RE: [agi] Solomonoff Induction is Not Universal and Probability is not Prediction
Note: Theorem 1.7.1 There eRectively exists a universal computer. If you copy and paste this declaration the ff gets replaced with a circle cap R :) Not sure how this shows up... John From: Ben Goertzel [mailto:b...@goertzel.org] Sent: Friday, July 09, 2010 8:50 AM To: agi Subject: Re: [agi] Solomonoff Induction is Not Universal and Probability is not Prediction To make this discussion more concrete, please look at http://www.vetta.org/documents/disSol.pdf Section 2.5 gives a simple version of the proof that Solomonoff induction is a powerful learning algorithm in principle, and Section 2.6 explains why it is not practically useful. What part of that paper do you think is wrong? thx ben On Fri, Jul 9, 2010 at 9:54 AM, Jim Bromer jimbro...@gmail.com wrote: On Fri, Jul 9, 2010 at 7:56 AM, Ben Goertzel b...@goertzel.org wrote: If you're going to argue against a mathematical theorem, your argument must be mathematical not verbal. Please explain one of 1) which step in the proof about Solomonoff induction's effectiveness you believe is in error 2) which of the assumptions of this proof you think is inapplicable to real intelligence [apart from the assumption of infinite or massive compute resources] Solomonoff Induction is not a provable Theorem, it is therefore a conjecture. It cannot be computed, it cannot be verified. There are many mathematical theorems that require the use of limits to prove them for example, and I accept those proofs. (Some people might not.) But there is no evidence that Solmonoff Induction would tend toward some limits. Now maybe the conjectured abstraction can be verified through some other means, but I have yet to see an adequate explanation of that in any terms. The idea that I have to answer your challenges using only the terms you specify is noise. Look at 2. What does that say about your Theorem. I am working on 1 but I just said: I haven't yet been able to find a way that could be used to prove that Solomonoff Induction does not do what Matt claims it does. Z What is not clear is that no one has objected to my characterization of the conjecture as I have been able to work it out for myself. It requires an infinite set of infinitely computed probabilities of each infinite string. If this characterization is correct, then Matt has been using the term string ambiguously. As a primary sample space: A particular string. And as a compound sample space: All the possible individual cases of the substring compounded into one. No one has yet to tell of his mathematical experiments of using a Turing simulator to see what a finite iteration of all possible programs of a given length would actually look like. I will finish this later. On Fri, Jul 9, 2010 at 7:49 AM, Jim Bromer jimbro...@gmail.com wrote: Abram, Solomoff Induction would produce poor predictions if it could be used to compute them. Solomonoff induction is a mathematical, not verbal, construct. Based on the most obvious mapping from the verbal terms you've used above into mathematical definitions in terms of which Solomonoff induction is constructed, the above statement of yours is FALSE. If you're going to argue against a mathematical theorem, your argument must be mathematical not verbal. Please explain one of 1) which step in the proof about Solomonoff induction's effectiveness you believe is in error 2) which of the assumptions of this proof you think is inapplicable to real intelligence [apart from the assumption of infinite or massive compute resources] Otherwise, your statement is in the same category as the statement by the protagonist of Dostoesvky's Notes from the Underground -- I admit that two times two makes four is an excellent thing, but if we are to give everything its due, two times two makes five is sometimes a very charming thing too. ;-) Secondly, since it cannot be computed it is useless. Third, it is not the sort of thing that is useful for AGI in the first place. I agree with these two statements -- ben G agi | https://www.listbox.com/member/archive/303/=now Archives https://www.listbox.com/member/archive/rss/303/ | Modify Your Subscription https://www.listbox.com/member/archive/rss/303/ https://www.listbox.com/member/archive/rss/303/ https://www.listbox.com/member/archive/rss/303/ agi | Archives | Modify Your Subscription https://www.listbox.com/member/archive/rss/303/ https://www.listbox.com/member/archive/rss/303/ -- Ben Goertzel, PhD CEO, Novamente LLC and Biomind LLC CTO, Genescient Corp Vice Chairman, Humanity+ Advisor, Singularity University and Singularity Institute External Research Professor, Xiamen University, China b...@goertzel.org When nothing seems to help, I go look at a stonecutter hammering away at his rock, perhaps a hundred times without as much as a crack showing in it. Yet at the hundred and first blow it will split in two,
Re: [agi] Solomonoff Induction is Not Universal and Probability is not Prediction
Abram, Solomoff Induction would produce poor predictions if it could be used to compute them. Secondly, since it cannot be computed it is useless. Third, it is not the sort of thing that is useful for AGI in the first place. You could experiment with finite possible ways to produce a string and see how useful the idea is, both as an abstraction and as an actual AGI tool. Have you tried this? An example is a word program that complete a word as you are typing. As far as Matt's complaint. I haven't yet been able to find a way that could be used to prove that Solomonoff Induction does not do what Matt claims it does, but I have yet to see an explanation of a proof that it does. When you are dealing with unverifiable pseudo-abstractions you are dealing with something that cannot be proven. All we can work on is whether or not the idea seems to make sense as an abstraction. As I said, the starting point would be to develop simpler problems and see how they behave as you build up more complicated problems. Jim On Thu, Jul 8, 2010 at 5:15 PM, Abram Demski abramdem...@gmail.com wrote: Yes, Jim, you seem to be mixing arguments here. I cannot tell which of the following you intend: 1) Solomonoff induction is useless because it would produce very bad predictions if we could compute them. 2) Solomonoff induction is useless because we can't compute its predictions. Are you trying to reject #1 and assert #2, reject #2 and assert #1, or assert both #1 and #2? Or some third statement? --Abram On Wed, Jul 7, 2010 at 7:09 PM, Matt Mahoney matmaho...@yahoo.com wrote: Who is talking about efficiency? An infinite sequence of uncomputable values is still just as uncomputable. I don't disagree that AIXI and Solomonoff induction are not computable. But you are also arguing that they are wrong. -- Matt Mahoney, matmaho...@yahoo.com -- *From:* Jim Bromer jimbro...@gmail.com *To:* agi agi@v2.listbox.com *Sent:* Wed, July 7, 2010 6:40:52 PM *Subject:* Re: [agi] Solomonoff Induction is Not Universal and Probability is not Prediction Matt, But you are still saying that Solomonoff Induction has to be recomputed for each possible combination of bit value aren't you? Although this doesn't matter when you are dealing with infinite computations in the first place, it does matter when you are wondering if this has anything to do with AGI and compression efficiencies. Jim Bromer On Wed, Jul 7, 2010 at 5:44 PM, Matt Mahoney matmaho...@yahoo.comwrote: Jim Bromer wrote: But, a more interesting question is, given that the first digits are 000, what are the chances that the next digit will be 1? Dim Induction will report .5, which of course is nonsense and a whole less useful than making a rough guess. Wrong. The probability of a 1 is p(0001)/(p()+p(0001)) where the probabilities are computed using Solomonoff induction. A program that outputs will be shorter in most languages than a program that outputs 0001, so 0 is the most likely next bit. More generally, probability and prediction are equivalent by the chain rule. Given any 2 strings x followed by y, the prediction p(y|x) = p(xy)/p(x). -- Matt Mahoney, matmaho...@yahoo.com -- *From:* Jim Bromer jimbro...@gmail.com *To:* agi agi@v2.listbox.com *Sent:* Wed, July 7, 2010 10:10:37 AM *Subject:* [agi] Solomonoff Induction is Not Universal and Probability is not Prediction Suppose you have sets of programs that produce two strings. One set of outputs is 00 and the other is 11. Now suppose you used these sets of programs to chart the probabilities of the output of the strings. If the two strings were each output by the same number of programs then you'd have a .5 probability that either string would be output. That's ok. But, a more interesting question is, given that the first digits are 000, what are the chances that the next digit will be 1? Dim Induction will report .5, which of course is nonsense and a whole less useful than making a rough guess. But, of course, Solomonoff Induction purports to be able, if it was feasible, to compute the possibilities for all possible programs. Ok, but now, try thinking about this a little bit. If you have ever tried writing random program instructions what do you usually get? Well, I'll take a hazard and guess (a lot better than the bogus method of confusing shallow probability with prediction in my example) and say that you will get a lot of programs that crash. Well, most of my experiments with that have ended up with programs that go into an infinite loop or which crash. Now on a universal Turing machine, the results would probably look a little different. Some strings will output nothing and go into an infinite loop. Some programs will output something and then either stop outputting anything or start outputting an infinite loop of the same substring. Other
