Re: [ontolog-forum] Axiomatic ontology

[Pat Hayes <phayes@xxxxxxx>]

At 12:08 PM -0600 2/12/08, Schiffel, Jeffrey A wrote:

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> A series of unfair coin tosses can easily be random.
> The distribution of the random variable will not be uniform (as is the
> case of a fair coin), but it may still be random.

It may be a random variable in the mathematical sense, of course.  (http://en.wikipedia.org/wiki/Random_variable)

But this rather begs the question, as this definition already incorporates the notion of probability. For example, a completely unfair dice which always gives '1' is a random variable in this sense.

But OK, let me accede to my own strictures. You are right, any variable which incorporates a degree of chance may be called 'random'; and Sean is right that a sampling of such values may well provide statistical information about likelihoods. Nevertheless, my original point still holds. When we are talking about what is usually called a 'random sequence' of numbers or bits, as spoken of in Kolmogoroff theory, what makes such a sequence have maximal information density is that there is no way to compute any one of its values from all the others.

Pat

> It is just that the probabilities must be described by
> a different distribution. Pink noise is a simple example, since the
> randomness compared to white noise is red-shifted.
>  
> -- Jeffrey Schiffel

---

From: Pat Hayes [mailto:phayes@xxxxxxx]
Sent: Tuesday, February 12, 2008 11:28 AM
To: Barker, Sean (UK)
Cc: [ontolog-forum]
Subject: Re: [ontolog-forum] Axiomatic ontology

At 10:14 AM +0000 2/12/08, Barker, Sean (UK) wrote:

Pat's claim "The definition of a random sequence is that no matter how
much of it you have, there is no way even in principle to compute any
information about the next item." is true only where you exclude
probabilistic estimates (which you might do depending on how you
interpret "information"). For example, if you encode the tosses of a
coin as a bit stream, as you continue to observe the bit stream, you
will be able to make increasing accurate estimates of the probability
that the next bit will be a 1. Given the additional knowledge that this
is the encoding of coin flips, you will also be able to estimate the

probability that it is a fair coin.

No, wait. A series of tosses of an unfair coin is not a random sequence. One gets randomness just when the actual probability of each toss being a head is 0.5 precisely.

What you say above is correct, of course, but it can be translated as: if a series of bits is not random, this can be detected with increasing accuracy as the series gets longer. Also, of course, if it is random, this can also be detected (if it were previously unknown), but that does not mean that any particular toss can be predicted.

Pat

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