Statistics

# More On Randomness

Anything could be at the end of this lane!

I took this picture yesterday (using my mom’s phone: mine can’t offload photos) at the start of the cul de sac, at the bottom of which lies the first job I ever had: washing dishes in a nursing home (in which my uncle now resides).

Who could have guessed I would have moved from there to where I am now?

Certainly not one of my old teachers, who was shocked when told by my parents of my career (such as it is). The teacher was cornered in the back of a grocery store. This same teacher would have taken in stride news that I was just coming up for parole.

This highlights one of the shades of meaning of random. It’s when an event was not just unpredictable, but that it happened almost against the evidence. “That was random,” we say.

This usage acknowledges random is a measure of information, which at least removes some of the mysticism the words has in scientific and mathematical contexts. But maybe not all. We sometimes almost have the idea Nature is working against our desires, though Her actions in this regard are weak.

Mysticism? Did you know that in classic statistical formula if some numbers aren’t imbued—nobody knows how—with randomness, the formula won’t work?

Oh, the formulas will still spit out answers, of course, but you won’t be allowed to use those answers. It’s kind of like sitting down to a feast and discovering the witch doctor didn’t give his prior blessing to the animals cooked, and therefore nobody is allowed to eat.

It’s not just frequentists who believe in magic, but most Bayesians, too. Whenever you hear somebody say, “X is randomly distributed normally” (or some other thing), you have heard an incantation. There is nothing in the world that makes a number “randomly normal” (or whatever). It’s only that our understanding might be quantified by a normal (or whatever).

How X knows it’s supposed to be normal (or whatever) is never specified either. It’s here that the Deadly Sin of Reification mixes with the mysticism of randomness. The formulas become realer than reality, and it’s the power of mysticism which does the deed. But again, nobody knows how. It’s a question which is never asked.

When I’m done with this mini-vacation, I’ll set out more specifically all the shades of meaning of random.

Categories: Statistics

### 14 replies »

1. DAV says:

Some random thoughts?

2. Art Kenney says:

So, if all randomness is due to ignorance, do you allow for the distinction between epistemic and aleatory uncertainties?

3. Briggs says:

Art,

I do not.

4. Nate says:

I really appreciate all of your insights on the idea of randomness. This is the kind of thought which is counter-intuitive: nothing happens “randomly” at all, but nearly everything that happens appears to be random. 🙂

5. I completely buy your arguments about “randomness”. What I would love somebody (you?) to elaborate about is what properties do this that we call random numbers have that makes them so usefull, for example when doing MCMC. What properties does a good random number generator have assuming it is not mystical “randomness”?

6. DAV says:

Rasmus,

What properties does a good random number generator have assuming it is not mystical â€œrandomnessâ€?

The short answer: It depends on the application.

In general, you don’t want a generator with a discernible pattern that would modulate the application. However, all pseudo generators have a pattern. This is a good thing for software testing and debugging but care must be taken.

Statistics is a can be viewed as a kind of engineering task. In engineering, “best” is whatever fulfills the requirements of the task at hand. Only you would know those requirements for your task.

There are many places on the web that discuss this.
Maybe start here: http://en.wikipedia.org/wiki/Statistical_randomness
but it’s rather short on links.

7. Gregory Chaitin has some interesting ideas on randomness: “The Unknowable”.
As I understand it (and it is pretty much above my pay grade), a string of numbers will be considered random if the algorithm to produce them is as long as the string.
Does that make sense? Enlighten me please.

8. DAV says:

Bob,

Does Chaitin mention what he means by algorithm length? I can think of several measures.

9. He does, but I’ll have to look it up… later?

10. Nullius in Verba says:

“What I would love somebody (you?) to elaborate about is what properties do this that we call random numbers have that makes them so useful”

Probability theory is a mathematical model of reality used to make predictions regarding the number of times events occur in (partially) defined circumstances. It has the mathematical form of a measure theory (the generalised theory of lengths, areas, and volumes), and all the formulas and theorems about it are derived from it having that form. (For example, Normal distributions arise from being a sum of many independent instances of some other ‘random’ quantity, subject to certain weak conditions. The Normal distribution formulas are just a ‘geometrical’ consequence of being such a sum.)

As with all models, the question is not whether it is true but whether it is useful. And it is. It has been found to make accurate predictions regarding the occurrences we observe in the world around us. Why it works so well is all of a piece with the question of why mathematics generally works so well in describing reality. You could ask the same question about why geometrical lengths and areas follow the rules they do – the same sort of rules as it happens. Why does the universe have the symmetries it does?

There is no way to tell for certain. Physically, we can only obtain information about the universe through observation, and construct models to fit. We judge the ‘truth’ of models by how well they work in making predictions, but generally many distinct models are possible that fit any finite collection of observations equally well, so we must also judge between them on aesthetic grounds such as simplicity and mathematical elegance. We cannot look directly at the machinery of the universe ‘behind the scenery’. We cannot see what happens where nobody is looking; what it actually does to give rise to our observations. Observation is like the wall of Plato’s cave – we cannot turn around to see directly what is casting the shadows.

All we can say is that when we build our own imaginary mathematical model of a universe in this way, and people it with imaginary observers, what we predict the modeled observers will see matches closely what we see in the real universe. Many people find it useful therefore to simply assume this is how our universe truly works, and operate on that basis. People are basically doing the same thing when they assume there is a universe out there at all, and it’s all not just an illusion. Technically, there’s no justification, other than pragmatism. But people would think you was nuts if you didn’t.

Pseudorandom numbers satisfy a set of measure theory axioms too. And because in this case we can look behind the scenes at the machinery that produces them, we can see why, as well. Conforming to measure theory is why the formulas work.

11. DAV says:

Bob,

Thanks. I haven’t the time at the moment to read the whole paper but he seems to be defining length by informational entropy content and likely means a Turing machine.

So we stipulate instead that H(A,B|C,D) is the size of the smallest self-delimiting program that produces A and B when it is given a minimal-size self-delimiting program for C and D

Not sure what the practical application would be in terms of actual algorithms.

The definition I use is pragmatic. If I (or rather my algorithm) can’t detect a pattern or be influenced by the output order then the generator is effectively random.

12. Peter Czerna says:

Thank you for an amusing and thought-provoking piece – best thing I’ve read this morning!

My thought: A number itself cannot be ‘random’ – it’s just a number. ‘Random’ either applies to the way it has come into existence, or, perhaps, to the (nonexistant) relationship to another number or set of numbers.

I’ll go back to drinking coffee now.

13. JH says:

Oooh, that street sign would fit rightly and look magnificent on my office door!

Rasmus BÃ¥Ã¥th,

Just search on Google using the keywords “Monte Carlo Simulation”!