Jim Bromer wrote: > Please give me a little more explanation why you say the fundamental method > is >that the probability of a string x is proportional to the sum of all programs >M >that output x weighted by 2^-|M|. Why is the M in a bracket?
By |M| I mean the length of the program M in bits. Why 2^-|M|? Because each bit means you can have twice as many programs, so they should count half as much. Being uncomputable doesn't make it wrong. The fact that there is no general procedure for finding the shortest program that outputs a string doesn't mean that you can never find it, or that for many cases you can't approximate it. You apply Solomonoff induction all the time. What is the next bit in these sequences? 1. 0101010101010101010101010101010 2. 1100100100001111110110101010001 In sequence 1 there is an obvious pattern with a short description. You can find a short program that outputs 0 and 1 alternately forever, so you predict the next bit will be 1. It might not be the shortest program, but it is enough that "alternate 0 and 1 forever" is shorter than "alternate 0 and 1 15 times followed by 00" that you can confidently predict the first hypothesis is more likely. The second sequence is not so obvious. It looks like random bits. With enough intelligence (or computation) you might discover that the sequence is a binary representation of pi, and therefore the next bit is 0. But the fact that you might not discover the shortest description does not invalidate the principle. It just says that you can't always apply Solomonoff induction and get the number you want. Perhaps http://en.wikipedia.org/wiki/Kolmogorov_complexity will make this clear. -- Matt Mahoney, [email protected] ________________________________ From: Jim Bromer <[email protected]> To: agi <[email protected]> Sent: Thu, July 22, 2010 5:06:12 PM Subject: Re: [agi] Comments On My Skepticism of Solomonoff Induction On Wed, Jul 21, 2010 at 8:47 PM, Matt Mahoney <[email protected]> wrote: The fundamental method is that the probability of a string x is proportional to the sum of all programs M that output x weighted by 2^-|M|. That probability is dominated by the shortest program, but it is equally uncomputable either way. Also, please point me to this mathematical community that you claim rejects Solomonoff induction. Can you find even one paper that refutes it? You give a precise statement of the probability in general terms, but then say that it is uncomputable. Then you ask if there is a paper that refutes it. Well, why would any serious mathematician bother to refute it since you yourself acknowledge that it is uncomputable and therefore unverifiable and therefore not a mathematical theorem that can be proven true or false? It isn't like you claimed that the mathematical statement is verifiable. It is as if you are making a statement and then ducking any responsibility for it by denying that it is even an evaluation. You honestly don't see the irregularity? My point is that the general mathematical community doesn't accept Solomonoff Induction, not that I have a paper that "refutes it," whatever that would mean. Please give me a little more explanation why you say the fundamental method is that the probability of a string x is proportional to the sum of all programs M that output x weighted by 2^-|M|. Why is the M in a bracket? On Wed, Jul 21, 2010 at 8:47 PM, Matt Mahoney <[email protected]> wrote: Jim Bromer wrote: >> The fundamental method of Solmonoff Induction is trans-infinite. > > >The fundamental method is that the probability of a string x is proportional >to >the sum of all programs M that output x weighted by 2^-|M|. That probability >is >dominated by the shortest program, but it is equally uncomputable either way. >How does this approximation invalidate Solomonoff induction? > > >Also, please point me to this mathematical community that you claim rejects >Solomonoff induction. Can you find even one paper that refutes it? > > -- Matt Mahoney, [email protected] > > > > > ________________________________ From: Jim Bromer <[email protected]> >To: agi <[email protected]> >Sent: Wed, July 21, 2010 3:08:13 PM > >Subject: Re: [agi] Comments On My Skepticism of Solomonoff Induction > > > >I should have said, It would be unwise to claim that this method could stand >as >an "ideal" for some valid and feasible application of probability. >Jim Bromer > > >On Wed, Jul 21, 2010 at 2:47 PM, Jim Bromer <[email protected]> wrote: > >The fundamental method of Solmonoff Induction is trans-infinite. Suppose you >iterate through all possible programs, combining different programs as you go. > >Then you have an infinite number of possible programs which have a >trans-infinite number of combinations, because each tier of combinations can >then be recombined to produce a second, third, fourth,... tier of >recombinations. >> >>Anyone who claims that this method is the "ideal" for a method of applied >>probability is unwise. >> Jim Bromer > >agi | Archives | Modify Your Subscription >agi | Archives | Modify Your Subscription agi | Archives | Modify Your Subscription ------------------------------------------- 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
