Re: [agi] My proposal for an AGI agenda
On Tue, 20 Mar 2007 15:29:06 -0400, Ben Goertzel [EMAIL PROTECTED] wrote: Well, **anything** can be dealt with in C++, it's just a matter of how awkward it is. nod :-) I don't want to become deeply involved in these language wars, because I cannot say honestly that my very limited experience in AI or AGI gives me qualification to speak authoritatively on this subject, but I'd guess C++ still rocks in this context. -gts - This list is sponsored by AGIRI: http://www.agiri.org/email To unsubscribe or change your options, please go to: http://v2.listbox.com/member/?list_id=303
Re: [agi] Priors and indefinite probabilities
On Wed, 14 Feb 2007 18:03:41 -0500, Ben Goertzel [EMAIL PROTECTED] wrote: Indeed, that is a cleaner and simpler argument than the various more concrete PI paradoxes... (wine/water, etc.) Yes. It seems to show convincingly that the PI cannot be consistently applied across the board, but can be heuristically applied to certain cases but not others as judged contextually appropriate. Cox addresses exactly what sort of cases in which it might be legitimately applied, and they are in his view rare and exceptional. Such cases exist for example in certain games of chance in which the necessary conditions for applying the PI are prescribed by the rules of the game or result from the design of the equipment. Those necessary conditions are in fact what the PI asks us to assume: not only must the possibilities be mutually exclusive and exhaustive, but they must also be *known a priori to be equiprobable*. We can say with confidence for example that each card in a shuffled deck is equally likely, but this is because in this trivial case equiprobability is prescribed by the rules of the game or result from the design of the equipment. The rest of the world is seldom so accommodating. The principle asks us to assume equiprobability when we have no a priori evidence of equiprobability -- that is its very function. So one might ask: what good is the PI if it can be invoked only when the possibilities are known a priori to be equiprobable? Cox writes of it only in a rhetorical sense, as if to say, You can invoke the PI but only if you already know that which it prescribes is true. -gts - This list is sponsored by AGIRI: http://www.agiri.org/email To unsubscribe or change your options, please go to: http://v2.listbox.com/member/?list_id=303
Re: [agi] Priors and indefinite probabilities
LEADING TO THE ONLY THING REALLY INTERESTING ABOUT THIS DISCUSSION: What interests me is that the Principle of Indifference is taken for granted by so many people as a logical truth when in reality it is fraught with logical difficulties. Gillies (2000) makes an analogy between the situation in probability theory concerning the Principle of Indifference and the situation that once existed in set theory concerning the Axiom of Comprehension. Like the Principle of Indifference, the Axiom of Comprehension seemed logical and intuitively obvious. That axiom states that all things which share a property form a set. What could be more logical and intuitively obvious? But the Axiom of Comprehension led to the Russell Paradox, and a crisis in set theory. Similarly the Principle of Indifference (and its predecessor the Principle of Insufficient Reason) led to numerous difficulties, (e.g., the Bertrand Paradoxes, and arguments such as Cox's). Subsequently we saw a schism in probability theory. The classical theory was discredited, including the classical interpretation of Bayes' Theorem, and replaced with at least four different alternative interpretations. Among bayesians, one might say De Finetti and Ramsey and the subjectivists helped rescue bayesianism from the jaws of (philosophical) death, by separating bayesianism from that albatross around its neck which is the Principle of Indifference. -gts - This list is sponsored by AGIRI: http://www.agiri.org/email To unsubscribe or change your options, please go to: http://v2.listbox.com/member/?list_id=303
Re: [agi] Priors and indefinite probabilities
On Thu, 15 Feb 2007 11:21:25 -0500, Ben Goertzel [EMAIL PROTECTED] wrote: I think it's been a pretty long time since the PI was taken by any serious thinkers as a logical truth, though... Objective bayesianism stands or falls (vs subjective bayesianism) on this question of whether the PI is a valid logical principle. And as far as I can tell objective bayesians certainly try to defend it as such. The PI is a main tenet of objective bayesianism; perhaps even its defining characteristic. Concerning physical entropy, the PI works well as a heuristic in certain related applications relevant to the physical sciences, which is why some physicists such as Jaynes were so fond of it. (Interestingly, though, Cox is a physicist and he is apparently not so fond of it.) Jaynes points out accurately that physicists have used the PI on numerous occasions to make accurate predictions, but Gillies points out that this heuristic success in no way proves the PI as a logical principle; if that were true then no empirical measurements would be needed to establish the veracity of their related hypotheses. One might ask why