How does knowledge affect randomness?

[loourr]

Suppose I have a true random number generator. It generates a string of 10 numbers. I then look at those numbers. Are they still random? If no then why, and would they be random for a bystander who did not see the numbers? If yes then consider this.

I am an omniscient being. This same random number generator generates another 10 numbers. I knew what they were going to be. Are they still random? If no, what has changed? Would they still be random for someone who did not know what they were going to be?

What If I created the random number generator with the knowledge of the numbers it was going to produce?

At what point does knowledge affect randomness? Are omniscience and randomness mutually exclusive? Is randomness relative? I'm very curious of what you think about this and what you think mathematics and science (particularly quantum mechanics) has to say about this.

 

[chiro]

Suppose I have a true random number generator. It generates a string of 10 numbers. I then look at those numbers. Are they still random? If no then why, and would they be random for a bystander who did not see the numbers? If yes then consider this.

I am an omniscient being. This same random number generator generates another 10 numbers. I knew what they were going to be. Are they still random? If no, what has changed? Would they still be random for someone who did not know what they were going to be?

What If I created the random number generator with the knowledge of the numbers it was going to produce?

At what point does knowledge affect randomness? Are omniscience and randomness mutually exclusive? Is randomness relative? I'm very curious of what you think about this and what you think mathematics and science (particularly quantum mechanics) has to say about this.

Hey loourr and welcome to the forums.

There are a variety of process that generate random looking numbers yet they are completely deterministic: they are called pseudo-random number generators.

These are used everywhere from statistical analysis, to gambling machines (think the slot machines), even in computer games!

There is the idea floating amongst some scientists (and mathematicians) that randomness found in nature (like the kind you see in the description of quantum mechanics), could in fact be "pseudo-random" and not "random". The idea is that the complexity of the output hides the fact of getting any kind of trivial pattern.

Personally I agree with these scientists views that the patterns of this process are more or less hidden in a world of great complexity.

One thing you have to remember is that for the most part human beings can not keep a lot of variables in their heads at once. In university most people have trouble with more than say 3 variables (some less!), and even the most complex models will most likely not go above a few hundred.

Now consider if we analyzed billions and billions of variables. To most people, even those with a lot of advanced mathematical training, this is just unfathomable to comprehend clearly.

Also you have to consider how primitive our mathematics is. Our elementary functions are not really that complex, and we don't have the analytical techniques that allows us to decompose systems with extremely complex patterns.

In terms of knowledge affecting randomness, this is one of the important aspects in Bayesian statistics. In this approach we can represent known information as a probability distribution and combine it with the standard likelihood.

So in terms of what you are asking, these random processes that are deterministic do exist (to be fair, they are "pseudo-random" and not "random"), and it might help if you look up that term to get a better idea of what I am talking about.

 

[pwsnafu]

If you look at Wikipedia's article on http://en.wikipedia.org/wiki/Random_sequence" , specifically the "Modern Approaches" section,

Kolmogrov's definition of a random string was that it is random if has no description shorter than itself via a universal Turing machine. (sic)

If you use that definition, then the existence of an omniscient being means nothing: doesn't change the fact that a UTM shorter than the string does not exist.

 

[vladb]

At what point does knowledge affect randomness? Are omniscience and randomness mutually exclusive? Is randomness relative? I'm very curious of what you think about this and what you think mathematics and science (particularly quantum mechanics) has to say about this.

I might be wrong about this, but I believe mathematics, science (and especially QM...) have nothing to say about this, because it is a philosophical question.

I believe it is De Finetti's book (Theory of Probability) that starts with "probability does not exist". Maybe you will find some philosophical ideas of interest to you there.

I am no philosopher, but I try to be open-minded, so I would ask (with actual interest, not with refutation goal) you: why ask such question? Is "omniscience" at all well defined? How do you define "knowledge" and "randomness"? Suppose someone answered all of the questions you asked - but what would it mean to have these answers? Is it not simply something depending on how you define things and not actually related to the 'real world'? It there really a correct (in what sense?) answer to this?

 

[loourr]

why ask such question? Is "omniscience" at all well defined? How do you define "knowledge" and "randomness"? Suppose someone answered all of the questions you asked - but what would it mean to have these answers? Is it not simply something depending on how you define things and not actually related to the 'real world'? It there really a correct (in what sense?) answer to this?

