. Perhaps a touch too simple but it's worth thinking on.

 

[entropy1]

Could one say that if we have two variables A and B that correlate for, say, 50%, that A (or B) is less random because knowledge of the outcome of B (or A) increases the likelyhood of a correct prediction of the outcome of A (or B)?

 

[FactChecker]

Could one say that if we have two variables A and B that correlate for, say, 50%, that A (or B) is less random because knowledge of the outcome of B (or A) increases the likelyhood of a correct prediction of the outcome of A (or B)?

That line of thought would open a can of worms. Every coin flip has an outcome that is completely determined by the down-side of the coin. It would not be correct to say that that makes the up-side outcome less random.

 

[entropy1]

That line of thought would open a can of worms. Every coin flip has an outcome that is completely determined by the down-side of the coin. It would not be correct to say that that makes the up-side outcome less random.

That seems to me comparable to predicting the outcome after observing it, which would not do justice to the notion of prediction.

Is there a definition of 'prediction' in this context?

 

[WWGD]

Would an intrinsically random process necessarily have correlation 0 with any other process?

 

[entropy1]

Would an intrinsically random process necessarily have correlation 0 with any other process?

That's a good question. I think so, yes. If there is dependence, correlation, I would suggest randomness has been limited. That is the issue I was getting at.

 

[WWGD]

That's a good question. I think so, yes. If there is dependence, correlation, I would suggest randomness has been limited. That is the issue I was getting at.

Wonder how this would pan out Mathematically and Physically.

 

[FactChecker]

That's a good question. I think so, yes. If there is dependence, correlation, I would suggest randomness has been limited. That is the issue I was getting at.

I think you are taking this in a direction that will not pay off. There are too many things that occur together, where you would not want to say that either one makes the other less random.

Example: Pick a random person out of a crowd. His height is related to the length of his left arm, right arm, left leg, right leg, weight, belt size, sex, age, etc., etc., etc. None of this makes anyone of them more or less random.
Yet they are correlated, so knowing one does help to make the others more predictable. But the one you need to know is itself random.

 

[andrewkirk]

A good question is whether there are phenomena that are somehow intrinsically unpredictable, i.e., not predictable within any system.

I wrote an essay about this a few years back, which you may find interesting:

https://wordpress.com/post/sageandonions.wordpress.com/75

My conclusion was that, unless we put artificial constraints on what counts as a theory, there is no such thing as intrinsically unpredictable, since we can imagine a theory that I call the 'M-law', which lists every event that happens anywhere in spacetime. No event is unpredictable under that theory. Such a theory would be unknowable by humans, but that's beside the point.

For fans of MWI, knowing the M-law is equivalent to knowing which of Everett's infinite set of parallel universes we are in. But I was not aware of that equivalence at the time of writing the essay.

I was unable to think of any constraint that excluded the M-law that wasn't just obviously constructed just for the purpose of excluding it. Any more general constraint rule I tried ended up excluding theories we would like to include, even down to Newton's laws of motion.

 

[atyy]

In quantum theory there is a definition of randomness that is inconsistent with certain types of determinism.

If we impose the condition that the deterministic theory is "local" (no faster than light propagation), then one can show that the randomness of quantum mechanics is incompatible with that type of determinism. Because in an operational sense, we believe that no one yet has technology that permits faster than light communication, quantum theory can guarantee randomness.

However, in a more general sense, if one allows the deterministic theory to be nonlocal, then the randomness of quantum theory is compatible with determinism, and ignorance of the initial conditions.

https://arxiv.org/abs/1708.00265
Certified randomness in quantum physics
Antonio Acín, Lluis Masanes
(Submitted on 1 Aug 2017)
The concept of randomness plays an important role in many disciplines. On one hand, the question of whether random processes exist is fundamental for our understanding of nature. On the other hand, randomness is a resource for cryptography, algorithms and simulations. Standard methods for generating randomness rely on assumptions on the devices that are difficult to meet in practice. However, quantum technologies allow for new methods for generating certified randomness. These methods are known as device-independent because do not rely on any modeling of the devices. Here we review the efforts and challenges to design device-independent randomness generators.

 

[entropy1]

I think you are taking this in a direction that will not pay off. There are too many things that occur together, where you would not want to say that either one makes the other less random.

Example: Pick a random person out of a crowd. His height is related to the length of his left arm, right arm, left leg, right leg, weight, belt size, sex, age, etc., etc., etc. None of this makes anyone of them more or less random.
Yet they are correlated, so knowing one does help to make the others more predictable. But the one you need to know is itself random.

