B What's the meaning of "random" in Mathematics?

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Ray Vickson

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Yay! Gotta love Cox & Jaynes!! Hooray to them! I somehow suspect that Kolmogorov and Cox/Jaynes are equivalent, as (I suspect) they come to the same conclusions through (I suspect) different axiomatic processes, but I did find Cox infinitely easier to grasp.
I think that Cox/Jaynes is "equivalent" to probability as done in volume I of Feller---and that is saying a lot. However, some "deeper" modern results (seem to) need the "measure-theoretic" apparatus, so would essentially be rejected by Jaynes. Certainly, the two approaches would no longer be equivalent in any easily-described sense.

Admittedly, some treatments in the modern way of doing things look like they might just be re-statements of results done in the old-fashioned way, but the resulting statements of the results are more cumbersome in the old way. For example, Feller, Vol. I, proves the so-called Strong Law of Large Numbers without using any measure theory or other more abstract methods. However, the result looks less appealing than the modern statement. The modern statement would amount to ##P(\lim_{n \to \infty} \bar{X}_n = \mu) = 1.## In pre-measure language the same result would say "For every ##\epsilon > 0## with probability 1 there occur only finitely many of the events ##|\bar{X}_n - \mu| > \epsilon##" How much nicer is the first way of saying it compared with the second way.
 

StoneTemplePython

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I think that Cox/Jaynes is "equivalent" to probability as done in volume I of Feller---and that is saying a lot.
I've long suspected something like this.. though I've only read part of Jaynes-- I couldn't shake the feeling while reading him that he was repackaging old ideas as new ones while using Feller as some kind of strawman to attack. There are some good ideas in Jaynes but I am leery of polemics these days. You're probably the one person on PF who has cited Feller even more than I have, so this seems satisfying.
 

Demystifier

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I don't think that anybody here came close to the crux of the problem, so let me try to direct the discussion into a different direction.

Are the digits of ##\pi## random?

Intuitively they are not because there is a deterministic algorithm that determines them, and yet all general tests of randomness suggest that they are random. It is this kind of problem that still seems to lack a satisfactory mathematical and/or philosophical solution.
 

Stephen Tashi

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Oh... randomness is interpretation...
The intuitive idea behind probability is that an event can have "tendencies" to occur in different possible ways, but only occurs in one of those ways. Yes, this intuitive idea is NOT implemented in the mathematical axioms of probability theory. So when people apply mathematical probability theory to situations and reason about various outcomes being "possible", but only one outcome being "actual", they are making an interpretation of probability theory that is not present in its mathematical formulation.

So a "random" variable is really just another variable, just like "time" is just a variable without anything different than say a "mass" variable.
No. There is a saying: "A random variable is not random and it is not a variable".

As mentioned above, the mathematical assumptions use in defining a "random variable" do not treat the concept of "random" in the intuitive and common language meaning of the word "random".

A "random variable" is not a variable in the same sense that a symbol representing time or mass is a variable nor is it a "variable" in the sense used in mathematical logic or in computer languages. The mathematical properties that define a random variable are stated in terms of functions called distributions. Of course, the definition of a function may contain variables (e.g. f(x) = 4x ). But the same function can be defined using different symbols. (e.g. f(x) = 4x and f(w) = 4w are definitions of the same function).

The inutuive idea of a "random variable" is that it is defined by a distribution function that can be use to compute "the fraction of times that particular sets of values of the random variable will occur". The logical problem with that interpretation is that "will occur" is a definite guarantee of something happening. Such a definite guarantee is a contradiction to the intuitive notion of "probability", which is a concept we apply when there are no definite guarantees. In mathematical probability theory, the distribution function can be used to calculate the probability of particular sets of values - without saying what physical interpretation we assign to "probability". (i.e. There is no mathematical definition of "probability" in terms of a "tendency" or "fraction of times" for an "actual" occurence).