objective bayesianism is still attractive to many. This I think is a very interesting question. I believe it has something to do with the sociology of science, where pragmatic considerations often take precedence over philosophy. Scientists, especially natural scientists, have a strong need to communicate mathematical ideas in an objective manner. Objective bayesianism offers the hope that a scientist can show his colleagues that a hypothesis is true at some *objective* level of credibility. That hope of objectivity is not present under subjective bayesianism, even if subjective bayesianism might have a more solid philosophical footing. For the same reason I think it's still true that most natural scientists eschew bayesianism whenever possible, preferring to think and communicate in terms of objectivist interpretations. -gts - This list is sponsored by AGIRI: http://www.agiri.org/email To unsubscribe or change your options, please go to: http://v2.listbox.com/member/?list_id=303
Re: [agi] Priors and indefinite probabilities
On Thu, 15 Feb 2007 12:21:22 -0500, Ben Goertzel [EMAIL PROTECTED] wrote: As I see it, science is about building **collective** subjective understandings among a group of rational individuals coping with a shared environment That is consistent with the views of de Finetti and other subjectivists. In their view our posteriors all converge in the end anyway, so it shouldn't matter if there are no 'objective' probabilities. However, my view is not the most common one, I would suppose... I'm quite sure you're correct about that. A minority subjectivist, attempting to communicating his bayesian conclusions to an non-subjectivist colleague in the majority, could be met with the disconcerting response that his numbers are mere statements about his psychology. :/ Thus there exists a strong disincentive to be subjectivist in the natural sciences, no matter the philosophical consequences. -gts - This list is sponsored by AGIRI: http://www.agiri.org/email To unsubscribe or change your options, please go to: http://v2.listbox.com/member/?list_id=303
Re: [agi] Priors and indefinite probabilities
So none of this is very new ;-) No. :) Also your idea of collective subjective understandings sounds similar to something I read about an 'inter-subjective' interpretation of probability theory, which purports to stand somewhere between objective bayesianism and subjective bayesianism. Lots of people with different ideas... By the way, did Lakatos take a stand on these questions? I.e., did he endorse any particular interpretation separate from any observations he may have made about their development? PS I've been getting multiple copies of your posts. Not sure if the problem is here or there but thought I would bring it to your attention. -gts - This list is sponsored by AGIRI: http://www.agiri.org/email To unsubscribe or change your options, please go to: http://v2.listbox.com/member/?list_id=303
Re: [agi] Enumeration of useful genetic biases for AGI
On Tue, 13 Feb 2007 21:28:53 -0500, Ben Goertzel [EMAIL PROTECTED] wrote: Toward that end, it would be interesting to have a systematic list somewhere of the genetic biases that are thought to be mostimportant for structuring human cognition. Does anyone know of a well-thought-out list of this sort. Of course I could make one by surveying the cognitive psych literature,but why reinvent the wheel? Your email acquaintance mentioned Kant. You may want to look at Kant's categories, in his Critique of Pure Reason. These are the 'Categories of the Understanding' by which Kant thought the mind structures cognition: Quantity *Unity *Plurality *Totality Quality *Reality *Negation *Limitation Relation *Inherence and Subsistence (substance and accident) *Causality and Dependence (cause and effect) *Community (reciprocity) Modality *Possibility *Existence *Necessity -gts - This list is sponsored by AGIRI: http://www.agiri.org/email To unsubscribe or change your options, please go to: http://v2.listbox.com/member/?list_id=303
Re: [agi] Priors and indefinite probabilities
Tying together recent threads on indefinite probabilities and prior distributions (PI, maxent, Occam)... For those who might not know, the PI (the principle of indifference) advises us, when confronted with n mutually exclusive and exhaustive possibilities, to assign probabilities of 1/n to each of them. In his book _The Algebra of Probable Inference_, R.T. Cox presents a convincing disproof of the PI when n = 2. I'm confident his argument applies for greater values of n, though of course the formalism would be more complicated. His argument is by reductio ad absurdum; Cox shows that the PI leads to an absurdity. (Not just an absurdity in his view, but a monstrous absurdity :-) The following quote is verbatim from his book, except that in the interest of clarity I have used the symbol to mean and instead of the dot used by Cox. The symbol v means or in the sense of and/or. Also there is an axiom used in the argument, referred to as Eq. (2.8 I). That axiom is (a v ~a) b = b. Cox writes, concerning two mutually exclusive and exhaustive propositions a and b... == ...it is supposed that a | a v ~a = 1/2 for