Then, perhaps, the non-random factor lies in picking a person rather than a cat, dog or snake? The properties are inherent to the person. If we take 'properties' as 'outcome', they are correlated. It would be like measuring circles on both sides and finding that they are both round.

 

[PeroK]

I wrote an essay about this a few years back, which you may find interesting:

https://wordpress.com/post/sageandonions.wordpress.com/75

My conclusion was that, unless we put artificial constraints on what counts as a theory, there is no such thing as intrinsically unpredictable, since we can imagine a theory that I call the 'M-law', which lists every event that happens anywhere in spacetime. No event is unpredictable under that theory. Such a theory would be unknowable by humans, but that's beside the point.

Your M-Law isn't compatible with Quantum Mechanics. A measurement of the spin of an electron will return a value which is not predictable. You would need to impose a specific interpretation of QM like the MWI and the existence of some meta-universe where the M-Law could operate across all worlds. But, MWI is only an interpretation and may not have any physical validity.

I think there is also a problem if the universe is spatially infinite, in that the M-Law would have to process an infinite amount of information. That can only be done if there is already a known pattern. I think there are computability issues in handling - dare I say it - a random set of initial conditions! Again, you would need to impose the theory that the universe's initial conditions could be predicted - precisely at every point in infinite space - by some prior law. And, even that appears not to work if the universe had no beginning. In that case, your M-Law needs to gather an infinite amount of data at some arbitrary initial time to get started.

I would say that the M-Law is roughly equivalent to God and I would question the existence of either.

PS the universe may be finite, have a defined beginning with a well-defined set of initial conditions and QM might be amenable to some sort of intrinsic predictability argument, but I don't believe we can assume any of those to be the case.

PPS in fact, in QM, talking about the precise position and momentum of every particle is not possible. Initial conditions in QM are intrinsically probabalistic.

 

[FactChecker]

Then, perhaps, the non-random factor lies in picking a person rather than a cat, dog or snake? The properties are inherent to the person. If we take 'properties' as 'outcome', they are correlated. It would be like measuring circles on both sides and finding that they are both round.

Now you are trying to isolate the cause of the random behavior of the selected person's right arm length (for example). That is possible but it does not change the fact that the selected arm length is random. A non-constant function of a random variable is a random variable.

Similarly, I could attempt to isolate the random behavior of a coin toss to the motion of the hand that flips the coin. That does not make the result of the coin toss any less random.

 

[entropy1]

Now you are trying to isolate the cause of the random behavior of the selected person's right arm length (for example). That is possible but it does not change the fact that the selected arm length is random. A non-constant function of a random variable is a random variable.

Similarly, I could attempt to isolate the random behavior of a coin toss to the motion of the hand that flips the coin. That does not make the result of the coin toss any less random.

Sorry, there is a misunderstanding I see: I was talking here, and here, about a correlation between two strings of data A and B. I see now that I never introduced that I was talking about that. Sorry.

 

[Stephen Tashi]

If you can't predict the next outcome, how come you can predict the average outcome?

We can't (with certainty) predict the average outcome. The "expected value" of a random variable has a mathematical definition. There is nothing in probability theory that guarantees that an actual set of outcomes will have an average equal to the expected value.

Attempts to connect the concept of probability in definite way with actual events have (so far) been unsuccessful. To guarantee some outcome will actually happen contradicts the notion that there is something probabilistic about it actually happening.

 

[FactChecker]

Sorry, there is a misunderstanding I see: I was talking here, and here, about a correlation between two strings of data A and B. I see now that I never introduced that I was talking about that. Sorry.

No, I understood that. I only brought up the example of a function because that can be the strongest correlation possible. The relationship between correlated variables is usually weaker than a functional relationship. If a function of a random variable is random, then we have to conclude that a correlated variable with a weaker relationship than a function is random.

 

[entropy1]

No, I understood that. I only brought up the example of a function because that can be the strongest correlation possible. The relationship between correlated variables is usually weaker than a functional relationship. If a function of a random variable is random, then we have to conclude that a correlated variable with a weaker relationship than a function is random.

Would you be willing to illustrate that mathematically a bit? I can't seem to see what you mean by text only.

 

[entropy1]

Do you mean sample_A=f(human_A) and sample_B=f(human_B), with human_A=g(general_human, random_value) and human_B=g(general_human, random_value)?

 

[FactChecker]

Would you be willing to illustrate that mathematically a bit? I can't seem to see what you mean by text only.