Mathematical probability theory is essentially circular. Given certain distributions , it tells how to compute other distributions. Given various probabilities, we can compute other probabilities. There is no breaking down of the concept of "probability" into more detailed concepts.

( It's amusing and understandable that the terminology used in mathematical statistics (e.g. "significance", "confidence", "uncertainty") strongly suggests that mathematical "probability" must have a specific interpretation along the lines of a "tendency" or "fraction of times". Applications of statistics were made before the invention of modern mathematical probability theory, so it developed such terminology on the basis of applications before the foundations of the subject were properly organized. )

I don't think that anybody here came close to the crux of the problem,
That depends on how you define "the problem".

If the problem is to state the content of mainstream mathematical probability theory (i.e. measure theory) , various posters have done this.

If the problem is to find an alternative mathematical theory that implements the intuitive notion of "randomness", then your question hints that such an approach can be founded on notions of computational complexity.
 

FactChecker

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@Demystifier
The original question was how a mathematician deals with "random". That is not as deep a question because a mathematician only needs to know that random is being assumed -- not that the assumption is physically correct. So the mathematician only has to know if we are assuming that the digits of ##\pi## are random or not.
I think that your question is different since it is asking if we should accept an assumption that the digits of ##\pi## are random. Suppose there is no statistical test that proves it is not random beyond a reasonable doubt. I would only consider it to be pseudo-random, just like any other algorithm for a pseudo-random number generator. A person as great as Einstein was able to retain the belief that there was no true randomness in the universe till his disagreement with Bohr became famous. When people as brilliant as they are disagree, I will stay out of the debate.
 
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I don't think that anybody here came close to the crux of the problem, so let me try to direct the discussion into a different direction.

Are the digits of ##\pi## random?

Intuitively they are not because there is a deterministic algorithm that determines them, and yet all general tests of randomness suggest that they are random. It is this kind of problem that still seems to lack a satisfactory mathematical and/or philosophical solution.

I'm not that surprised with that... these "randomness" tests also pass pseudo-random numbers, even if they are fully deterministic... I'd suspect that these "randomness" tests are incomplete, and pass as random sequences that are deterministic.
 

FactChecker

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I'm not that surprised with that... these "randomness" tests also pass pseudo-random numbers, even if they are fully deterministic... I'd suspect that these "randomness" tests are incomplete, and pass as random sequences that are deterministic.
I would like to point out that any truely random sequence can be turned into a deterministic sequence just by recording the random numbers and replaying them as deterministic. Therefore, it is not possible to have a statistical test that would distinguish between the two. There are large tables of "random" numbers available and used.
 

FactChecker

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I would like to point out that any truely random sequence can be turned into a deterministic sequence just by recording the random numbers and replaying them as deterministic. Therefore, it is not possible to have a statistical test that would distinguish between the two. There are large tables of "random" numbers available and used.
I suppose that one could change the question to asking if a process generates random numbers. In that case, repeating the process of replaying the table of numbers would produce the exact same sequence of numbers and be easily identified as deterministic.
 

jbriggs444

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I'm not that surprised with that... these "randomness" tests also pass pseudo-random numbers, even if they are fully deterministic... I'd suspect that these "randomness" tests are incomplete, and pass as random sequences that are deterministic.
One can go down the road of Kolmogorov randomness. I am no expert on it.

Roughly speaking, if you take this approach, the information content of a string is the length of the shortest computer program that can produce the string as output. In the context of infinite strings (such as pi) one has to get a little fancier and talk about a program that can generate the output stream from some input stream. If no program can produce output bytes at a better than 1 to 1 ratio to input bytes then the output stream is "random".

The gotcha with this approach is that "finding the best program" is not a feasible problem in general. Kolmorogorov randomness can be defined but it can not always be determined.

Edit: If you have an output stream produced by a loaded die then the Kolmogorov definition will (with probability 1) match the Shannon notion of information content in the stream.
 