Re: [agi] Solomonoff Induction is Not Universal and Probability is not Prediction
On Fri, Jul 9, 2010 at 7:49 AM, Jim Bromer jimbro...@gmail.com wrote: Abram, Solomoff Induction would produce poor predictions if it could be used to compute them. Solomonoff induction is a mathematical, not verbal, construct. Based on the most obvious mapping from the verbal terms you've used above into mathematical definitions in terms of which Solomonoff induction is constructed, the above statement of yours is FALSE. If you're going to argue against a mathematical theorem, your argument must be mathematical not verbal. Please explain one of 1) which step in the proof about Solomonoff induction's effectiveness you believe is in error 2) which of the assumptions of this proof you think is inapplicable to real intelligence [apart from the assumption of infinite or massive compute resources] Otherwise, your statement is in the same category as the statement by the protagonist of Dostoesvky's Notes from the Underground -- I admit that two times two makes four is an excellent thing, but if we are to give everything its due, two times two makes five is sometimes a very charming thing too. ;-) Secondly, since it cannot be computed it is useless. Third, it is not the sort of thing that is useful for AGI in the first place. I agree with these two statements -- ben G --- 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=8660244id_secret=8660244-6e7fb59c Powered by Listbox: http://www.listbox.com
Re: [agi] Solomonoff Induction is Not Universal and Probability is not Prediction
Ben Goertzel wrote: Secondly, since it cannot be computed it is useless. Third, it is not the sort of thing that is useful for AGI in the first place. I agree with these two statements The principle of Solomonoff induction can be applied to computable subsets of the (infinite) hypothesis space. For example, if you are using a neural network to make predictions, the principle says to use the smallest network that computes the past training data. -- Matt Mahoney, matmaho...@yahoo.com From: Ben Goertzel b...@goertzel.org To: agi agi@v2.listbox.com Sent: Fri, July 9, 2010 7:56:53 AM Subject: Re: [agi] Solomonoff Induction is Not Universal and Probability is not Prediction On Fri, Jul 9, 2010 at 7:49 AM, Jim Bromer jimbro...@gmail.com wrote: Abram, Solomoff Induction would produce poor predictions if it could be used to compute them. Solomonoff induction is a mathematical, not verbal, construct. Based on the most obvious mapping from the verbal terms you've used above into mathematical definitions in terms of which Solomonoff induction is constructed, the above statement of yours is FALSE. If you're going to argue against a mathematical theorem, your argument must be mathematical not verbal. Please explain one of 1) which step in the proof about Solomonoff induction's effectiveness you believe is in error 2) which of the assumptions of this proof you think is inapplicable to real intelligence [apart from the assumption of infinite or massive compute resources] Otherwise, your statement is in the same category as the statement by the protagonist of Dostoesvky's Notes from the Underground -- I admit that two times two makes four is an excellent thing, but if we are to give everything its due, two times two makes five is sometimes a very charming thing too. ;-) Secondly, since it cannot be computed it is useless. Third, it is not the sort of thing that is useful for AGI in the first place. I agree with these two statements -- ben G 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=8660244id_secret=8660244-6e7fb59c Powered by Listbox: http://www.listbox.com
Re: [agi] Solomonoff Induction is Not Universal and Probability is not Prediction
On Fri, Jul 9, 2010 at 8:38 AM, Matt Mahoney matmaho...@yahoo.com wrote: Ben Goertzel wrote: Secondly, since it cannot be computed it is useless. Third, it is not the sort of thing that is useful for AGI in the first place. I agree with these two statements The principle of Solomonoff induction can be applied to computable subsets of the (infinite) hypothesis space. For example, if you are using a neural network to make predictions, the principle says to use the smallest network that computes the past training data. Yes, of course various versions of Occam's Razor are useful in practice, and we use an Occam bias in MOSES inside OpenCog for example But as you know, these are not exactly the same as Solomonoff Induction, though they're based on the same idea... -- Ben --- 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=8660244id_secret=8660244-6e7fb59c Powered by Listbox: http://www.listbox.com
Re: [agi] Solomonoff Induction is Not Universal and Probability is not Prediction
On Fri, Jul 9, 2010 at 7:56 AM, Ben Goertzel b...@goertzel.org wrote: If you're going to argue against a mathematical theorem, your argument must be mathematical not verbal. Please explain one of 1) which step in the proof about Solomonoff induction's effectiveness you believe is in error 2) which of the assumptions of this proof you think is inapplicable to real intelligence [apart from the assumption of infinite or massive compute resources] Solomonoff Induction is not a provable Theorem, it is therefore a conjecture. It cannot be computed, it cannot be verified. There are many mathematical theorems that require the use of limits to prove them for example, and I accept those proofs. (Some people might not.) But there is no evidence that Solmonoff Induction would tend toward some limits. Now maybe the conjectured abstraction can be verified through some other means, but I have yet to see an adequate explanation of that in any terms. The idea that I have to answer your challenges using only the terms you specify is noise. Look at 2. What does that say about your Theorem. I am working on 1 but I just said: I haven't yet been able to find a way that could be used to prove that Solomonoff Induction does not do what Matt claims it does. Z What is not clear is that no one has objected to my characterization of the conjecture as I have been able to work it out for myself. It requires an infinite set of infinitely computed probabilities of each infinite string. If this characterization is correct, then Matt has been using the term string ambiguously. As a primary sample space: A particular string. And as a compound sample space: All the possible individual cases of the substring compounded into one. No one has yet to tell of his mathematical experiments of using a Turing simulator to see what a finite iteration of all possible programs of a given length would actually look like. I will finish this later. On Fri, Jul 9, 2010 at 7:49 AM, Jim Bromer jimbro...@gmail.com wrote: Abram, Solomoff Induction would produce poor predictions if it could be used to compute them. Solomonoff induction is a mathematical, not verbal, construct. Based on the most obvious mapping from the verbal terms you've used above into mathematical definitions in terms of which Solomonoff induction is constructed, the above statement of yours is FALSE. If you're going to argue against a mathematical theorem, your argument must be mathematical not verbal. Please explain one of 1) which step in the proof about Solomonoff induction's effectiveness you believe is in error 2) which of the assumptions of this proof you think is inapplicable to real intelligence [apart from the assumption of infinite or massive compute resources] Otherwise, your statement is in the same category as the statement by the protagonist of Dostoesvky's Notes from the Underground -- I admit that two times two makes four is an excellent thing, but if we are to give everything its due, two times two makes five is sometimes a very charming thing too. ;-) Secondly, since it cannot be computed it is useless. Third, it is not the sort of thing that is useful for AGI in the first place. I agree with these two statements -- ben G *agi* | Archives https://www.listbox.com/member/archive/303/=now https://www.listbox.com/member/archive/rss/303/ | Modifyhttps://www.listbox.com/member/?;Your Subscription http://www.listbox.com/ --- 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=8660244id_secret=8660244-6e7fb59c Powered by Listbox: http://www.listbox.com