Yes, I do think this is largely a philosophical question, and yes it does depend on how things are defined. However by asking these questions about mathematically defined terms I was hoping to gain a deeper understanding of the implications of such definitions. Omniscience is poorly defined here but what if it was a computer programmer who knew the seed to his program? So no there is no correct answer, but by choosing a definition of randomness and thinking about it in terms of this thought experiment different conclusions will be reached based on ones personal understanding of randomness.

 

[Stephen Tashi]

The use of conditional probabilities shows that information can change probability. In this thread, what distinction is there between probability and "randomness"?

In the last post of the thread https://www.physicsforums.com/showthread.php?t=524198, I speculated that information can turn a known probability into an unknown one. I expected some philosopher to object. Perhaps philosophers don't read threads that get into specifics.

 

[lavinia]

Fitting patterns to the past does not tell you whether the process that generates the pattern is random or not.There is a difference between describing a pattern in a fixed known sequence and being able to predict the pattern given all knowledge available including the previous history of the series. A process is random if current knowledge gives you no help in predicting the next outcome. For instance in a random walk, the history may in fact show a pattern but there is no way to use this pattern to predict the next step. Generally, processes like these are called Martingales and they are good equilibrium models for many real world processes. the key idea of a Martingale is that randomness is relative to the available knowledge. If for some observers their knowledge does not tell them anything about the future then the process is random for them. If there were an omniscient being who actually determined the future then his knowledge would make the process not random.

 

[loourr]

If there were an omniscient being who actually determined the future then his knowledge would make the process not random.

would they be random for a third party?

 

[lavinia]

would they be random for a third party?

right. I meant that it would not be random for him.

 

[vladb]

A process is random if current knowledge gives you no help in predicting the next outcome.

Since this is a mathematics forum:
A process can still be random even if current/past knowledge does give some help in predicting the outcome (think of markov chains of order > 1). Actually, if the initial state is random, but future changes are fully deterministic given most current state, the process is still random.
What you said sounds more like '(stationary) independent increments'.

For instance in a random walk, the history may in fact show a pattern but there is no way to use this pattern to predict the next step. Generally, processes like these are called Martingales ...

This is not true. Martingales have something to do with conditional expectation of the value of the process, it is not about predictability of the next step. In fact, the conditional distribution may very well depend on past patterns, but as long as the conditional expectation of next incerement is zero, it is a martingale.

... the key idea of a Martingale is that randomness is relative to the available knowledge. ...

Where does this claim come from? Any reference?


Please don't get me wrong. I am interested in a discussion, but (in my opinion) we have to keep the definitions well fixed. Otherwise it's discussing apples and oranges using the vague term 'fruits'.

 

[Phrak]

Can anyone give a good argument that a random sequence is a mathemtical object rather than the result of a physical process?

 

[vladb]

Can anyone give a good argument that a random sequence is a mathemtical object rather than the result of a physical process?

For me a random sequence is clearly a mathematical object. It is a concept. We define a probability space, and also a mapping that maps values of my abstract sample space to sequences. Given a probability measure, I can work with this sequency (mapping) according to certain rules, proving different results. To conveniently communicate my ideas to other people, I use the short term 'random process' instead of describing the object I'm dealing with each times starting from the axioms.

To think of a random process as a result of physical process is difficult. Do you define it to be random because *you* can't predict it exactly? What if you're just missing some parts in your description of the phenomenon in question?

Of course in practice we often refer to physical processes as being random. But what we (in my opinion) really mean by this is that given a chosen description (laws of physics) of the process, this process is well and usefully (but perhaps still approximately) described using our mathematical object called "random process".

Edit: I noticed I used "random process" and "random sequence" somewhat interchangeably, But I hope my thought is still clear despite of this inaccuracy.

 

[Stephen Tashi]

But what we (in my opinion) really mean by this is that given a chosen description (laws of physics) of the process, this process is well and usefully (but perhaps still approximately) described using our mathematical object called "random process".

However, one can also say that "mass" and "energy" are merely useful descriptions of physical properties, so using "probability" in this manner doesn't mean it is less physically real than mass or energy. Those quantities can change in value depending on the state of an observer. So the fact that probabilities can be differ depending on the observer doesn't mean that probability is less real than mass or energy. What makes probability a more nebulous idea than mass or energy is that is it not a unified concept. We can talk of the probability of a change in momentum, of a change in color, of a coin landing heads. By contrast, mass and energy are associated to particular physical systems, not to things like drawing an ace from a deck of cards.