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FactChecker

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I suppose that one could change the question to asking if a process generates random numbers. In that case, repeating the process of replaying the table of numbers would produce the exact same sequence of numbers and be easily identified as deterministic.
We could say that a process generates sequences of random numbers if there are no statistical tests on repetitions of the process which would identify it as deterministic beyond a reasonable doubt. Then we could call any output sequence of the process as random. That opens several cans of worms, one of which is that "no statistical tests" is poorly defined and it is concievable that there will always be a statistical test that a given process will fail.
 

Buzz Bloom

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I'm not that surprised with that... these "randomness" tests also pass pseudo-random numbers, even if they are fully deterministic... I'd suspect that these "randomness" tests are incomplete, and pass as random sequences that are deterministic.
Hi fbs:

A randomness test gives a value that estimates the degree of close approximation a particular pseudo-random number sequence (e.g., the digits in pi) has to true random numbers. It is not intended to be complete.

Regards,
Buzz
 
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Buzz Bloom

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Appreciate it. I was stuck with the matter that logic is completely deterministic.
Hi fbs:

A deterministic process is by definition non-random. It is possible to voice logical statements about randomness, but such statements are not processes for generating random values. What they might be are statements that describe requirements about a process which are necessary for the process to create random values. For example, it is logical to say with respect to QM that measurements of the spin of a sequence of non-entangled particles produces a sequence of random values each of which is either "up" or "down".

Regards,
Buzz
 

WWGD

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I don't know if this is too basic, but a variable is random if its outcome cannot be predicted with certainty. At best we can describe the distribution of the outcomes.
 

FactChecker

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I don't know if this is too basic, but a variable is random if its outcome cannot be predicted with certainty. At best we can describe the distribution of the outcomes.
That certainly has advantages. For one thing, it is simple and as usable as any other definition. For another thing, it allows one to discuss the aspect that lack of available information plays in the inability to predict. One can reasonably think of randomness and probabilities in terms of the ability to guess and predict, given the information available. So the question of whether a result is really physically random or not is no longer appropriate -- it becomes a question of whether enough information is known to make the prediction. The second question is easier to agree on.
 

WWGD

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That certainly has advantages. For one thing, it is simple and as usable as any other definition. For another thing, it allows one to discuss the aspect that lack of available information plays in the inability to predict. One can reasonably think of randomness and probabilities in terms of the ability to guess and predict, given the information available. So the question of whether a result is really physically random or not is no longer appropriate -- it becomes a question of whether enough information is known to make the prediction. The second question is easier to agree on.
Well, true, I am doing a good amount of assuming/black boxing. We may need to conduct tests on whether the variable ( and not a single output) is random. But at least these are some guide posts/ goalposts, and, yes, pretty tricky: what information , if any, would be needed to do a better approximation or estimation of the output? We may also have trouble with (somewhat_- pathological cases like Cauchy variables with infinite variance.

I can, tho, think of genuinely random variables when, e.g., using a pendulum oscillating between ( more than 1) magnets and seeing where it settles.
 
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Wow, many exquisite insights!!

Let me ask you mathematicians this question: a pseudo-random sequence will repeat, even if after an absurd amount of time. But the digits of π do not repeat.

We know that neither of them are truly random, given that number n-th can be exactly calculated, although both cases "seem" to be random up to some measure.

But, given the fact that the digits of π do not repeat, does that make that sequence a bit more "random-ny" than a pseudo-random generator, if there's such thing? I fear I'm butchering proper mathematic-inglish again! :biggrin:
 

FactChecker

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Wow, many exquisite insights!!

Let me ask you mathematicians this question: a pseudo-random sequence will repeat, even if after an absurd amount of time.
That is a property of some specific methods of generating pseudo-random numbers. It is not a consequence of "pseudo-random" alone. Evan very good methods may fail sophisticated statistical tests, but they are adequate for most uses.
 