Re: [agi] Solomonoff Induction is Not Universal and Probability is not Prediction
To make this discussion more concrete, please look at http://www.vetta.org/documents/disSol.pdf Section 2.5 gives a simple version of the proof that Solomonoff induction is a powerful learning algorithm in principle, and Section 2.6 explains why it is not practically useful. What part of that paper do you think is wrong? thx ben On Fri, Jul 9, 2010 at 9:54 AM, Jim Bromer jimbro...@gmail.com wrote: On Fri, Jul 9, 2010 at 7:56 AM, Ben Goertzel b...@goertzel.org wrote: If you're going to argue against a mathematical theorem, your argument must be mathematical not verbal. Please explain one of 1) which step in the proof about Solomonoff induction's effectiveness you believe is in error 2) which of the assumptions of this proof you think is inapplicable to real intelligence [apart from the assumption of infinite or massive compute resources] Solomonoff Induction is not a provable Theorem, it is therefore a conjecture. It cannot be computed, it cannot be verified. There are many mathematical theorems that require the use of limits to prove them for example, and I accept those proofs. (Some people might not.) But there is no evidence that Solmonoff Induction would tend toward some limits. Now maybe the conjectured abstraction can be verified through some other means, but I have yet to see an adequate explanation of that in any terms. The idea that I have to answer your challenges using only the terms you specify is noise. Look at 2. What does that say about your Theorem. I am working on 1 but I just said: I haven't yet been able to find a way that could be used to prove that Solomonoff Induction does not do what Matt claims it does. Z What is not clear is that no one has objected to my characterization of the conjecture as I have been able to work it out for myself. It requires an infinite set of infinitely computed probabilities of each infinite string. If this characterization is correct, then Matt has been using the term string ambiguously. As a primary sample space: A particular string. And as a compound sample space: All the possible individual cases of the substring compounded into one. No one has yet to tell of his mathematical experiments of using a Turing simulator to see what a finite iteration of all possible programs of a given length would actually look like. I will finish this later. On Fri, Jul 9, 2010 at 7:49 AM, Jim Bromer jimbro...@gmail.com wrote: Abram, Solomoff Induction would produce poor predictions if it could be used to compute them. Solomonoff induction is a mathematical, not verbal, construct. Based on the most obvious mapping from the verbal terms you've used above into mathematical definitions in terms of which Solomonoff induction is constructed, the above statement of yours is FALSE. If you're going to argue against a mathematical theorem, your argument must be mathematical not verbal. Please explain one of 1) which step in the proof about Solomonoff induction's effectiveness you believe is in error 2) which of the assumptions of this proof you think is inapplicable to real intelligence [apart from the assumption of infinite or massive compute resources] Otherwise, your statement is in the same category as the statement by the protagonist of Dostoesvky's Notes from the Underground -- I admit that two times two makes four is an excellent thing, but if we are to give everything its due, two times two makes five is sometimes a very charming thing too. ;-) Secondly, since it cannot be computed it is useless. Third, it is not the sort of thing that is useful for AGI in the first place. I agree with these two statements -- ben G *agi* | Archives https://www.listbox.com/member/archive/303/=now https://www.listbox.com/member/archive/rss/303/ | Modifyhttps://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/ | Modifyhttps://www.listbox.com/member/?;Your Subscription http://www.listbox.com -- Ben Goertzel, PhD CEO, Novamente LLC and Biomind LLC CTO, Genescient Corp Vice Chairman, Humanity+ Advisor, Singularity University and Singularity Institute External Research Professor, Xiamen University, China b...@goertzel.org “When nothing seems to help, I go look at a stonecutter hammering away at his rock, perhaps a hundred times without as much as a crack showing in it. Yet at the hundred and first blow it will split in two, and I know it was not that blow that did it, but all that had gone before.” --- 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=8660244id_secret=8660244-6e7fb59c Powered by Listbox: http://www.listbox.com
Re: [agi] Solomonoff Induction is Not Universal and Probability is not Prediction
Although I haven't studied Solomonoff induction yet, although I plan to read up on it, I've realized that people seem to be making the same mistake I was. People are trying to find one silver bullet method of induction or learning that works for everything. I've begun to realize that its OK if something doesn't work for everything. As long as it works on a large enough subset of problems to be useful. If you can figure out how to construct justifiable methods of induction for enough problems that you need to solve, then that is sufficient for AGI. This is the same mistake I made and it was the point I was trying to make in the recent email I sent. I kept trying to come up with algorithms for doing things and I could always find a test case to break it. So, now I've begun to realize that it's ok if it breaks sometimes! The question is, can you define an algorithm that breaks gracefully and which can figure out what problems it can be applied to and what problems it should not be applied to. If you can do that, then you can solve the problems where it is applicable, and avoid the problems where it is not. This is perfectly OK! You don't have to find a silver bullet method of induction or inference that works for everything! Dave On Fri, Jul 9, 2010 at 10:49 AM, Ben Goertzel b...@goertzel.org wrote: To make this discussion more concrete, please look at http://www.vetta.org/documents/disSol.pdf Section 2.5 gives a simple version of the proof that Solomonoff induction is a powerful learning algorithm in principle, and Section 2.6 explains why it is not practically useful. What part of that paper do you think is wrong? thx ben On Fri, Jul 9, 2010 at 9:54 AM, Jim Bromer jimbro...@gmail.com wrote: On Fri, Jul 9, 2010 at 7:56 AM, Ben Goertzel b...@goertzel.org wrote: If you're going to argue against a mathematical theorem, your argument must be mathematical not verbal. Please explain one of 1) which step in the proof about Solomonoff induction's effectiveness you believe is in error 2) which of the assumptions of this proof you think is inapplicable to real intelligence [apart from the assumption of infinite or massive compute resources] Solomonoff Induction is not a provable Theorem, it is therefore a conjecture. It cannot be computed, it cannot be verified. There are many mathematical theorems that require the use of limits to prove them for example, and I accept those proofs. (Some people might not.) But there is no evidence that Solmonoff Induction would tend toward some limits. Now maybe the conjectured abstraction can be verified through some other means, but I have yet to see an adequate explanation of that in any terms. The idea that I have to answer your challenges using only the terms you specify is noise. Look at 2. What does that say about your Theorem. I am working on 1 but I just said: I haven't yet been able to find a way that could be used to prove that Solomonoff Induction does not do what Matt claims it does. Z What is not clear is that no one has objected to my characterization of the conjecture as I have been able to work it out for myself. It requires an infinite set of infinitely computed probabilities of each infinite string. If this characterization is correct, then Matt has been using the term string ambiguously. As a primary sample space: A particular string. And as a compound sample space: All the possible individual cases of the substring compounded into one. No one has yet to tell of his mathematical experiments of using a Turing simulator to see what a finite iteration of all possible programs of a given length would actually look like. I will finish this later. On Fri, Jul 9, 2010 at 7:49 AM, Jim Bromer jimbro...@gmail.com wrote: Abram, Solomoff Induction would produce poor predictions if it could be used to compute them. Solomonoff induction is a mathematical, not verbal, construct. Based on the most obvious mapping from the verbal terms you've used above into mathematical definitions in terms of which Solomonoff induction is constructed, the above statement of yours is FALSE. If you're going to argue against a mathematical theorem, your argument must be mathematical not verbal. Please explain one of 1) which step in the proof about Solomonoff induction's effectiveness you believe is in error 2) which of the assumptions of this proof you think is inapplicable to real intelligence [apart from the assumption of infinite or massive compute resources] Otherwise, your statement is in the same category as the statement by the protagonist of Dostoesvky's Notes from the Underground -- I admit that two times two makes four is an excellent thing, but if we are to give everything its due, two times two makes five is sometimes a very charming thing too. ;-) Secondly, since it cannot be computed it is useless. Third, it is not the sort of
