Jim, Fair enough.
Oh, and Matt: kudos for being better at patiently explaining details than me. --Abram On Mon, Jul 26, 2010 at 4:11 PM, Jim Bromer <jimbro...@gmail.com> wrote: > Abram, > We all have some misconceptions about this, and about related issues. Let > me think about this more carefully and I will answer during the next week - > or two, (I have already reached my quota of egregious errors in this > thread). > Jim > > On Mon, Jul 26, 2010 at 2:54 PM, Abram Demski <abramdem...@gmail.com>wrote: > >> Jim, >> >> I'll argue that solomonoff probabilities are in fact like Pi, that is, >> computable in the limit. >> >> I still do not understand why you think these combinations are necessary. >> It is not necessary to make some sort of ordering of the sum to get it to >> converge: ordering only matters for infinite sums which include negative >> numbers. (Perhaps that's where you're getting the idea?) >> >> Here's my proof, rewritten from an earlier post, using the properties of >> infinite sums of non-negative numbers. >> >> (preliminaries) >> >> Define the computation as follows: we start with a string S which we want >> to know the Solomonoff probability of. We are given a time-limit T. We start >> with P=0, where P is a real number with precision 2*log_4(T) or more. We use >> some binary encoding for programs which (unlike normal programming >> languages) does not contain syntactically invalid programs, but still will >> (of course) contain infinite loops and so on. We run program "0" and "1" for >> T/4 each, "00", "01", "10" and "11" for T/16 each, and in general run each >> program of length N for floor[T/(4^N)] until T/(4^N) is less than 1. Each >> time we run a program and the result is S, we add 1/(4^N) to P. >> >> (assertion) >> >> P converges to some value as T is increased. >> >> (proof) >> >> If every single program were to output S, then T would converge to 1/4 + >> 1/4 + 1/16 + 1/16 + 1/16 + 1/16 + ... that is, 2*(1/(4^1)) + 4*(1/(4^2)) + >> 8*(1/(4^3)) + ... which comes to 1/2 + 1/4 + 1/8 + 1/16 + .. i.e. 1/(2^1) + >> 1/(2^2) + 1/(2^3) + ... ; it is well-known that this sequence converges to >> 1. Thus 1 is an upper bound for P: it could only get that high if every >> single program were to output S. >> >> 0 is a lower bound, since we start there and never subtract anything. In >> fact, every time we add more to P, we have a new lower bound: P will never >> go below a number once it reaches it. The sum can only increase. Infinite >> sums with this property must either converge to a finite number, or go to >> infinity. However, we already know that 1 is an upper bound for P; so, it >> cannot go to infinity. Hence, it must converge. >> >> --Abram >> >> On Mon, Jul 26, 2010 at 9:14 AM, Jim Bromer <jimbro...@gmail.com>wrote: >> >>> As far as I can tell right now, my theories that Solomonoff Induction >>> is trans-infinite were wrong. Now that I realize that the mathematics do >>> not support these conjectures, I have to acknowledge that I would not be >>> able to prove or even offer a sketch of a proof of my theories. Although I >>> did not use rigourous mathematics while I have tried to make an assessment >>> of the Solomonoff method, the first principle of rigourous mathematics is to >>> acknowledge that the mathematics does not support your supposition when they >>> don't. >>> >>> Solomonoff Induction isn't a mathematical theory because the desired >>> results are not computable. As I mentioned before, there are a great many >>> functions that are not computable but which are useful and important because >>> they tend toward a limit which can be seen in with a reasonable number of >>> calculations using the methods available. Pi is one such function. (I am >>> presuming that pi would require an infinite expansion which seems right.) >>> >>> I have explained, and I think it is a correct explanation, that there is >>> no way that you could make an apriori computation of all possible >>> combinations taken from infinite values. So you could not even come up with >>> a theoretical construct that could take account of that level of >>> complexity. It is true that you could come up with a theoretical >>> computational method that could take account of any finite number of values, >>> and that is what we are talking about when we talk about the infinite, but >>> in this context it only points to a theoretical paradox. Your theoretical >>> solution could not take the final step of computing a probability for a >>> string until it had run through the infinite combinations and this is >>> impossible. The same problem does not occur for pi because the function >>> that produces it tends toward a limit. >>> >>> The reason I thought Solomonoff Induction was trans-infinite was because >>> I thought it used the Bayesian notion to compute the probability using all >>> possible programs that produced a particular substring following a given >>> prefix. Now I understand that the desired function is the computation of >>> only the probability of a particular string (not all possible substrings >>> that are identical to the string) following a given prefix. I want to study >>> the method a little further during the next few weeks, but I just wanted to >>> make it clear that, as far as now understand the program, that I do not >>> think that it is trans-infinite. >>> >>> Jim Bromer >>> *agi* | Archives <https://www.listbox.com/member/archive/303/=now> >>> <https://www.listbox.com/member/archive/rss/303/> | >>> Modify<https://www.listbox.com/member/?&;>Your Subscription >>> <http://www.listbox.com/> >>> >> >> >> >> -- >> Abram Demski >> http://lo-tho.blogspot.com/ >> http://groups.google.com/group/one-logic >> *agi* | Archives <https://www.listbox.com/member/archive/303/=now> >> <https://www.listbox.com/member/archive/rss/303/> | >> Modify<https://www.listbox.com/member/?&;>Your Subscription >> <http://www.listbox.com/> >> > > *agi* | Archives <https://www.listbox.com/member/archive/303/=now> > <https://www.listbox.com/member/archive/rss/303/> | > Modify<https://www.listbox.com/member/?&;>Your Subscription > <http://www.listbox.com> > -- Abram Demski http://lo-tho.blogspot.com/ http://groups.google.com/group/one-logic ------------------------------------------- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: https://www.listbox.com/member/?member_id=8660244&id_secret=8660244-6e7fb59c Powered by Listbox: http://www.listbox.com
