What is the definition of randomness in mathematics and physics?

[xxxx0xxxx]

That's why I asked for an accurate definition.

I was interested at the idea to distinguish between random and pseudo-random events. For example, think you have a radiotelescope and receive a signal which can be interpreted as a possible signal coming from an extraterrestrial intelligence. But you don't know the language, those beings might communicate with a set of symbols we interpret as random when in reality they are only pseudo-random. Is there any way to distinguish between the two? I don't think so, because it is knowledge- and code-dependent. Therefore "randomness" is a subjective concept, like the word "ignorance" or "beauty", not an objective or "intrinsic" property of things or a process.

Ah well, we're talking about a different kind of animal than randomness, signals always contain extra information which we call "noise." In this case you must be able to distinquish the part of the signal containing useful information from the part that is just noise (another word for the part we're not interested in detecting). The way that it is done, is you have to know something about the information in the signal a priori, eg its modulation scheme.

It is a peculiar property of signals that the more noisy, i.e. random, they seem to be, the more information they contain.

 

[Bacle]

Still, re recovering noisy signals without knowing the original, I have heard of
some techniques using maximum-likelihood estimation (with the two parameters being mean and variance, tho cheating, in that variance is ultimately held fixed) , in which the parameter
value (re the mean) that maximizes the likelihood of the received signal is that of the average of a collection of samples of signals taken at different receivers. This technique is used at a lower level with cell-phones, in which noisy signals are received, and then an approximation to the original signal is made by averaging out the signals received at different towers. I don't know how well this would extend to signals coming from the outside, but it seems like something to start with.

 

[xxxx0xxxx]

Still, re recovering noisy signals without knowing the original, I have heard of
some techniques using maximum-likelihood estimation (with the two parameters being mean and variance, tho cheating, in that variance is ultimately held fixed) , in which the parameter
value (re the mean) that maximizes the likelihood of the received signal is that of the average of a collection of samples of signals taken at different receivers. This technique is used at a lower level with cell-phones, in which noisy signals are received, and then an approximation to the original signal is made by averaging out the signals received at different towers. I don't know how well this would extend to signals coming from the outside, but it seems like something to start with.

Yes I think that that's a power management technique but sometimes the actual information is modulated using a so-called "spread spectrum" technique; the modulation scheme depends on some pseudo-random sequencing of power, frequency, phase, or amplitude. Provided the receiver knows the sequencing scheme, the signal can be detected from the noise by averaging. GPS for instance works along these lines, as well as CDMA (3G) cell phones (although the power is high enough on cell phones that they can use a rake receiver to pull the signal out immediately; this is primarily for combating fading due to multipath propagation).

 

[Aidyan]

I think the problem with the symbols may not be so much in whether they are random or not, but more in interpreting and assigning some meaning to them.

In fact. Think of the decimal digits sequence of pi which are compeltely random, but they have a clear defined meaning. But only if you know what that strange number pi is. And yet there is that deeply ingraved pre-conception that randomness = no meaning, no purpose, no conscious choice, etc.

 

[Aidyan]

Ah well, we're talking about a different kind of animal than randomness, signals always contain extra information which we call "noise." In this case you must be able to distinquish the part of the signal containing useful information from the part that is just noise (another word for the part we're not interested in detecting). The way that it is done, is you have to know something about the information in the signal a priori, eg its modulation scheme.

As I understand it "noise" is considered a random process. The separation of noise from "useful information" is knowledge/observer dependent. It is the type of knowledge a priori which makes the difference. If we know nothing all might appear noisy, i.e., random.

 

[MarcoD]

In fact. Think of the decimal digits sequence of pi which are compeltely random, but they have a clear defined meaning. But only if you know what that strange number pi is. And yet there is that deeply ingraved pre-conception that randomness = no meaning, no purpose, no conscious choice, etc.

By the Kolmogorov definition, pi therefor isn't a random sequence. I didn't know that.

 

[Aidyan]

By the Kolmogorov definition, pi therefor isn't a random sequence. I didn't know that.

