The omitted information in this approach is information with a high Kolmogorov complexity, which is omitted in favor of information with low Kolmogorov complexity. A very rough analogy would be to describe humans as having a bias towards ideas expressible in few words of English in favor of ideas that need many words of English to express. Using Kolmogorov complexity for sequence prediction instead of English language for ideas in the construction gets rid of the very many problems of rigor involved in the latter, but the basic idea is pretty much the same. You look into things that are briefly expressible in favor of things that must be expressed in length. The information isn’t permanently omitted, it’s just depriorized. The algorithm doesn’t start looking at the stuff you need long sentences to describe before it has convinced itself that there are no short sentences that describe the observations it wants to explain in a satisfactory way.
One bit of context that is assumed is that the surrounding universe is somewhat amenable to being Kolmogorov-compressed. That is, there are some recurring regularities that you can begin to discover. The term “lawful universe” sometimes thrown around in LW probably refers to something similar.
Solomonoff’s universal induction would not work in a completely chaotic universe, where there are no regularities for Kolmogorov compression to latch on. You’d also be unlikely to find any sort of native intelligent entities in such universes. I’m not sure if this means that the Solomonoff approach is philosophically untenable, but needing to have some discoverable regularities to begin with before discovering regularities with induction becomes possible doesn’t strike me as that great a requirement.
If the problem of context is about exactly where you draw the data for the sequence which you will then try to predict with Solomonoff induction, in a lawless universe you wouldn’t be able to infer things no matter which simple instrumentation you picked, while in a lawful universe you could pick all sorts of instruments, tracking the change of light during time, tracking temperature, tracking the luminousity of the Moon, for simple examples, and you’d start getting Kolmogorov-compressible data where the induction system could start figuring repeating periods.
The core thing “independent of context” in all this is that all the universal induction systems are reduced to basically taking a series of numbers as input, and trying to develop an efficient predictor for what the next number will be. The argument in the paper is that this construction is basically sufficient for all the interesting things an induction solution could do, and that all the various real-world cases where induction is needed can be basically reduced into such a system by describing the instrumentation which turns real-world input into a time series of numbers.
Okay. In this case, the article does seem to begin to make sense. Its connection to the problem of induction is perhaps rather thin. The idea of using low Kolmogorov complexity as justification for an inductive argument cannot be deduced as a theorem of something that’s “surely true”, whatever that might mean. And if it were taken as an axiom, philosophers would say: “That’s not an axiom. That’s the conclusion of an inductive argument you made! You are begging the question!”
However, it seems like advancements in computation theory have made people able to do at least remotely practical stuff on areas, that bear resemblance to more inert philosophical ponderings. That’s good, and this article might even be used as justification for my theory RP—given that the use of Kolmogorov complexity is accepted. I was not familiar with the concept of Kolmogorov complexity despite having heard of it a few times, but my intuitive goal was to minimize the theory’s Kolmogorov complexity by removing arbitrary declarations and favoring symmetry.
I would say, that there are many ways of solving the problem of induction. Whether a theory is a solution to the problem of induction depends on whether it covers the entire scope of the problem. I would say this article covers half of the scope. The rest is not covered, to my knowledge, by anyone else than Robert Pirsig and experts of Buddhism, but these writings are very difficult to approach analytically. Regrettably, I am still unable to publish the relativizability article, which is intended to succeed in the analytic approach.
In any case, even though the widely rejected “statistical relevance” and this “Kolmogorov complexity relevance” share the same flaw, if presented as an explanation of inductive justification, the approach is interesting. Perhaps, even, this paper should be titled: “A Formalization of Occam’s Razor Principle”. Because that’s what it surely seems to be. And I think it’s actually an achievement to formalize that principle—an achievement more than sufficient to justify the writing of the article.
“When artificial intelligence researchers attempted to capture everyday statements of inference using classical logic they began to realize this was a difficult if not impossible task.”
I hope nobody’s doing this anymore. It’s obviously impossible. “Everyday statements of inference”, whatever that might mean, are not exclusively statements of first-order logic, because Russell’s paradox is simple enough to be formulated by talking about barbers. The liar paradox is also expressible with simple, practical language.
Wait a second. Wikipedia already knows this stuff is a formalization of Occam’s razor. One article seems to attribute the formalization of that principle to Solomonoff, another one to Hutter. In addition, Solomonoff induction, that is essential for both, is not computable. Ugh. So Hutter and Rathmanner actually have the nerve to begin that article by talking about the problem of induction, when the goal is obviously to introduce concepts of computation theory? And they are already familiar with Occam’s razor, and aware of it having, at least probably, been formalized?
Okay then, but this doesn’t solve the problem of induction. They have not even formalized the problem of induction in a way that accounts for the logical structure of inductive inference, and leaves room for various relevance operators to take place. Nobody else has done that either, though. I should get back to this later.
“When artificial intelligence researchers attempted to capture everyday statements of inference using classical logic they began to realize this was a difficult if not impossible task.”
I hope nobody’s doing this anymore. It’s obviously impossible. “Everyday statements of inference”, whatever that might mean, are not exclusively statements of first-order logic, because Russell’s paradox is simple enough to be formulated by talking about barbers. The liar paradox is also expressible with simple, practical language.