arbitrary meanings of a. In disproof of this supposition, let us consider the probability of the conjunction a b on each of the two hypotheses, a v ~a and b v ~b. We have a v b | a v ~a = (a | a v ~a)[b | (a v ~a) a] By Eq (2.8 I) (a v ~a) a = a and therefore a b | a v ~a = (a | a v ~a) (b | a) Similarly a b | b v ~b = (b | b v ~b) (a | b) But, also by Eq. (2.8 I), a v ~a and b v ~b are each equal to (a v ~a) (b v ~b) and each is therefore equal to the other. Thus a b | b v ~b = a b | a v ~a and hence (a | a v ~a) (b | a) = (b | b v ~b) (a | b) If then a | a v ~a and b | b v ~b were each equal to 1/2, it would follow that b | a = a | b for arbitrary meanings of and b. This would be a monstrous conclusion, because b | a and a | b can have any ratio from zero to infinity. Instead of supposing that a | a v ~a = 1/2, we may more reasonably conclude, when the hypothesis is the truism, that all probabilities are entirely undefined except these of the truism itself and its contradictory, the absurdity. This conclusion agrees with common sense and might perhaps have been reached without formal argument, because the knowledge of a probability, though it is knowledge of a particular and limited kind, is still knowledge, and it would be surprising if it could be derived from the truism, which is the expression of complete ignorance, asserting nothing. === -gts - This list is sponsored by AGIRI: http://www.agiri.org/email To unsubscribe or change your options, please go to: http://v2.listbox.com/member/?list_id=303
Re: [agi] Betting and multiple-component truth values
On Sat, 10 Feb 2007 21:40:28 -0500, Benjamin Goertzel [EMAIL PROTECTED] wrote: Sorry it took me so long to get to this message... About the Principle of Indifference and probability theory... The question is what should an AGI system does when the data available to it appears to support multiple contradictory conclusions. It has to decide, somehow. Yes, I understand. The PI is one way to decide... Yes, and there is nothing particularly wrong with the PI, I think, provided that one understands it is not some kind of a priori 'logical truth' handed down to us from the heavens. That is, I think there is no logical sin in using some other method, just as subjective bayesians have been telling the world since about 1926, much to the chagrin of objective bayesians. I note that the Occam prior connects more closely to neuroscience than the PI, in that there are plausible arguments the brain uses an energy minimization heuristic in some cases. Read Montague makes an argument in this direction in: I would need to learn more about Montague's idea to understand what he means about the neuroscience connection, but it sounds reasonable. However, when multiple choices seem to have roughly equivalent complexity, then the Occam prior basically degenerates to the PI. This goes back to my earlier idea that equivalent complexity (or equivalent information) takes on a different practical meaning in the special case in which there is no information at all, i.e., when one is in a state of total ignorance, which is the case when the PI might be invoked. Under such special circumstances I think one might say All bets are off. Think as thou wilt, within the bounds of reason. This at least seems to me a reasonable position for humans to take here (and it is consistent with Cox, I think). What this idea might mean for AGI is a different question, of course, and I understand that is the question on your mind. I'll need also to read the paper by Zurek and others... thanks. And, just as with the PI, these more sophisticated approaches must be applied correctly and intelligently to be useful. Yes. -gts - This list is sponsored by AGIRI: http://www.agiri.org/email To unsubscribe or change your options, please go to: http://v2.listbox.com/member/?list_id=303
Re: [agi] conjunction fallacy
On Sun, 11 Feb 2007 11:41:31 -0500, Richard Loosemore [EMAIL PROTECTED] wrote: P.S. This isn't the first time this topic has come up. For a now famous example, see my essay at http://sl4.org/archive/0605/14748.html and the follow-up at http://sl4.org/archive/0605/14773.html. The link to your essay didn't work. I read the thread at the second link. In general I found myself in agreement with your detractors, for example Eliezer and Ben. -gts - This list is sponsored by AGIRI: http://www.agiri.org/email To unsubscribe or change your options, please go to: http://v2.listbox.com/member/?list_id=303
Re: [agi] Betting and multiple-component truth values