Hmm... google, google... ah, yes. You are right. Pi isn't completely random in Kolmogorov sense. Anyhow, as I understand his theory, his definition of randomness will not bring us further as to the distinction between randomness and pseudo-randomness.

 

[SW VandeCarr]

Hmm... google, google... ah, yes. You are right. Pi isn't completely random in Kolmogorov sense. Anyhow, as I understand his theory, his definition of randomness will not bring us further as to the distinction between randomness and pseudo-randomness.

The short answer to this seemingly endless philosophical talk is that you can't know if an apparently random (no discernible pattern) string of digits is random or determined unless you know how it's generated. A generator based on nuclear decay for example is considered to be random, not pseudorandom.

 

[Aidyan]

The short answer to this seemingly endless philosophical talk is that you can't know if an apparently random (no discernible pattern) string of digits is random or determined unless you know how it's generated.

And once you know how a (previously) apparently random string was generated will you still call it “random”?

 

[SW VandeCarr]

And once you know how a (previously) apparently random string was generated will you still call it “random”?

I wouldn't if it was generated by an algorithm that is shorter than the string.

 

[disregardthat]

Randomness doesn't have a definition in the sense that we have a rule for pointing out something as random. But if a probabilistic model works, then it is justifiable to call it random.

The discussions of whether something really is random are pointless, not only because randomness has no definitive sense, but that it fails to acknowledge the way the word random is used. It doesn't refer to some fundamental aspect of a situation at all.

 

[xxxx0xxxx]

As I understand it "noise" is considered a random process. The separation of noise from "useful information" is knowledge/observer dependent. It is the type of knowledge a priori which makes the difference. If we know nothing all might appear noisy, i.e., random.

I'm talking about the electromagnetic environment of the receiver.

The antenna sits in a sea of electrical noise that it amplifies and sends to the demodulator.

The demodulator has the task of picking out a particular type of electromagnetic "noise" that corresponds to a modulation scheme known by the receiver. It rejects all the other noise, but this does not mean that the rejected noise contains no information or is random.

The rejected information is not useful, because the receiver has been designed to reject it. But this rejected "noise" contains everything else in the electromagnetic spectrum at the antenna; it consists of all the other electromagnetic signals, man-made or natural.

 

[lavinia]

Just piling on, there's at least two problems here:
- Useless is far too strong a term.
- "Next outcome" rules out continuous random processes.

A process is "random" if the future evolution of the process is not uniquely determined by any (knowable) set of initial data.

To be picky, there is no reason to exclude processes whose evolution over time is uniquely determined by initial data. For example, given the probability space {4}, I guarantee that every time you randomly select from this space you will get a four.

Next outcome does not rule out continuous processes because one can sample a continuous process on discrete time intervals.

Generally if a random process does not have expectation zero it will tend to drift deterministically and then the randomness will be the uncertainty around the deterministic drift. No amount of information can help predict the next value of this uncertain part.

 

[lavinia]

Thanks, interesting podcast. Finally it confirms, randomness is a 'slippery' thing: there are sequences of symbols one produces deterministically (nice was the example of pi and prime numbers) and yet they pass all the definitions of randomness statisticians could think of. Obviously, simply because randomness is not an intrinsic objective property of things or processes, it does not have any concrete existence in itself, but is a relative subjective mental category. And also the idea to connect "lack of purpose", or "lack of will", or "lack of conscious choice" to random events is an unwarranted logical inference. Finally the good old Democritus was right.

In Quantum Mechanics the wave function evolves according to a Markov like process. Presuambly there is no way to ever improve our knowledge of the evolution of amplitudes so the process is intrinsically random. It is not subjective.

 

[Aidyan]

In Quantum Mechanics the wave function evolves according to a Markov like process. Presuambly there is no way to ever improve our knowledge of the evolution of amplitudes so the process is intrinsically random. It is not subjective.

Isn't the idea of intrinisc randomness as a process on which we have "no way to ever improve our knowledge" already a subjective category?

