Re: probabilities & measures & computable universes

From: Wei Dai <weidai.domain.name.hidden>
Date: Sat, 24 Jan 2004 04:49:41 -0500

On Fri, Jan 23, 2004 at 09:04:20PM -0800, Hal Finney wrote:
> Do you think it would come out differently with a universal distribution?

There are an infinite number of universal distributions. Some of them
assign greater probability to even integers, some of them assign greater
probability to odd integers.

For those who think that a theory of everything should specify a unique
prior over universes, observers, or observer-moments, I think this
multiplicity of universal distributions is a big problem. My view is that
a unique prior is not necessary. Instead the prior can be thought of as a
representation of how much one cares about each universe, observer, or
observer-moment, and therefore is a purely subjective preference.

> The more conventional interpretation would use the probability computed
> over all numbers less than n, and take the limit as n approaches infinity.
> This would say that the probability of being even is 1/2. I think this
> is how such results are derived as the one mentioned earlier by Bruno,
> that the probability of two random integers being coprime is 6/pi^2.

These kinds of results are useful when you have a uniform distribution
over all integers less than n, with n large. Then you can use these
results to approximate probabilities under the actual distribution.
I don't think you can use these results to say that somehow the
*real* probability of being even is 1/2.

> I'd imagine that this result would not hold using a universal
> distribution. Are these mathematical results fundamentally misguided,
> or is this an example where the UD is not the best tool for the job?

I'm not sure what you mean here.
Received on Sat Jan 24 2004 - 04:50:46 PST