Yesterday I received from amazon.com a copy of Cox's book _The Algebra of Probable Inference_. (Thanks for the recommendation, Ben.) In his preface Cox expresses his indebtedness to Keynes, and Keynes' influence is obvious throughout. For this reason I was expecting to find somewhere within the text a Keynesian-like attempt to rehabilitate the Principle of Indifference. However in this respect Cox breaks clearly from Keynes. Cox offers a strong and clear argument against the principle, starting at the bottom of page 31 and extending to about the middle of page 33 (in my paperback edition). Briefly, his argument is that the conditions necessary for applying the principle of indifference are exceptional and rare. They are present only for example in such trivial cases as certain games of chance in which the necessary conditions are prescribed by the rules of the game or result from the design of the equipment. Cox offers a formal disproof of the principle in the case in which there exist two mutually exclusive outcomes and nothing else is known. In such situations the principle prescribes that we assign prior probabilities of .5 to each outcome. Cox shows this to be absurd and unfounded, and writes this about his own conclusion: This conclusion agrees with common sense and might perhaps have been reached without formal argument, because the knowledge of a probability, though it is knowledge of a particular and limited kind, is still knowledge, and it would be surprising if it could be derived from the truism, which is the expression of complete ignorance, asserting nothing. Indeed! -gts - This list is sponsored by AGIRI: http://www.agiri.org/email To unsubscribe or change your options, please go to: http://v2.listbox.com/member/?list_id=303
Re: [agi] conjunction fallacy
Eliezer offered this apparent example of the fallacy: ** Two independent sets of professional analysts at the Second International Congress on Forecasting were asked to rate, respectively, the probability of A complete suspension of diplomatic relations between the USA and the Soviet Union, sometime in 1983 or A Russian invasion of Poland, and a complete suspension of diplomatic relations between the USA and the Soviet Union, sometime in 1983. The second set of analysts responded with significantly higher probabilities. ** I think it's worth noting here that while this example suggests very strongly that humans are susceptible to the conjunction fallacy (and I agree that they are) no individual analyst in this example can be *proved* to have actually committed the fallacy. This is another way of saying that no analyst was incoherent in the De Finetti sense, i.e., no analyst made himself susceptible to a dutch book. I think it was Pei who pointed out that the situation would be much different if the analysts in either group were asked to assign probabilities to *both* hypotheses. In that case some fraction of the analysts would likely have committed the fallacy in a provable way; some of them would have failed to satisfy the coherency constraint and thus made themselves vulnerable to a dutch book. I fail to see why an AGI must be so vulnerable, even with modest resources. An AGI could atomize the second hypothesis into its two constituent hypotheses: A: suspension of relations and B: invasion of Poland and then apply coherent probabilistic reasoning in such a way that the constraint P(A) P(AB) P(B) is not violated. -gts - This list is sponsored by AGIRI: http://www.agiri.org/email To unsubscribe or change your options, please go to: http://v2.listbox.com/member/?list_id=303
Re: [agi] conjunction fallacy
On Sat, 10 Feb 2007 13:15:19 -0500, Benjamin Goertzel [EMAIL PROTECTED] wrote: Look, the susceptibility of humans to dutch books is clear... I was not arguing to the contrary, Ben. As I wrote below: I think it's worth noting here that while this example suggests very strongly that humans are susceptible to the conjunction fallacy (and I agree that they are)... I agree humans are susceptible to the conjunction fallacy. -gts You are correct that the specific protocol underlying the psych experiment Eli cited did not show individuals being personally incoherent, but there are many other psych experiments that do. I don't feel like digging up the references, but they are there in the heuristics and biases literature. -- Ben On 2/10/07, gts [EMAIL PROTECTED] wrote: Eliezer offered this apparent example of the fallacy: ** Two independent sets of professional analysts at the Second International Congress on Forecasting were asked to rate, respectively, the probability of A complete suspension of diplomatic relations between the USA and the Soviet Union, sometime in 1983 or A Russian invasion of Poland, and a complete suspension of diplomatic relations between the USA and the Soviet Union, sometime in 1983. The second set of analysts responded with significantly higher probabilities. ** I think it's worth noting here that while this example suggests very strongly that humans are susceptible to the conjunction fallacy (and I agree that they are) no individual analyst in this example can be *proved* to have actually committed the fallacy. This is another way of saying that no analyst was incoherent in the De Finetti sense, i.e., no analyst made himself susceptible to a dutch book. I think it was Pei who pointed out that the situation would be much different if the analysts in either group were asked to assign probabilities to *both* hypotheses. In that case some fraction of the analysts would likely have committed the fallacy in a provable way; some of them would have failed to satisfy the coherency constraint and thus made themselves vulnerable to a dutch book. I fail to see why an AGI must be so vulnerable, even with modest resources. An AGI could atomize the second hypothesis into its two constituent hypotheses: A: suspension of relations and B: invasion of Poland and then apply coherent probabilistic reasoning in such a way that the constraint P(A) P(AB) P(B) is not violated. -gts - This list is sponsored by AGIRI: http://www.agiri.org/email To unsubscribe or change your options, please go to: http://v2.listbox.com/member/?list_id=303 - This list is sponsored by AGIRI: http://www.agiri.org/email To unsubscribe or change your options, please go to: http://v2.listbox.com/member/?list_id=303 - This list is sponsored by AGIRI: http://www.agiri.org/email To unsubscribe or change your options, please go to: http://v2.listbox.com/member/?list_id=303
