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entropy1
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Is there a definition of "random(ness)"? Is it defined?
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.As I understand , a Random process is one that cannot be predicted but can be described probabilistically.
Are you talking about the population average or a sample average?If you can't predict the next outcome, how come you can predict the average outcome?
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.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.
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.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 seems to me comparable to predicting the outcome after observing it, which would not do justice to the notion of prediction.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'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.Would an intrinsically random process necessarily have correlation 0 with any other process?
Wonder how this would pan out Mathematically and Physically.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.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 wrote an essay about this a few years back, which you may find interesting:A good question is whether there are phenomena that are somehow intrinsically unpredictable, i.e., not predictable within any system.
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.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 any one 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.
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.
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.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.
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.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.
If you can't predict the next outcome, how come you can predict the average outcome?
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.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.