Suppose we have random variables X, Y = 2X and Z = Y + ε = 2X + ε, where ε is an independent random variable. All three variables X, Y, Z, are correlated. Y is a function of X, but is still considered a random variable. Z is related to X, but is not a deterministic function of X. It is still considered a random variable. It's connection (correlation) to X is weaker than Y's is. There is no reason to say that Y or Z are less random than X. We could have just as easily started with Y, X = 0.5Y, Z = Y + ε.

 

[andrewkirk]

A measurement of the spin of an electron will return a value which is not predictable.

It is predictable by the M-law.

Later on you say that the M-law is roughly equivalent to [some mainstream concepts of] God. It has one similarity, which is that of omniscience. But there is a lot more to mainstream concepts of God than just omniscience. The M-law could just be a list written down by some 25-dimensional being that is observing our spacetime, multiverse or whatever from the outside, transcending our space and time - but presumably having its own meta-time. It doesn't need to be supreme. It could in its turn be being observed by a 52-dimensional being, and that by a 104-dimensional being, and so on ad infinitum.

I would question the existence of either.

So do I. But the question is not whether there exists a theory or being that could predict the spin of the electron, but rather, is it in principle impossible that such a theory or being could exist? Of course we can't say, even though we may strongly suspect that no such theory or being exists in fact.

We could avoid the difficulty by limiting the allowable means of prediction to those that we currently know exist and to be accessible to humans. But if we go down that path, what is predictable changes over time. As I point out in the essay, that means that a thousand years ago the year of the next appearance of Halley's comment was unpredictable. And in practice, that's what people mean by predictable. That definition is purely epistemological. It's only when people try to talk about fundamental or ontological unpredictability that they run into trouble.

 

[PeroK]

It is predictable by the M-law.

You can't sweep away the evidence of QM and will a deterministic theory of everything into existence by writing those six words.

There is, in fact, an interesting quotation from Griffith's book on QM: "even God doesn't know which electron is which". If God doesn't know, then neither do your hypothetical 25-dimensional beings.

 

[andrewkirk]

There is, in fact, an interesting quotation from Griffith's book on QM: "even God doesn't know which electron is which". If God doesn't know, then neither do your hypothetical 25-dimensional beings.

Indistinguishability of particles has nothing to do with the predictability of future observations.

Predictions under QM are probabilistic. But there is nothing in QM - including Bell's Theorem - that says QM can't be part of, and consistent with, some larger theory that is not probabilistic.

 

[FactChecker]

Do you mean sample_A=f(human_A) and sample_B=f(human_B), with human_A=g(general_human, random_value) and human_B=g(general_human, random_value)?

Perhaps what you are looking for is this:
Suppose X and Y are correlated and P(X∈A | Y∈B) ≠ P(X∈A), so the events A and B are not independent events. Then knowing Y∈B has changed the probabilities of X. There are examples where knowing Y∈B can increase or decrease the probabilities of X∈A. That is, there are examples where P(X∈A | Y∈B) > P(X∈A) and other examples where P(X∈A | Y∈B) < P(X∈A).

Furthermore, if X and Y are correlated real valued random variables, there are examples where var( X | Y∈B ) < var( X ) and other examples where var( X | Y∈B ) > var( X ) . So knowing Y∈ B can either decrease or increase the random variability of X, depending on whether knowing Y∈B increased or decreased the predictability of X.

The bottom line is that it is not possible to make a general rule about how "random" a variable is simply because you know that it is correlated with another variable.

 

[entropy1]

Is it true that if X and Y are random variables, that X given Y, P(X=1|Y=1), is random too?

 

[FactChecker]

Is it correct that if X and Y are random variables, that X given Y, P(X=1|Y=1) is random too?

P(X=1|Y=1) is a specific probability number, not a random variable. (If P(Y=1)=0, then P(X=1|Y=1) is not even a valid probability.)
Suppose X and Y are discrete random variables where (X,Y) are the results of an experiment, E, and P(Y=1) > 0. Then you could define a related experiment, E_{till Y=1} where experiment E is repeated till the result has Y=1. The X value result of E_{till Y=1} would be a random variable, Z = X_{till Y=1}.

 

[nomadreid]

As the original question was about the definition (or lack thereof) of randomness, I am surprised no one has mentioned the work of Gregory Chaitin; e.g., Part III of https://www.cs.auckland.ac.nz/~chaitin/ait/index.html

 

[Sherwood Botsford]

In passing, random data is incompressible. Sometimes this means that the compression algorithm isn't clever enough to exploit the redundency present.

So: If the data is compressible, it's not random. If it's incompressible it *might* be random.

You may be able to come up with a calculable degree of randomness using entropy/enthalpy calculations.