Re: [agi] conjunction fallacy
On Sat, 10 Feb 2007 13:41:33 -0500, Richard Loosemore [EMAIL PROTECTED] wrote: The meat of this argument is all in what exact type of AGI you claim is the best, of the two suggested above. The best AGI in this context would be one capable of avoiding the conjunction fallacy, of course, but neither of those you described even addressed the question of whether the two outcomes together have a greater, lesser, or equal probability than either of them separately. The conjunction fallacy is a sort of mental illusion, brought about by our mistaken use of certain heuristics. Heuristics are all very well and good, but I should think any sophisticated AGI would not take them as gospel in situations in which they contradict of the axioms of probability. -gts - This list is sponsored by AGIRI: http://www.agiri.org/email To unsubscribe or change your options, please go to: http://v2.listbox.com/member/?list_id=303
Re: [agi] Betting and multiple-component truth values
On Sat, 10 Feb 2007 13:59:27 -0500, Jef Allbright [EMAIL PROTECTED] wrote: gts wrote: I'm not expecting essentially perfect coherency in AGI. I understand perfection is out of reach. Thanks for quoting me here, Jef. I think Ben may have thought I believe something differently. I understand and agree with everyone here that perfect coherency is not feasible in AGI. My question to you was whether, as a professed C++ developer, you are familiar with the well-known impracticality of certifying a non-trivial software product to be essentially free of unexpected failure modes, and if so, do you see a similarity to your question of coherent reasoning by machine intelligence? Sure, an analogy can be made. In a similar vein, do you think you understand Ben's comment about the problem being NP-hard? Sure... ...our differences here seem to be a matter of degree I am less optimistic about the possibility of developing a smart, accurate, probabilistic AGI than I am about developing one that totally *smokes* humanity in measures of probabilistic (De Finetti) coherency. By the way, De Finetti used the word coherent in the very standard sense meaning that all the pieces must fit together from all possible points of view (within all possible contexts.) I was explaining that here yesterday. This same concept of coherence is the basis of the axioms of probability... Yes. ... and the principle of indifference. No. Understand this underlying concept and you may understand the others. I understand it, Jef. But do you? The principle of indifference is not derived from or implied in any way by De Finetti coherency. De Finetti had no use for the idea. Neither do I. -gts - This list is sponsored by AGIRI: http://www.agiri.org/email To unsubscribe or change your options, please go to: http://v2.listbox.com/member/?list_id=303
Re: [agi] Correction: Betting and multiple-component truth values
Correction: Needed to add [the idea that] below. - Jef Got it. I understand it, Jef. But do you? The principle of indifference is not derived from or implied in any way by De Finetti coherency. De Finetti had no use for the idea. Neither do I. That's like saying you have no use for [the idea that] a balance scale reads zero when both pans are empty. Your beef is not just with me; it is with Bruno De Finetti and Frank P. Ramsey and their modern followers in the subjectivist school of probability theory, most of whom call themselves subjective bayesians. At the risk of mixing metaphors: A subjectivist has no use for the idea of an empty scale no matter how it might balance, because after all there is nothing there to weigh. -gts - This list is sponsored by AGIRI: http://www.agiri.org/email To unsubscribe or change your options, please go to: http://v2.listbox.com/member/?list_id=303
Re: [agi] Correction: Betting and multiple-component truth values
On Sat, 10 Feb 2007 15:26:01 -0500, Benjamin Goertzel [EMAIL PROTECTED] wrote: This is true, but, subjective Bayesianism does not give you any suggestion as to what prior distribution to use in place of the maximum-entropy prior. It just says that you can use any prior you want so long as you use it consistently... Yes. So, for AGI purposes, the subjective Bayesian approach is not enough... Seems that way, but on the other hand, subjective bayesianism seems to me to be closer to the way humans actually think. Subjective bayesians are not constrained under some supposed force of logic to make their probabilistic judgements conform to an idealized objective standard. -gts - This list is sponsored by AGIRI: http://www.agiri.org/email To unsubscribe or change your options, please go to: http://v2.listbox.com/member/?list_id=303