The omitted information in this approach is information with a high Kolmogorov complexity, which is omitted in favor of information with low Kolmogorov complexity. A very rough analogy would be to describe humans as having a bias towards ideas expressible in few words of English in favor of ideas that need many words of English to express. Using Kolmogorov complexity for sequence prediction instead of English language for ideas in the construction gets rid of the very many problems of rigor involved in the latter, but the basic idea is pretty much the same. You look into things that are briefly expressible in favor of things that must be expressed in length. The information isn’t permanently omitted, it’s just depriorized. The algorithm doesn’t start looking at the stuff you need long sentences to describe before it has convinced itself that there are no short sentences that describe the observations it wants to explain in a satisfactory way.
One bit of context that is assumed is that the surrounding universe is somewhat amenable to being Kolmogorov-compressed. That is, there are some recurring regularities that you can begin to discover. The term “lawful universe” sometimes thrown around in LW probably refers to something similar.
Solomonoff’s universal induction would not work in a completely chaotic universe, where there are no regularities for Kolmogorov compression to latch on. You’d also be unlikely to find any sort of native intelligent entities in such universes. I’m not sure if this means that the Solomonoff approach is philosophically untenable, but needing to have some discoverable regularities to begin with before discovering regularities with induction becomes possible doesn’t strike me as that great a requirement.
If the problem of context is about exactly where you draw the data for the sequence which you will then try to predict with Solomonoff induction, in a lawless universe you wouldn’t be able to infer things no matter which simple instrumentation you picked, while in a lawful universe you could pick all sorts of instruments, tracking the change of light during time, tracking temperature, tracking the luminousity of the Moon, for simple examples, and you’d start getting Kolmogorov-compressible data where the induction system could start figuring repeating periods.
The core thing “independent of context” in all this is that all the universal induction systems are reduced to basically taking a series of numbers as input, and trying to develop an efficient predictor for what the next number will be. The argument in the paper is that this construction is basically sufficient for all the interesting things an induction solution could do, and that all the various real-world cases where induction is needed can be basically reduced into such a system by describing the instrumentation which turns real-world input into a time series of numbers.
Okay. In this case, the article does seem to begin to make sense. Its connection to the problem of induction is perhaps rather thin. The idea of using low Kolmogorov complexity as justification for an inductive argument cannot be deduced as a theorem of something that’s “surely true”, whatever that might mean. And if it were taken as an axiom, philosophers would say: “That’s not an axiom. That’s the conclusion of an inductive argument you made! You are begging the question!”
However, it seems like advancements in computation theory have made people able to do at least remotely practical stuff on areas, that bear resemblance to more inert philosophical ponderings. That’s good, and this article might even be used as justification for my theory RP—given that the use of Kolmogorov complexity is accepted. I was not familiar with the concept of Kolmogorov complexity despite having heard of it a few times, but my intuitive goal was to minimize the theory’s Kolmogorov complexity by removing arbitrary declarations and favoring symmetry.
I would say, that there are many ways of solving the problem of induction. Whether a theory is a solution to the problem of induction depends on whether it covers the entire scope of the problem. I would say this article covers half of the scope. The rest is not covered, to my knowledge, by anyone else than Robert Pirsig and experts of Buddhism, but these writings are very difficult to approach analytically. Regrettably, I am still unable to publish the relativizability article, which is intended to succeed in the analytic approach.
In any case, even though the widely rejected “statistical relevance” and this “Kolmogorov complexity relevance” share the same flaw, if presented as an explanation of inductive justification, the approach is interesting. Perhaps, even, this paper should be titled: “A Formalization of Occam’s Razor Principle”. Because that’s what it surely seems to be. And I think it’s actually an achievement to formalize that principle—an achievement more than sufficient to justify the writing of the article.
Commenting the article:
“When artificial intelligence researchers attempted to capture everyday statements of inference using classical logic they began to realize this was a difficult if not impossible task.”
I hope nobody’s doing this anymore. It’s obviously impossible. “Everyday statements of inference”, whatever that might mean, are not exclusively statements of first-order logic, because Russell’s paradox is simple enough to be formulated by talking about barbers. The liar paradox is also expressible with simple, practical language.
Wait a second. Wikipedia already knows this stuff is a formalization of Occam’s razor. One article seems to attribute the formalization of that principle to Solomonoff, another one to Hutter. In addition, Solomonoff induction, that is essential for both, is not computable. Ugh. So Hutter and Rathmanner actually have the nerve to begin that article by talking about the problem of induction, when the goal is obviously to introduce concepts of computation theory? And they are already familiar with Occam’s razor, and aware of it having, at least probably, been formalized?
Okay then, but this doesn’t solve the problem of induction. They have not even formalized the problem of induction in a way that accounts for the logical structure of inductive inference, and leaves room for various relevance operators to take place. Nobody else has done that either, though. I should get back to this later.
Commenting the article:
“When artificial intelligence researchers attempted to capture everyday statements of inference using classical logic they began to realize this was a difficult if not impossible task.”
I hope nobody’s doing this anymore. It’s obviously impossible. “Everyday statements of inference”, whatever that might mean, are not exclusively statements of first-order logic, because Russell’s paradox is simple enough to be formulated by talking about barbers. The liar paradox is also expressible with simple, practical language.