I didn't go through all that but as to QM and Markov processes this seems to be debatable:

http://pra.aps.org/abstract/PRA/v49/i3/p1607_1
http://pra.aps.org/abstract/PRA/v56/i4/p3301_1
http://pra.aps.org/abstract/PRA/v54/i2/p1737_1

 

[lavinia]

Isn't the idea of intrinisc randomness as a process on which we have "no way to ever improve our knowledge" already a subjective category?

I didn't go through all that but as to QM and Markov processes this seems to be debatable:

http://pra.aps.org/abstract/PRA/v49/i3/p1607_1
http://pra.aps.org/abstract/PRA/v56/i4/p3301_1
http://pra.aps.org/abstract/PRA/v54/i2/p1737_1

The evolution of the wave function is not a Markov process but it is similar. Instead of probabilities evolving, there are amplitudes. I don't know what a "subjective category" is but it seems like what you are saying is that everything is subjective - which I guess is true philosophically - but not really the point here.

 

[Aidyan]

The evolution of the wave function is not a Markov process but it is similar. Instead of probabilities evolving, there are amplitudes. I don't know what a "subjective category" is but it seems like what you are saying is that everything is subjective - which I guess is true philosophically - but not really the point here.

It is probably that dissimilarity that makes the difference. In QM the evolution of the wavefunction can not be interpreted as in CM, i.e. as "those states the system goes through in time while we are not observing it".

Randomness is not a physical intrinsic objective property of things as could be the amplitude of a signal which is independent from what we know of that signal. Randomness is a mental construct which measures our ignorance of that process, it depends from what we know. This unawareness leads to the unjustified question how it could be possible to distinguish between truly random and pseudo-random processes? It is unjustified because only the latter is measured, while the former does not exist in itself, if not in our minds.

 

[xxxx0xxxx]

It is probably that dissimilarity that makes the difference. In QM the evolution of the wavefunction can not be interpreted as in CM, i.e. as "those states the system goes through in time while we are not observing it".

Randomness is not a physical intrinsic objective property of things as could be the amplitude of a signal which is independent from what we know of that signal. Randomness is a mental construct which measures our ignorance of that process, it depends from what we know. This unawareness leads to the unjustified question how it could be possible to distinguish between truly random and pseudo-random processes? It is unjustified because only the latter is measured, while the former does not exist in itself, if not in our minds.

Certain physical processes, such as radioactive decay, spontaneous emission, etc. are totally unpredictable, and are used as sources of entropy for random number generation. Some classical field properties cannot be measured simultaneously due to the uncertainty principle (phase and amplitude for instance), these are all directly observable sources of randomness, but no computational process can be shown to be a source of randomness.

 

[lavinia]

It is probably that dissimilarity that makes the difference. In QM the evolution of the wavefunction can not be interpreted as in CM, i.e. as "those states the system goes through in time while we are not observing it".

Randomness is not a physical intrinsic objective property of things as could be the amplitude of a signal which is independent from what we know of that signal. Randomness is a mental construct which measures our ignorance of that process, it depends from what we know. This unawareness leads to the unjustified question how it could be possible to distinguish between truly random and pseudo-random processes? It is unjustified because only the latter is measured, while the former does not exist in itself, if not in our minds.

I don't necessarily agree with this since the evolution of the wave function seems intrinsically random. How is is not intrinsic? It is a theorem that there are no hidden variables that can improve our knowledge.

 

[Oldfart]

I have sort of an offshoot question for you folks:

Would the first 100 decimal places of pi (minus the decimal point, or 314159...) be considered a series of random numbers? What about the first 100 decimal places of pi/2?

 

[SW VandeCarr]

I have sort of an offshoot question for you folks:

Would the first 100 decimal places of pi (minus the decimal point, or 314159...) be considered a series of random numbers? What about the first 100 decimal places of pi/2?

This has been discussed ad nauseum in this thread and I might re-ignite it by responding, but I'll give you the view to which I subscribe.

The decimal expansions of irrational numbers cannot be random as they are completely determined by an algorithm (see Kolmogorov). However, many use intervals of such sequences as "random numbers" because they have no apparent pattern. That is, they can pass statistical tests for randomness.