Re: [agi] Betting and multiple-component truth values
On Sat, 10 Feb 2007 15:27:23 -0500, Jef Allbright [EMAIL PROTECTED] wrote: On the contrary, a subjectivist understands that even to pose a question, one must have some prior. That observation does not speak to the question at hand concerning the principle of indifference. The principle of indifference is seen as a 'logical principle' only under objective bayesianism. Under subjective bayesianism it is at most a heuristic device. Subjectivists know better than to believe they are bound by some 'universal principle of logic' (to use your term) to invoke the principle of indifference under conditions of total ignorance about the true state of nature, which is of course the only condition under which it can be invoked. You were wrong to suggest earlier that the principle of indifference can be derived from De Finetti coherence. The axioms of probability can be derived from coherence but the principle of indifference is certainly not one of them. You're also confusing zero with nothing. Nope. -gts - This list is sponsored by AGIRI: http://www.agiri.org/email To unsubscribe or change your options, please go to: http://v2.listbox.com/member/?list_id=303
Re: [agi] Betting and multiple-component truth values
Jef, I understand it, Jef. But do you? The principle of indifference is not derived from or implied in any way by De Finetti coherency. De Finetti had no use for the idea. Neither do I. That's like saying you have no use for [the idea that] a balance scale reads zero when both pans are empty. Here is something Frank P. Ramsey wrote about the principle of indifference after discovering (independently of Bruno De Finetti) that coherence was sufficient to derive the axioms of probability: Secondly, the Principle of Indifference can now be altogether dispensed with; we do not regard it as belonging to formal logic to say what should be a man's expectation of drawing a white or a black ball from an urn; his original expectations may within the limits of consistency be any he likes; all we have to point out is that if he has certain expectations he is bound in consistency to have certain others. This is simply bringing probability into line with ordinary formal logic, which does not criticize premisses but merely declares that certain conclusions are the only ones consistent with them. To be able to turn the Principle of Indifference out of formal logic is a great advantage; for it is fairly clearly impossible to lay down purely logical conditions for its validity, as is attempted by Mr Keynes. -F.P. Ramsey (1926) Truth and Probability Ramsey was a great genius, in my opinion. I suggest you read his paper above. It's available on the net if you look for it. I provided a link in some earlier message here. -gts - This list is sponsored by AGIRI: http://www.agiri.org/email To unsubscribe or change your options, please go to: http://v2.listbox.com/member/?list_id=303
Re: [agi] Betting and multiple-component truth values
On Wed, 07 Feb 2007 18:37:52 -0500, Ben Goertzel [EMAIL PROTECTED] wrote: This is simply a re-post of my prior post, with corrected terminology, but unchanged substance: Thanks! Very helpful. Now that you have a better understanding of dutch books, I wonder if you still feel the De Finetti coherence constraint is as formidable as you may have first thought. I haven't seen your code but I would be surprised if Novamente is really incoherent. Probably you can show that the prices of the bets set by Gambler and Meta-gambler respectively are consistent and related in such a way that the House cannot make a dutch book against the Gambler and Meta-Gambler seen as a team; that is, that the House cannot force Novamente to lose automatically no matter what is true. The House might attempt, for example, buying the operational subjective probability bet (p) from Gambler while simultaneously selling the g bet to Meta-Gambler, or vice versa, in such a way as to force the team of Gamblers to lose money. These transactions could perhaps take place at separate times. For example the House might attempt to buy one bet after n observations of S and sell the other after n+x observations of S. This doesn't really add anything practical to the indefinite probabilities framework as already formulated, it just makes clearer the interpretation of the indefinite probabilities in terms of de Finetti style betting games. Yes, thanks for the illustration. Note that coherency does not constrain one to be especially accurate in one's judgemental probabilities. A coherent entity needn't be very smart about the true state of nature. The coherency constraint merely defines the outer limits of what one may rationally consider possible. -gts - This list is sponsored by AGIRI: http://www.agiri.org/email To unsubscribe or change your options, please go to: http://v2.listbox.com/member/?list_id=303
Re: [agi] Betting and multiple-component truth values
