RE: probabilities & measures & computable universes
The notion of complex-valued or even quaternionic or octonionic probabilities has been considered; see http://physics.bu.edu/~youssef/quantum/quantum_refs.html for some pointers into the literature. -- Ben Goertzel > -Original Message- > From: scerir [mailto:[EMAIL PROTECTED] > Sent: Friday, January 23, 2004 9:23 AM > To: [EMAIL PROTECTED] > Subject: Re: probabilities & measures & computable universes > > > Are probabilities always and necessarily positive-definite? > > I'm asking this because there is a thread, started by Dirac > and Feynman, saying the only difference between the classical > and quantum cases is that in the former we assume the probabilities > are positive-definite. > > Thus, speaking of MWI, we could also ask: what is the joint > probability of finding ourselves in a universe alpha and of > finding ourselves in a universe beta, which is 180 degrees > out of phase with the first one (whatever that could mean)? > > s. > >
Re: probabilities & measures & computable universes
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.
Re: probabilities & measures & computable universes
Juergen Schmidhuber writes: > What is the probability of an integer being, say, > a square? This question does not make sense without > a prior probability distribution on the integers. > > This prior cannot be uniform. Try to find one! > Under _any_ distribution some integers must be > more likely than others. > > Which prior is good? Is there a `best' or > `universal' prior? Yes, there is. It assigns to > each integer n as much probability as any other > computable prior, save for a constant factor that > does not depend on n. (A computable prior can be > encoded as a program that takes n as input and > outputs n's probability, e.g., a program that > implements Bernoulli's formula, etc.) What is the probability that an integer is even? Suppose we use a distribution where integer n has probability 1/2^n. As is appropriate for a probability distribution, this has the property that it sums to 1 as n goes from 1 to infinity. The even integers would then have probability 1/2^2 + 1/2^4 + 1/2^6 ... which works out to 1/3. So under this distribution, the probability that an integer is even is 1/3, and odd is 2/3. Do you think it would come out differently with a universal distribution? 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. 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? Hal Finney
Re: probabilities & measures & computable universes
Are probabilities always and necessarily positive-definite? I'm asking this because there is a thread, started by Dirac and Feynman, saying the only difference between the classical and quantum cases is that in the former we assume the probabilities are positive-definite. Thus, speaking of MWI, we could also ask: what is the joint probability of finding ourselves in a universe alpha and of finding ourselves in a universe beta, which is 180 degrees out of phase with the first one (whatever that could mean)? s.