 

[Oldfart]

Thanks, SW! But wouldn't that eliminate any series of numbers as being random? This because one could (in theory) endlessly multiply or divide the digits of pi by different numbers (a determinant process), eventually yielding every possible combination of 100 digits. Or a zillion digits.

What am I missing here? Duhh...

 

[SW VandeCarr]

Thanks, SW! But wouldn't that eliminate any series of numbers as being random? This because one could (in theory) endlessly multiply or divide the digits of pi by different numbers (a determinant process), eventually yielding every possible combination of 100 digits. Or a zillion digits.

What am I missing here? Duhh...

Well for one thing, multiplication and division are algorithmic procedures, so you are taking a string produced by an algorithm and transforming it by additional algorithmic procedures. By the Kolmogorov definition, a random sequence must be generated by a random process (not pseudorandom). Such a process may from time to time generate strings that will fail a statistical test for randomness.

As was stated in this thread, you can't tell if a string is random in the Kolmogorov sense unless you know how it is generated.

 

[bbbeard]

I don't necessarily agree with this since the evolution of the wave function seems intrinsically random. How is is not intrinsic? It is a theorem that there are no hidden variables that can improve our knowledge.

I know I'm jumping in late to respond to a post you put up a month ago, but...

It seems to me that there is nothing random about the time evolution of a quantum state. If you know the initial state |psi₀>, you just have to compute all the components <k|psi₀> in the energy eigenbasis {|k>}. Then each component evolves as exp(-iω_kt) where ω_k=E_k/hbar.

In other words, the evolution is not random. The evolution is deterministic. But if you measure the energy at a later time, the result of the measurement will be probabilistic (unless the initial state is an energy eigenstate, in which case the later measurement should give the same energy).

 

[PatrickPowers]

The Oxford English Dictionary defines 'random' as: "Having no definite aim or purpose; not sent or guided in a particular direction; made, done, occurring, etc., without method or conscious choice". However, if we intend randomness as events with equal frequency probability this can't be. Think for example of the frequency of symbol sequences of a crypted text file. So I'm wondering if there exists a rigorous definition of randomness in mathematics and/or physics which can be interpreted as the above definition?

No one owns the word so it is used with several different meanings intended.

The definition I like the most is simply "unpredictable." It is perfectly OK for something to be predictable to one person with more knowledge and unpredictable to another. It is subjective.

Often it is used to mean "unpredictable with every possibility is equally likely."

Physicists tend to use it as "as far as we know no one can predict this." Sometimes they seem to be saying that "no one will ever be able to predict this" which seems to me like overreaching. I think a better definition would be "don't try to get a Phd by trying to figure this process out because we are pretty sure you won't succeed."

"Pseudorandom" may be used for something that is unpredictable the first time but repeats so it is predictable subsequently. "Stochastic" means something that is both unpredictable and doesn't repeat.

So is something that is predictable 99.99% of the time predictable or is it random? Well, if you predict "no hurricanes" every day you may be right 99.99% per cent of the time and wrong only 0.01%, but your prediction is nonetheless worthless.

 

[lavinia]

I know I'm jumping in late to respond to a post you put up a month ago, but...

It seems to me that there is nothing random about the time evolution of a quantum state. If you know the initial state |psi₀>, you just have to compute all the components <k|psi₀> in the energy eigenbasis {|k>}. Then each component evolves as exp(-iω_kt) where ω_k=E_k/hbar.

In other words, the evolution is not random. The evolution is deterministic. But if you measure the energy at a later time, the result of the measurement will be probabilistic (unless the initial state is an energy eigenstate, in which case the later measurement should give the same energy).

you are right but I just wrote too hastily and used the wrong words. Quantum mechanical amplitudes evolve according to a Markov like process.This Markov process describes the world as intrinsically random although i suppose - I don't know - there might be another description of the world where things are merely intrinsically unpredicatble. The Shroedinger equation for a free particle is a complex heat equation so it is no surprise that it describes a random process. A great description of this can be found in Feynmann's third volume of Lectures on Physics.