On Fri, 09 Feb 2007 11:19:52 -0500, Ben Goertzel [EMAIL PROTECTED] wrote: Note that coherency does not constrain one to be especially accurate in one'sjudgemental probabilities. A coherent entity needn't be very smart about thetrue state of nature. The coherency constraint merely defines the outer limitsof what one may rationally consider possible. This is incorrect, I believe. Coherency requires one to be reasonably consistent in one's assignment of probabilities to various interdependent outcomes, otherwise a dutch book can be made against one. That would depend on the meaning of reasonably consistent but in any case I believe this is at the root of our differences of opinion about De Finetti coherence. You may mean something else by coherence, but as I understand De Finetti it does not entail anything like in-depth knowledge or omniscience about the world of complex interdependences. To be coherent one need only avoid self-contradiction. Here is a quote from a source I've found very helpful in understanding De Finetti coherence: Naturally, coherence does not determine a single degree of rational belief but leaves open a wide variety of choices... The idea here is that we have to make sure our various degrees of belief fit together so to avoid the 'contradiction' of a Dutch book being made against us. The term 'coherence' is now generally preferred... [1] Thus, to be coherent, we need to ensure that our beliefs fit together (logically). This is separate from considerations about whether those beliefs are actually true. This coherency constraint is entirely subjective, a sort of first order rational constraint which comes before other logical constraints which might be related to what is actually true 'out there' in the world of complex interdependencies, which I certainly do not deny exists. Guaranteed losses to dutch books in De Finetti-style arguments are not evidence of a lack of knowledge about the complex interdependencies in the world --- they are evidence of self-contradiction, evidence of incoherent thinking on the part of the better no matter his degree of knowledge about the world. To avoid a dutch book, an entity need only check first before acting to make sure its relevant assumptions are logically compatible. And in the case where it has no relevant assumptions then no book can be made against it. [Concerning the interesting conjunction fallacy post by Eliezer, I should read it again but under the assumptions given, (concerning Kolmogorov complexity and so forth), it seemed to me that the example as stated was not actually an example of fallacious reasoning.] 1. D. Gillies (2000)_Philosophical Theories of Probability_, pg 59 -gts - This list is sponsored by AGIRI: http://www.agiri.org/email To unsubscribe or change your options, please go to: http://v2.listbox.com/member/?list_id=303
Re: [agi] Betting and multiple-component truth values
Well, although I am not an AI developer, I am a C++ application developer and I know I or any reasonably skilled developer could write task-specific applications that would be extremely coherent in the De Finetti sense (applicable to making probabilistic judgements in horse-racing, casinos, the stockmarket, whatever). These applications would make mincemeat of humans in any test of coherence. Such applications already exist, come to think of it. So I think people should be optimistic about coherence in AGI, not pessimistic. -gts - This list is sponsored by AGIRI: http://www.agiri.org/email To unsubscribe or change your options, please go to: http://v2.listbox.com/member/?list_id=303
Re: [agi] Betting and multiple-component truth values
On Wed, 07 Feb 2007 20:40:27 -0500, Charles D Hixson [EMAIL PROTECTED] wrote: I suspect you of mis-analyzing the goals and rewards of casino gamblers I'm not sure whether or not this speaks to the points that you are attempting to raise, but it certainly calls into question comments about stupid bets. Well, the lottery isn't a casino, so perhaps you are correct, but I would be suspicious about calculating values based solely on the money. The point I was making, and it applies equally well to lottery bets as it does to casino bets, is that such bets are not evidence of incoherence where incoherence is defined (by De Finetti) as vulnerability to dutch books. A dutch book occurs when an incoherent thinker is forced to lose as a result of his inconsistent judgmental probabilities, no matter the outcome. Such bets are worse than stupid. :) I gave an example of a Dutch book in a post to Russell in which an incoherent thinker assigns a higher probability to intelligent life on Mars than to mere life on Mars. Since the first hypothesis can be true only if the second is true, it is incoherent to assign a higher probability to the first than to the second. Coherence is basically just common sense applied to probabilistic reasoning. I'm dismayed to learn from Ben that coherence is so difficult to achieve in AGI. -gts - This list is sponsored by AGIRI: http://www.agiri.org/email To unsubscribe or change your options, please go to: http://v2.listbox.com/member/?list_id=303
Re: [agi] Betting and multiple-component truth values