Re: [agi] Solomonoff Induction is Not Universal and Probability is not Prediction
The same goes for inference. There is no silver bullet method that is completely general and can infer anything. There is no general inference method. Sometimes it works, sometimes it doesn't. That is the nature of the complex world we live in. My current theory is that the more we try to find a single silver bullet, the more we will just break against the fact that none exists. On Fri, Jul 9, 2010 at 11:35 AM, David Jones davidher...@gmail.com wrote: Although I haven't studied Solomonoff induction yet, although I plan to read up on it, I've realized that people seem to be making the same mistake I was. People are trying to find one silver bullet method of induction or learning that works for everything. I've begun to realize that its OK if something doesn't work for everything. As long as it works on a large enough subset of problems to be useful. If you can figure out how to construct justifiable methods of induction for enough problems that you need to solve, then that is sufficient for AGI. This is the same mistake I made and it was the point I was trying to make in the recent email I sent. I kept trying to come up with algorithms for doing things and I could always find a test case to break it. So, now I've begun to realize that it's ok if it breaks sometimes! The question is, can you define an algorithm that breaks gracefully and which can figure out what problems it can be applied to and what problems it should not be applied to. If you can do that, then you can solve the problems where it is applicable, and avoid the problems where it is not. This is perfectly OK! You don't have to find a silver bullet method of induction or inference that works for everything! Dave On Fri, Jul 9, 2010 at 10:49 AM, Ben Goertzel b...@goertzel.org wrote: To make this discussion more concrete, please look at http://www.vetta.org/documents/disSol.pdf Section 2.5 gives a simple version of the proof that Solomonoff induction is a powerful learning algorithm in principle, and Section 2.6 explains why it is not practically useful. What part of that paper do you think is wrong? thx ben On Fri, Jul 9, 2010 at 9:54 AM, Jim Bromer jimbro...@gmail.com wrote: On Fri, Jul 9, 2010 at 7:56 AM, Ben Goertzel b...@goertzel.org wrote: If you're going to argue against a mathematical theorem, your argument must be mathematical not verbal. Please explain one of 1) which step in the proof about Solomonoff induction's effectiveness you believe is in error 2) which of the assumptions of this proof you think is inapplicable to real intelligence [apart from the assumption of infinite or massive compute resources] Solomonoff Induction is not a provable Theorem, it is therefore a conjecture. It cannot be computed, it cannot be verified. There are many mathematical theorems that require the use of limits to prove them for example, and I accept those proofs. (Some people might not.) But there is no evidence that Solmonoff Induction would tend toward some limits. Now maybe the conjectured abstraction can be verified through some other means, but I have yet to see an adequate explanation of that in any terms. The idea that I have to answer your challenges using only the terms you specify is noise. Look at 2. What does that say about your Theorem. I am working on 1 but I just said: I haven't yet been able to find a way that could be used to prove that Solomonoff Induction does not do what Matt claims it does. Z What is not clear is that no one has objected to my characterization of the conjecture as I have been able to work it out for myself. It requires an infinite set of infinitely computed probabilities of each infinite string. If this characterization is correct, then Matt has been using the term string ambiguously. As a primary sample space: A particular string. And as a compound sample space: All the possible individual cases of the substring compounded into one. No one has yet to tell of his mathematical experiments of using a Turing simulator to see what a finite iteration of all possible programs of a given length would actually look like. I will finish this later. On Fri, Jul 9, 2010 at 7:49 AM, Jim Bromer jimbro...@gmail.comwrote: Abram, Solomoff Induction would produce poor predictions if it could be used to compute them. Solomonoff induction is a mathematical, not verbal, construct. Based on the most obvious mapping from the verbal terms you've used above into mathematical definitions in terms of which Solomonoff induction is constructed, the above statement of yours is FALSE. If you're going to argue against a mathematical theorem, your argument must be mathematical not verbal. Please explain one of 1) which step in the proof about Solomonoff induction's effectiveness you believe is in error 2) which of the assumptions of this proof you think is inapplicable to real
Re: [agi] Solomonoff Induction is Not Universal and Probability is not Prediction
I don't think Solomonoff induction is a particularly useful direction for AI, I was just taking issue with the statement made that it is not capable of correct prediction given adequate resources... On Fri, Jul 9, 2010 at 11:35 AM, David Jones davidher...@gmail.com wrote: Although I haven't studied Solomonoff induction yet, although I plan to read up on it, I've realized that people seem to be making the same mistake I was. People are trying to find one silver bullet method of induction or learning that works for everything. I've begun to realize that its OK if something doesn't work for everything. As long as it works on a large enough subset of problems to be useful. If you can figure out how to construct justifiable methods of induction for enough problems that you need to solve, then that is sufficient for AGI. This is the same mistake I made and it was the point I was trying to make in the recent email I sent. I kept trying to come up with algorithms for doing things and I could always find a test case to break it. So, now I've begun to realize that it's ok if it breaks sometimes! The question is, can you define an algorithm that breaks gracefully and which can figure out what problems it can be applied to and what problems it should not be applied to. If you can do that, then you can solve the problems where it is applicable, and avoid the problems where it is not. This is perfectly OK! You don't have to find a silver bullet method of induction or inference that works for everything! Dave On Fri, Jul 9, 2010 at 10:49 AM, Ben Goertzel b...@goertzel.org wrote: To make this discussion more concrete, please look at http://www.vetta.org/documents/disSol.pdf Section 2.5 gives a simple version of the proof that Solomonoff induction is a powerful learning algorithm in principle, and Section 2.6 explains why it is not practically useful. What part of that paper do you think is wrong? thx ben On Fri, Jul 9, 2010 at 9:54 AM, Jim Bromer jimbro...@gmail.com wrote: On Fri, Jul 9, 2010 at 7:56 AM, Ben Goertzel b...@goertzel.org wrote: If you're going to argue against a mathematical theorem, your argument must be mathematical not verbal. Please explain one of 1) which step in the proof about Solomonoff induction's effectiveness you believe is in error 2) which of the assumptions of this proof you think is inapplicable to real intelligence [apart from the assumption of infinite or massive compute resources] Solomonoff Induction is not a provable Theorem, it is therefore a conjecture. It cannot be computed, it cannot be verified. There are many mathematical theorems that require the use of limits to prove them for example, and I accept those proofs. (Some people might not.) But there is no evidence that Solmonoff Induction would tend toward some limits. Now maybe the conjectured abstraction can be verified through some other means, but I have yet to see an adequate explanation of that in any terms. The idea that I have to answer your challenges using only the terms you specify is noise. Look at 2. What does that say about your Theorem. I am working on 1 but I just said: I haven't yet been able to find a way that could be used to prove that Solomonoff Induction does not do what Matt claims it does. Z What is not clear is that no one has objected to my characterization of the conjecture as I have been able to work it out for myself. It requires an infinite set of infinitely computed probabilities of each infinite string. If this characterization is correct, then Matt has been using the term string ambiguously. As a primary sample space: A particular string. And as a compound sample space: All the possible individual cases of the substring compounded into one. No one has yet to tell of his mathematical experiments of using a Turing simulator to see what a finite iteration of all possible programs of a given length would actually look like. I will finish this later. On Fri, Jul 9, 2010 at 7:49 AM, Jim Bromer jimbro...