Buzz Bloom

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But, given the fact that the digits of π do not repeat, does that make that sequence a bit more "random-ny" than a pseudo-random generator, if there's such thing? I fear I'm butchering proper mathematic-inglish again!
Hi fbs:

I am guessing that your question is whether it is possible for a randomness test method to determine if one pseudo-random number generator is better than another. I am also guessing that the answer is yes, and I am pretty confident this second guess is correct.

You should keep in mind that pseudo-random number generators are used with a specific purpose. It is desired that the pseudo random number generator used will give a result adequately close to what would be obtained using a true random number generator, e.g., something based on QM. If you can know theoretically what statistical result to expect form a sequence of random numbers used for a specific purpose, e.g., Monte-Carlo calculations, then the results of performing several Monte-Carlo runs with different pseudo-random number generators can be compared to determine which is the best for that particular purpose.

Regards,
Buzz
 

andrewkirk

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Wow, many exquisite insights!!

Let me ask you mathematicians this question: a pseudo-random sequence will repeat, even if after an absurd amount of time. But the digits of π do not repeat.

We know that neither of them are truly random, given that number n-th can be exactly calculated, although both cases "seem" to be random up to some measure.

But, given the fact that the digits of π do not repeat, does that make that sequence a bit more "random-ny" than a pseudo-random generator, if there's such thing? I fear I'm butchering proper mathematic-inglish again! :biggrin:
A pseudo-random sequence of numbers is just a sequence that has no readily apparent pattern, but is generated by a finite algorithm that takes no external inputs after the generation of the first M values (M is a fixed positive integer). Since the digits of pi are generated by an algorithm and its calculation takes no external inputs, they are a pseudo-random sequence.

So far as I am aware, it is not the case that a pseudo-random sequence must eventually repeat. It is just that non-repeating sequences become too slow to compute, the longer they continue, so the ones used by computers tend to be repeating for efficiency reasons. One way that I think will make a sequence non-repeating is as follows:

First, a definition: we say that a sequence of numbers, which may be finite or infinite, is 'repeating with lag L' if, for any j that is an index of the sequence such that j+L is also an index, the j-th and the (j+L)th element are equal.

The algorithm will use a simple, repeating, pseudo-random generator as its base. It generates positive integers.

Say we have already generated the first n numbers. Then for every k from 2 to (n+1), let m(n,k) be the quotient from dividing (n+1) by k and let r(n,k) be the quotient on dividing k by 2. For each k, there will be zero or one numbers such that, if that number is chosen, the last r(n,k) . m(n,k) elements out of the first (n+1) sequence elements, is repeating with lag m(n,k). Such a number is a 'forbidden number', as it creates a temporary repeat for a significant tail of the sequence thus far. Let F(n) be the set of all forbidden numbers at the step for generating the (n+1)th number.

Then generate a new number h(n+1) from the simple generator. Now chose as the (n+1)th number the closest positive integer to h(n+1) that is not in F(n).

I believe such a sequence will never repeat. But it will become progressively slower, because the number of checks that need to be made for each number increases as the sequence progresses.

But the sequence is pseudo-random because it satisfies the definition.
 
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Hmm... true point!! I'm a programmer, so I should have seen that too, appreciate you pointing out.

Another example of a generator that will (eventually, and very very very slowly) not repeat is this one, for say 8-bit integers (or whatever size integers we choose):

Code:
initialize a[i] = 0 for i in 1..N
repeat
    calculate randomness of a[]
    for i = 1..N, for j = 1 to 8:
         invert bit j of a[i]
         recalculate randomness of a[]
         if new sequence has lower randomness, revert the bit back
   until randomness of a[] > desired-target
that algorithm (a brute-force search, actually) will eventually generate a non-repeating, whatever-length sequence that will pass whatever randomness criteria we set... at a cost of taking ages to process... hmm... actually this is probably the slowest random number generator ever... I should get a prize or something for that :biggrin:

hmm... actually I'm a bad programmer.. I should check for the thing hitting a hole, where any one-bit changes does not increase randomness; if one checks for valleys and holes and backtracks, the thing gets even slower...
 

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