[extropy-chat] Coin Flip Paradox

[gts]

On Sun, 28 Jan 2007 23:59:59 -0500, The Avantguardian  
<avantguardian2020 at yahoo.com> wrote:

> The *actual* frequency of any real random
> sequence would be more accurately described to
> chaotically orbit the probability, like a strange
> attractor, rather than approach it as any kind of
> deterministic limit in a classical calculus sense.

I totally disagree, and wonder where you came up with the unusual idea  
that frequencies "chaotically orbit the probability like a strange  
attractor". Do you have mathematical or empirical evidence to support that  
claim?

Frequentists have plenty of evidence, both empirical and mathematical, to  
support their much more boring claim that frequencies converge in an  
ordinary way as n increases.

But let's talk a bit about the meaning of randomness.

I surmise that you see an ambiguity in the conventional view of randomness  
that I also see, but that you are expressing your displeasure about it in  
ways that make no sense to me.

As I mentioned and you agreed, randomness and entropy are closely related  
ideas, but the ideas should (perhaps) be kept apart.

Rafal objected, for example, when I wrote that a sequence of flips of a  
heavily weighted coin is still a completely random sequence. It seems his  
intuition was telling him that a weighted coin should produce a sequence  
less random than a fair coin.

I think Rafal really meant that such a heavily weighted sequence has lower  
*entropy*, not lower *randomness*. I think people are sometimes confused  
about the two terms because of their close meanings.

As probability theorists normally use the word (at least in my experience)  
randomness is mainly about the independence (or exchangeability) of  
individual trials/observations, not about the measure of disorder in the  
sequence of trials/observations.

The situation is made more cloudy (or perhaps more clear, depending on  
your perspective) by algorithmic definitions of randomness.

Consider a binary sequence generated by an idealized perfectly random fair  
coin, where Heads=1 and Tails=0. What if this unlikely sequence came up?

11111111111111111111

20 heads in a row! Is this freaky sequence still random? It certainly  
doesn't *look* random, but how could it not still *be* random? After all  
we stipulated in advance that it was generated by an idealized perfectly  
random coin-flip process.

Well, according to the algorithmic definition of randomness, randomness is  
a property of the *sequence*, not a property of the *process*. So this  
sequence of 20 heads is extremely un-random by that definition even though  
it was obtained via a purely random process. This is a sort of marriage of  
entropy to randomness, for better or worse.

-gts