@gmail.comwrote: Abram, Solomoff Induction would produce poor predictions if it could be used to compute them. Solomonoff induction is a mathematical, not verbal, construct. Based on the most obvious mapping from the verbal terms you've used above into mathematical definitions in terms of which Solomonoff induction is constructed, the above statement of yours is FALSE. If you're going to argue against a mathematical theorem, your argument must be mathematical not verbal. Please explain one of 1) which step in the proof about Solomonoff induction's effectiveness you believe is in error 2) which of the assumptions of this proof you think is inapplicable to real intelligence [apart from the assumption of infinite or massive compute resources] Otherwise, your statement is in the same category as the statement by the protagonist of Dostoesvky's
Re: [agi] Solomonoff Induction is Not Universal and Probability is not Prediction
On Fri, Jul 9, 2010 at 11:37 AM, Ben Goertzel b...@goertzel.org wrote: I don't think Solomonoff induction is a particularly useful direction for AI, I was just taking issue with the statement made that it is not capable of correct prediction given adequate resources... Pi is not computable. It would take infinite resources to compute it. However, because Pi approaches a limit, the theory of limits can be used to show that it can be refined to any limit that is possible and since it consistently approaches a limit it can be used in general theorems that can be proven through induction. You can use *computed values* of pi in a general theorem as long as you can show that the usage is valid by using the theory of limits. I think I figured out a way, given infinite resources, to write a program that could compute Solomonoff Induction. However, since it cannot be shown (or at least I don't know anyone who has ever shown) that the probabilities approaches some value (or values) as a limit (or limits), this program (or a variation on this kind of program) could not be used to show that it can be: 1. computed to any specified degree of precision within some finite number of steps. 2. proven through the use of mathematical induction. The proof is based on the diagonal argument of Cantor, but it might be considered as variation of Cantor's diagonal argument. There can be no one to one *mapping of the computation to an usage* as the computation approaches infinity to make the values approach some limit of precision. For any computed values there is always a *possibility* (this is different than Cantor) that there are an infinite number of more precise values (of the probability of a string (primary sample space or compound sample space)) within any two iterations of the computational program (formula). So even though I cannot disprove what Solomonoff Induction might be given infinite resources, if this superficial analysis is right, without a way to compute the values so that they tend toward a limit for each of the probabilities needed, it is not a usable mathematical theorem. What uncomputable means is that any statement (most statements) drawn from it are matters of mathematical conjecture or opinion. It's like opinioning that the Godel sentence, given infinite resources, is decidable. I don't think the question of whether it is valid for infinite resources or not can be answered mathematically for the time being. And conclusions drawn from uncomputable results have to be considered dubious. However, it certainly leads to other questions which I think are more interesting and more useful. What is needed to promote greater insight about the problem of conditional probabilities in complicated situations where the probability emitters and the elementary sample space may be obscured by the use of complicated interactions and a preliminary focus on compound sample spaces? Are there theories, which like asking questions about the givens in a problem, that could lead toward a greater detection of the relation between the givens and the primary probability emitters and the primary sample space? Can a mathematical theory be based solely on abstract principles even though it is impossible to evaluate the use of those abstractions with examples from the particulars (of the abstractions)? How could those abstract principles be reliably defined so that they aren't too simplistic? Jim Bromer --- 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=8660244id_secret=8660244-6e7fb59c Powered by Listbox: http://www.listbox.com
Re: [agi] Solomonoff Induction is Not Universal and Probability is not Prediction
On Fri, Jul 9, 2010 at 1:12 PM, Jim Bromer jimbro...@gmail.com wrote: The proof is based on the diagonal argument of Cantor, but it might be considered as variation of Cantor's diagonal argument. There can be no one to one *mapping of the computation to an usage* as the computation approaches infinity to make the values approach some limit of precision. For any computed values there is always a *possibility* (this is different than Cantor) that there are an infinite number of more precise values (of the probability of a string (primary sample space or compound sample space)) within any two iterations of the computational program (formula). Ok, I didn't get that right, but there is enough there to get the idea. For any computed values there is always a *possibility* (I think this is different than Cantor) that there are an infinite number of more precise values (of the probability of a string (primary sample space or compound sample space)) that may fall outside the limits that could be derived from any finite sequence of iterations of the computational program (formula). On Fri, Jul 9, 2010 at 1:12 PM, Jim Bromer jimbro...@gmail.com wrote: On Fri, Jul 9, 2010 at 11:37 AM, Ben Goertzel b...@goertzel.org wrote: I don't think Solomonoff induction is a particularly useful direction for AI, I was just taking issue with the statement made that it is not capable of correct prediction given adequate resources... Pi is not computable. It would take infinite resources to compute it. However, because Pi approaches a limit, the theory of limits can be used to show that it can be refined to any limit that is possible and since it consistently approaches a limit it can be used in general theorems that can be proven through induction. You can use *computed values* of pi in a general theorem as long as you can show that the usage is valid by using the theory of limits. I think I figured out a way, given infinite resources, to write a program that could compute Solomonoff Induction. However, since it cannot be shown (or at least I don't know anyone who has ever shown) that the probabilities approaches some value (or values) as a limit (or limits), this program (or a variation on this kind of program) could not be used to show that it can be: 1. computed to any specified degree of precision within some finite number of steps. 2. proven through the use of mathematical induction. The proof is based on the diagonal argument of Cantor, but it might be considered as variation of Cantor's diagonal argument. There can be no one to one *mapping of the computation to an usage* as the computation approaches infinity to make the values approach some limit of precision. For any computed values there is always a *possibility* (this is different than Cantor) that there are an infinite number of more precise values (of the probability of a string (primary sample space or compound sample space)) within any two iterations of the computational program (formula). So even though I cannot disprove what Solomonoff Induction might be given infinite resources, if this superficial analysis is right, without a way to compute the values so that they tend toward a limit for each of the probabilities needed, it is not a usable mathematical theorem. What uncomputable means is that any statement (most statements) drawn from it are matters of mathematical conjecture or opinion. It's like opinioning that the Godel sentence, given infinite resources, is decidable. I don't think the question of whether it is valid for infinite resources or not can be answered mathematically for the time being. And conclusions drawn from uncomputable results have to be considered dubious. However, it certainly leads to other questions which I think are more interesting and more useful. What is needed to promote greater insight about the problem of conditional probabilities in complicated situations where the probability emitters and the elementary sample space may be obscured by the use of complicated interactions and a preliminary focus on compound sample spaces? Are there theories, which like asking questions about the givens in a problem, that could lead toward a greater detection of the relation between the givens and the primary probability emitters and the primary sample space? Can a mathematical theory be based solely on abstract principles even though it is impossible to evaluate the use of those abstractions with examples from the particulars (of the abstractions)? How could those abstract principles be reliably defined so that they aren't too simplistic? Jim Bromer --- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/ Modify Your Subscription:
