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> From [EMAIL PROTECTED] Sat Mar 3 18:05:53 2001 > From: "Saibal Mitra" <[EMAIL PROTECTED]> > Jürgen wrote: > ----- Original Message ----- > From: <[EMAIL PROTECTED]> > To: <[EMAIL PROTECTED]> > Sent: Thursday, February 22, 2001 5:32 PM > Subject: Re: on formally describable universes and measures > > > > Saibal Mitra wrote: > > > > > I think the source of the problem is equation 1 of Juergen's paper. This > > > equation supposedly gives the probability that I am in a particular > > > universe, but it ignores that multiple copies of me might exist in > > > one universe. Let's consider a simple example. The prior probability > > > of universe i (i>0) is denoted as P(i), and i copies of me exist in > > > universe i. In this case, Juergen computes the propability that if you > > > pick a universe at random, sampled with the prior P, you pick universe > > > i. This probability is, of course, P(i). Therefore Juergen never has > > > to identify how many times I exist in a particular universe, and can > > > ignore what consciousness actually is. Surely an open universe where an > > > infinite number of copies of me exist is infinitely more likely than a > > > closed universe where I don't have any copies, assuming that the priors > > > are of the same order? > > > > To respond, let me repeat the context of eq. 1 [In which universe am I?] > > Let h(y) represent a property of any possibly infinite bitstring y, say, > > h(y)=1 if y represents the history of a universe inhabited by yourself > > and h(y)=0 otherwise. According to the weak anthropic principle, the > > conditional probability of finding yourself in a universe compatible with > > your existence equals 1. But there may be many y's satisfying h(y)=1. > > What is the probability that y=x, where x is a particular universe > > satisfying h(x)=1? According to Bayes, > > P(x=y | h(y)=1) = > > (P(h(y)=1 | x=y) P(x = y)) / (sum_{z:h(z)=1} P(z)) > > propto P(x), > > where P(A | B) denotes the probability of A, given knowledge of B, and > > the denominator is just a normalizing constant. So the probability of > > finding yourself in universe x is essentially determined by P(x), the > > prior probability of x. > > > > Universes without a single copy of yourself are ruled out by the weak > > anthropic principle. But the others indeed suggest the question: what can > > we say about the distribution on the copies within a given universe U > (maybe > > including those living in virtual realities running on various computers > in U)? > > I believe this is the issue you raise - please correct me if I am wrong! > > (Did you really mean to write "i copies in universe i?") > > > I did mean to write i copies in universe i, maybe it would have been better > to write > n(i) copies in universe i. Anyway, according to equation 1 the probability > of universe x > given that n(x) >0 is proportional to P(x), which is also intuitively > logical. My point is > that from the perspective of the observer, of which there are n(x) copies in > universe x, things > look different. Intuitively, it seems that the measure of the observer > should be n(x)* P(x). > E.g. suppose there exist x1 and x2 such that P(x1) = P(x2) and n(x1) > > n(x2) > 0. > It seems to me that the observer is more likely to find himself in universe > x1 compared to > universe x2. >From an algorithmic TOE perspective the only important thing is that the measure is computable in the limit - a bit more below. > > Intuitively, some copies might be more likely than others. But what > > exactly does that mean? If the copies were identical in the sense no > > outsider could distinguish them, then the concept of multiple copies > > wouldn't make sense - there simply would not be any multiple copies. So > > there must be detectable differences between copies, such as those > > embodied by their different environments. > > > > So my answer would be: as soon as you have a method for identifying and > > separating various observer copies within a universe U, each > distinguishable > > copy_i is different in the sense that it lives in a different universe > > U_i, just like you and me can be viewed as living in different universes > > because your inputs from the environment are not identical to mine. > > > > In general, the pair (U_i, copy_i) conveys more information than U by > > itself (information is needed to separate them). The appropriate domain > > of universes x (to use the paper's notation) would be the set of all > possible > > pairs of the form (separate universe, separate observer). > > > > Equation 1 above is perfectly applicable to this domain. > > > Okay, but since I don't know which of the copies I am, the probability > that I am one of the copies inside universe i is given as: > Sum_{i = 1}^{n(U)} P(U_i) > > Is this proportional to P(U) or is it > proportional to n(U) P(U) ? > > Saibal To say something about some observer's future, given the past, we need a probability distribution on the possible futures. Does it really make sense to speak of several different copies of some observer within a given universe? Not really, because the observers in general will have different pasts or futures - otherwise they won't be different. To distinguish observers we have to look at their entire histories (as opposed to just their current states). Apart from this, algorithmic TOEs correspond to distributions on histories and futures that are computable in the limit. If your n(U) and P(U) are computable then n(U)P(U) is computable, too, and the basic results hold as well, e.g., any complex future without short description is unlikely. Juergen