Re: [agi] Solomonoff Induction is Not Universal and Probability is not Prediction
Solomonoff Induction is not a mathematical conjecture. We can talk about a function which is based on all mathematical functions, but since we cannot define that as a mathematical function it is not a realizable function. --- 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=8660244id_secret=8660244-6e7fb59c Powered by Listbox: http://www.listbox.com
Re: [agi] Solomonoff Induction is Not Universal and Probability is not Prediction
I guess the Godel Theorem is called a theorem, so Solomonoff Induction would be called a theorem. I believe that Solomonoff Induction is computable, but the claims that are made for it are not provable because there is no way you could prove that it approaches a stable limit (stable limits). You can't prove that it does just because the sense of all possible programs is so ill-defined that there is not enough to go on. Whether my outline of a disproof could actually be used to find an adequate disproof, I don't know. My attempt to disprove it may just be an unprovable theorem (or even wrong). Jim Bromer --- 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=8660244id_secret=8660244-6e7fb59c Powered by Listbox: http://www.listbox.com
Re: [agi] Solomonoff Induction is Not Universal and Probability is not Prediction
Yes, Jim, you seem to be mixing arguments here. I cannot tell which of the following you intend: 1) Solomonoff induction is useless because it would produce very bad predictions if we could compute them. 2) Solomonoff induction is useless because we can't compute its predictions. Are you trying to reject #1 and assert #2, reject #2 and assert #1, or assert both #1 and #2? Or some third statement? --Abram On Wed, Jul 7, 2010 at 7:09 PM, Matt Mahoney matmaho...@yahoo.com wrote: Who is talking about efficiency? An infinite sequence of uncomputable values is still just as uncomputable. I don't disagree that AIXI and Solomonoff induction are not computable. But you are also arguing that they are wrong. -- Matt Mahoney, matmaho...@yahoo.com -- *From:* Jim Bromer jimbro...@gmail.com *To:* agi agi@v2.listbox.com *Sent:* Wed, July 7, 2010 6:40:52 PM *Subject:* Re: [agi] Solomonoff Induction is Not Universal and Probability is not Prediction Matt, But you are still saying that Solomonoff Induction has to be recomputed for each possible combination of bit value aren't you? Although this doesn't matter when you are dealing with infinite computations in the first place, it does matter when you are wondering if this has anything to do with AGI and compression efficiencies. Jim Bromer On Wed, Jul 7, 2010 at 5:44 PM, Matt Mahoney matmaho...@yahoo.com wrote: Jim Bromer wrote: But, a more interesting question is, given that the first digits are 000, what are the chances that the next digit will be 1? Dim Induction will report .5, which of course is nonsense and a whole less useful than making a rough guess. Wrong. The probability of a 1 is p(0001)/(p()+p(0001)) where the probabilities are computed using Solomonoff induction. A program that outputs will be shorter in most languages than a program that outputs 0001, so 0 is the most likely next bit. More generally, probability and prediction are equivalent by the chain rule. Given any 2 strings x followed by y, the prediction p(y|x) = p(xy)/p(x). -- Matt Mahoney, matmaho...@yahoo.com -- *From:* Jim Bromer jimbro...@gmail.com *To:* agi agi@v2.listbox.com *Sent:* Wed, July 7, 2010 10:10:37 AM *Subject:* [agi] Solomonoff Induction is Not Universal and Probability is not Prediction Suppose you have sets of programs that produce two strings. One set of outputs is 00 and the other is 11. Now suppose you used these sets of programs to chart the probabilities of the output of the strings. If the two strings were each output by the same number of programs then you'd have a .5 probability that either string would be output. That's ok. But, a more interesting question is, given that the first digits are 000, what are the chances that the next digit will be 1? Dim Induction will report .5, which of course is nonsense and a whole less useful than making a rough guess. But, of course, Solomonoff Induction purports to be able, if it was feasible, to compute the possibilities for all possible programs. Ok, but now, try thinking about this a little bit. If you have ever tried writing random program instructions what do you usually get? Well, I'll take a hazard and guess (a lot better than the bogus method of confusing shallow probability with prediction in my example) and say that you will get a lot of programs that crash. Well, most of my experiments with that have ended up with programs that go into an infinite loop or which crash. Now on a universal Turing machine, the results would probably look a little different. Some strings will output nothing and go into an infinite loop. Some programs will output something and then either stop outputting anything or start outputting an infinite loop of the same substring. Other programs will go on to infinity producing something that looks like random strings. But the idea that all possible programs would produce well distributed strings is complete hogwash. Since Solomonoff Induction does not define what kind of programs should be used, the assumption that the distribution would produce useful data is absurd. In particular, the use of the method to determine the probability based given an initial string (as in what follows given the first digits are 000) is wrong as in really wrong. The idea that this crude probability can be used as prediction is unsophisticated. Of course you could develop an infinite set of Solomonoff Induction values for each possible given initial sequence of digits. Hey when you're working with infeasible functions why not dream anything? I might be wrong of course. Maybe there is something you guys haven't been able to get across to me. Even if you can think for yourself you can still make mistakes. So if anyone has actually tried writing a program to output all possible programs (up to some feasible point) on a Turing Machine
[agi] Solomonoff Induction is Not Universal and Probability is not Prediction
Suppose you have sets of programs that produce two strings. One set of outputs is 00 and the other is 11. Now suppose you used these sets of programs to chart the probabilities of the output of the strings. If the two strings were each output by the same number of programs then you'd have a .5 probability that either string would be output. That's ok. But, a more interesting question is, given that the first digits are 000, what are the chances that the next digit will be 1? Dim Induction will report .5, which of course is nonsense and a whole less useful than making a rough guess. But, of course, Solomonoff Induction purports to be able, if it was feasible, to compute the possibilities for all possible programs. Ok, but now, try thinking about this a little bit. If you have ever tried writing random program instructions what do you usually get? Well, I'll take a hazard and guess (a lot better than the bogus method of confusing shallow probability with prediction in my example) and say that you will get a lot of programs that crash. Well, most of my experiments with that have ended up with programs that go into an infinite loop or which crash. Now on a universal Turing machine, the results would probably look a little different. Some strings will output nothing and go into an infinite loop. Some programs will output something and then either stop outputting anything or start outputting an infinite loop of the same substring. Other programs will go on to infinity producing something that looks like random strings. But the idea that all possible programs would produce well distributed strings is complete hogwash. Since Solomonoff Induction does not define what kind of programs should be used, the assumption that the distribution would produce useful data is absurd. In particular, the use of the method to determine the probability based given an initial string (as in what follows given the first digits are 000) is wrong as in really wrong. The idea that this crude probability can be used as prediction is unsophisticated. Of course you could develop an infinite set of Solomonoff Induction values for each possible given initial sequence of digits. Hey when you're working with infeasible functions why not dream anything? I might be wrong of course. Maybe there is something you guys haven't been able to get across to me. Even if you can think for yourself you can still make mistakes. So if anyone has actually tried writing a program to output all possible programs (up to some feasible point) on a Turing Machine simulator, let me know how it went. Jim Bromer --- 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=8660244id_secret=8660244-6e7fb59c Powered by Listbox: http://www.listbox.com
Re: [agi] Solomonoff Induction is Not Universal and Probability is not Prediction
Jim, I am unable to find the actual objection to Solomonoff in what you wrote (save for that it's wrong as in really wrong). It's true that a lot of programs won't produce any output. That just means they won't alter the prediction. It's also true that a lot of programs will produce random-looking or boring-looking output. This just means that Solomonoff will have some expectation of those things. To use your example, given 000, the chances that the next digit will be 0 will be fairly high thanks to boring programs which just output lots of zeros. (Not sure why you mention the idea that it might be .5? This sounds like no induction rather than dim induction...) --Abram On Wed, Jul 7, 2010 at 10:10 AM, Jim Bromer jimbro...@gmail.com wrote: Suppose you have sets of programs that produce two strings. One set of outputs is 00 and the other is 11. Now suppose you used these sets of programs to chart the probabilities of the output of the strings. If the two strings were each output by the same number of programs then you'd have a .5 probability that either string would be output. That's ok. But, a more interesting question is, given that the first digits are 000, what are the chances that the next digit will be 1? Dim Induction will report .5, which of course is nonsense and a whole less useful than making a rough guess. But, of course, Solomonoff Induction purports to be able, if it was feasible, to compute the possibilities for all possible programs. Ok, but now, try thinking about this a little bit. If you have ever tried writing random program instructions what do you usually get? Well, I'll take a hazard and guess (a lot better than the bogus method of confusing shallow probability with prediction in my example) and say that you will get a lot of programs that crash. Well, most of my experiments with that have ended up with programs that go into an infinite loop or which crash. Now on a universal Turing machine, the results would probably look a little different. Some strings will output nothing and go into an infinite loop. Some programs will output something and then either stop outputting anything or start outputting an infinite loop of the same substring. Other programs will go on to infinity producing something that looks like random strings. But the idea that all possible programs would produce well distributed strings is complete hogwash. Since Solomonoff Induction does not define what kind of programs should be used, the assumption that the distribution would produce useful data is absurd. In particular, the use of the method to determine the probability based given an initial string (as in what follows given the first digits are 000) is wrong as in really wrong. The idea that this crude probability can be used as prediction is unsophisticated. Of course you could develop an infinite set of Solomonoff Induction values for each possible given initial sequence of digits. Hey when you're working with infeasible functions why not dream anything? I might be wrong of course. Maybe there is something you guys haven't been able to get across to me. Even if you can think for yourself you can still make mistakes. So if anyone has actually tried writing a program to output all possible programs (up to some feasible point) on a Turing Machine simulator, let me know how it went. Jim Bromer *agi* | Archives https://www.listbox.com/member/archive/303/=now https://www.listbox.com/member/archive/rss/303/ | Modifyhttps://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=8660244id_secret=8660244-6e7fb59c Powered by Listbox: http://www.listbox.com
Re: [agi] Solomonoff Induction is Not Universal and Probability is not Prediction
Abram, I don't think you are right. The reason is that Solomonoff Induction does not produce a true universal probability for any given first digits. To do so it would have to be capable of representing the probability of any (computable) sequence that follows any (computable) string of given first digits. Yes, if a high proportion of programs produce 00, it will be able to register that as string as more probable, but the information on what the next digits will be, given some input, will not be represented in anything that resembled compression. For instance, if you had 62 bits and wanted to know what the probability of the next two bits were, you would have to have done the infinite calculations of a Solomonoff Induction for each of the 2^62 possible combination of bits that represented the possible input to your problem. I might be wrong, but I don't see where all this is information is being hidden if I am. On the other hand, if I am right (or even partially right) I don't understand why seemingly smart people are excited about this as a possible AGI method. We in AGI specifically want to know the answer to the kind of question: Given some partially defined situation, how could a computer best figure out what is going on. Most computer situations are going to be represented by kilobytes or megabytes these days, not in strings of 32 bits or less. If there was an abstraction that could help us think about these things, it could help even if the ideal would be way beyond any feasible technology. And there is an abstraction like this that can help us. Applied probability. We can think about these ideas in the terms of strings if we want to but the key is that WE have to work out the details because we see the problems differently. There is nothing that I have seen in Solomonoff Induction that suggests that this is an adequate or even useful method to use. On the other hand I would not be talking about this if it weren't for Solomonoff so maybe I just don't share your enthusiasm. If I have misunderstood something then all I can say is that I am still waiting for someone to explain it in a way that I can understand. Jim On Wed, Jul 7, 2010 at 1:58 PM, Abram Demski abramdem...@gmail.com wrote: Jim, I am unable to find the actual objection to Solomonoff in what you wrote (save for that it's wrong as in really wrong). It's true that a lot of programs won't produce any output. That just means they won't alter the prediction. It's also true that a lot of programs will produce random-looking or boring-looking output. This just means that Solomonoff will have some expectation of those things. To use your example, given 000, the chances that the next digit will be 0 will be fairly high thanks to boring programs which just output lots of zeros. (Not sure why you mention the idea that it might be .5? This sounds like no induction rather than dim induction...) --Abram On Wed, Jul 7, 2010 at 10:10 AM, Jim Bromer jimbro...@gmail.com wrote: Suppose you have sets of programs that produce two strings. One set of outputs is 00 and the other is 11. Now suppose you used these sets of programs to chart the probabilities of the output of the strings. If the two strings were each output by the same number of programs then you'd have a .5 probability that either string would be output. That's ok. But, a more interesting question is, given that the first digits are 000, what are the chances that the next digit will be 1? Dim Induction will report .5, which of course is nonsense and a whole less useful than making a rough guess. But, of course, Solomonoff Induction purports to be able, if it was feasible, to compute the possibilities for all possible programs. Ok, but now, try thinking about this a little bit. If you have ever tried writing random program instructions what do you usually get? Well, I'll take a hazard and guess (a lot better than the bogus method of confusing shallow probability with prediction in my example) and say that you will get a lot of programs that crash. Well, most of my experiments with that have ended up with programs that go into an infinite loop or which crash. Now on a universal Turing machine, the results would probably look a little different. Some strings will output nothing and go into an infinite loop. Some programs will output something and then either stop outputting anything or start outputting an infinite loop of the same substring. Other programs will go on to infinity producing something that looks like random strings. But the idea that all possible programs would produce well distributed strings is complete hogwash. Since Solomonoff Induction does not define what kind of programs should be used, the assumption that the distribution would produce useful data is absurd. In particular, the use of the method to determine the probability based given an initial string (as in what follows given the
Re: [agi] Solomonoff Induction is Not Universal and Probability is not Prediction
Jim Bromer wrote: But, a more interesting question is, given that the first digits are 000, what are the chances that the next digit will be 1? Dim Induction will report .5, which of course is nonsense and a whole less useful than making a rough guess. Wrong. The probability of a 1 is p(0001)/(p()+p(0001)) where the probabilities are computed using Solomonoff induction. A program that outputs will be shorter in most languages than a program that outputs 0001, so 0 is the most likely next bit. More generally, probability and prediction are equivalent by the chain rule. Given any 2 strings x followed by y, the prediction p(y|x) = p(xy)/p(x). -- Matt Mahoney, matmaho...@yahoo.com From: Jim Bromer jimbro...@gmail.com To: agi agi@v2.listbox.com Sent: Wed, July 7, 2010 10:10:37 AM Subject: [agi] Solomonoff Induction is Not Universal and Probability is not Prediction Suppose you have sets of programs that produce two strings. One set of outputs is 00 and the other is 11. Now suppose you used these sets of programs to chart the probabilities of the output of the strings. If the two strings were each output by the same number of programs then you'd have a .5 probability that either string would be output. That's ok. But, a more interesting question is, given that the first digits are 000, what are the chances that the next digit will be 1? Dim Induction will report .5, which of course is nonsense and a whole less useful than making a rough guess. But, of course, Solomonoff Induction purports to be able, if it was feasible, to compute the possibilities for all possible programs. Ok, but now, try thinking about this a little bit. If you have ever tried writing random program instructions what do you usually get? Well, I'll take a hazard and guess (a lot better than the bogus method of confusing shallow probability with prediction in my example) and say that you will get a lot of programs that crash. Well, most of my experiments with that have ended up with programs that go into an infinite loop or which crash. Now on a universal Turing machine, the results would probably look a little different. Some strings will output nothing and go into an infinite loop. Some programs will output something and then either stop outputting anything or start outputting an infinite loop of the same substring. Other programs will go on to infinity producing something that looks like random strings. But the idea that all possible programs would produce well distributed strings is complete hogwash. Since Solomonoff Induction does not define what kind of programs should be used, the assumption that the distribution would produce useful data is absurd. In particular, the use of the method to determine the probability based given an initial string (as in what follows given the first digits are 000) is wrong as in really wrong. The idea that this crude probability can be used as prediction is unsophisticated. Of course you could develop an infinite set of Solomonoff Induction values for each possible given initial sequence of digits. Hey when you're working with infeasible functions why not dream anything? I might be wrong of course. Maybe there is something you guys haven't been able to get across to me. Even if you can think for yourself you can still make mistakes. So if anyone has actually tried writing a program to output all possible programs (up to some feasible point) on a Turing Machine simulator, let me know how it went. Jim Bromer 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=8660244id_secret=8660244-6e7fb59c Powered by Listbox: http://www.listbox.com
Re: [agi] Solomonoff Induction is Not Universal and Probability is not Prediction
Matt, But you are still saying that Solomonoff Induction has to be recomputed for each possible combination of bit value aren't you? Although this doesn't matter when you are dealing with infinite computations in the first place, it does matter when you are wondering if this has anything to do with AGI and compression efficiencies. Jim Bromer On Wed, Jul 7, 2010 at 5:44 PM, Matt Mahoney matmaho...@yahoo.com wrote: Jim Bromer wrote: But, a more interesting question is, given that the first digits are 000, what are the chances that the next digit will be 1? Dim Induction will report .5, which of course is nonsense and a whole less useful than making a rough guess. Wrong. The probability of a 1 is p(0001)/(p()+p(0001)) where the probabilities are computed using Solomonoff induction. A program that outputs will be shorter in most languages than a program that outputs 0001, so 0 is the most likely next bit. More generally, probability and prediction are equivalent by the chain rule. Given any 2 strings x followed by y, the prediction p(y|x) = p(xy)/p(x). -- Matt Mahoney, matmaho...@yahoo.com -- *From:* Jim Bromer jimbro...@gmail.com *To:* agi agi@v2.listbox.com *Sent:* Wed, July 7, 2010 10:10:37 AM *Subject:* [agi] Solomonoff Induction is Not Universal and Probability is not Prediction Suppose you have sets of programs that produce two strings. One set of outputs is 00 and the other is 11. Now suppose you used these sets of programs to chart the probabilities of the output of the strings. If the two strings were each output by the same number of programs then you'd have a .5 probability that either string would be output. That's ok. But, a more interesting question is, given that the first digits are 000, what are the chances that the next digit will be 1? Dim Induction will report .5, which of course is nonsense and a whole less useful than making a rough guess. But, of course, Solomonoff Induction purports to be able, if it was feasible, to compute the possibilities for all possible programs. Ok, but now, try thinking about this a little bit. If you have ever tried writing random program instructions what do you usually get? Well, I'll take a hazard and guess (a lot better than the bogus method of confusing shallow probability with prediction in my example) and say that you will get a lot of programs that crash. Well, most of my experiments with that have ended up with programs that go into an infinite loop or which crash. Now on a universal Turing machine, the results would probably look a little different. Some strings will output nothing and go into an infinite loop. Some programs will output something and then either stop outputting anything or start outputting an infinite loop of the same substring. Other programs will go on to infinity producing something that looks like random strings. But the idea that all possible programs would produce well distributed strings is complete hogwash. Since Solomonoff Induction does not define what kind of programs should be used, the assumption that the distribution would produce useful data is absurd. In particular, the use of the method to determine the probability based given an initial string (as in what follows given the first digits are 000) is wrong as in really wrong. The idea that this crude probability can be used as prediction is unsophisticated. Of course you could develop an infinite set of Solomonoff Induction values for each possible given initial sequence of digits. Hey when you're working with infeasible functions why not dream anything? I might be wrong of course. Maybe there is something you guys haven't been able to get across to me. Even if you can think for yourself you can still make mistakes. So if anyone has actually tried writing a program to output all possible programs (up to some feasible point) on a Turing Machine simulator, let me know how it went. Jim Bromer *agi* | Archives https://www.listbox.com/member/archive/303/=now https://www.listbox.com/member/archive/rss/303/ | Modifyhttps://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/ | Modifyhttps://www.listbox.com/member/?;Your Subscription http://www.listbox.com/ --- 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=8660244id_secret=8660244-6e7fb59c Powered by Listbox: http://www.listbox.com