On Wed, 07 Feb 2007 16:51:18 -0500, Ben Goertzel [EMAIL PROTECTED] wrote: In fact I have been thinking about how one might attempt a dutch book against Novamente involving your multiple component values, but I do not yet fully understand b. My impression at the moment is that b is similar to 'power' in conventional statistics -- a real number from 0 to 1 that roughly speaking acts as a measure of the robustness of the analysis. Fair comparison? The power of a statistical hypothesis test measures the test's ability to reject the null hypothesis when it is actually false ... this has very little to do with indefinite or imprecise probabilities... Let me ask you in a different way: Can b be regarded as a measure of Novamente's confidence in p? All other things being equal, does b increase with N? -gts - This list is sponsored by AGIRI: http://www.agiri.org/email To unsubscribe or change your options, please go to: http://v2.listbox.com/member/?list_id=303
Re: [agi] Betting and multiple-component truth values
On Thu, 08 Feb 2007 09:26:28 -0500, Pei Wang [EMAIL PROTECTED] wrote: In simple cases like the above one, an AGI should achieve coherence with little difficulty. What an AGI cannot do is to guarantee coherence in all situations, which is impossible for human beings, neither --- think about situations where the incoherence of a bet setting needs many steps of inference, as well as necessary domain knowledge, to reveal. Yes, but as I wrote to Ben yesterday, it is not possible to make a dutch book against an AGI that does not pretend to have knowledge it does not have. So an AGI can be perfectly coherent, to *some* degree of knowledge, provided it knows its own bounds. And such a modest AGI would certainly be more trustworthy, especially if it were employed in such fields as national defense, where incoherent reasoning could lead to disaster. -gts - This list is sponsored by AGIRI: http://www.agiri.org/email To unsubscribe or change your options, please go to: http://v2.listbox.com/member/?list_id=303
Re: [agi] Betting and multiple-component truth values
On Thu, 08 Feb 2007 10:22:19 -0500, Ben Goertzel [EMAIL PROTECTED] wrote: Well, if the scope of a mind is narrowed enough, then it can be more coherent. Right, I understand there is a definite trade-off here between knowledge or scope and coherency, due mainly to resource limitations. The best we can hope for is that an AGI might be more coherent than us, but this is by no means assured. On a slightly different but closely related subject... Last night I was out having pizza with some others, trying to pretend to be interested in the conversation, while actually thinking about the posts we had exchanged earlier in the day. :) While munching on onions and pepperoni it occurred to me that the problem of achieving complete or near-complete coherency in AGI is closely related to the epistemological problem of obtaining knowledge where knowledge is defined as 'justified true belief'. Karl Popper's arguments against that possibility strike me as similar to and closely related to your arguments against the possibility of complete probabilistic coherency in AGI: any such attempt must lead to an infinite regress. So then I wondered to myself how Popper's alternative, non-justificationist epistemology might be applicable to AGI. Any thoughts on that subject? (I won't presume to educate you about Popper; I recall that you studied Philosophy of Science and so should know all about him.) -gts - This list is sponsored by AGIRI: http://www.agiri.org/email To unsubscribe or change your options, please go to: http://v2.listbox.com/member/?list_id=303
Re: [agi] Betting and multiple-component truth values
re: the right order of definition De Finetti's (and Ramsey's) main contribution was in showing that the formal axioms of probability can be derived entirely from considerations about people betting on their subjective beliefs under the relatively simple constraint of coherency. No other rational/logical constraints are needed, which is contrary to the suppositions of for example Keynes. This was I think a pretty remarkable discovery! I'm not yet sure what it means to derive the 'axioms of Novamente' in the same way, but I think it's pretty cool that Ben is attempting it. :) -gts - This list is sponsored by AGIRI: http://www.agiri.org/email To unsubscribe or change your options, please go to: http://v2.listbox.com/member/?list_id=303
Re: [agi] Betting and multiple-component truth values
On Tue, 06 Feb 2007 20:02:11 -0500, Ben Goertzel [EMAIL PROTECTED] wrote: Consistency in the sense of de Finetti or Cox is out of reach for a modest-resources AGI, in principle... Sorry to be the one to break the news... You used the word consistency instead of the word coherency that I was using, but assuming you mean them as synonyms, and assuming you're correct, then I think that really is terrible news for AGI and I wonder why you're even bothering with it. Coherency in the De Finetti sense is not very much different from coherency as the word is used in normal conversation, as when evaluating the words and mental states of people. Incoherent people are in worse shape than stupid. We put incoherent people in psychiatric facilities. -gts - This list is sponsored by AGIRI: http://www.agiri.org/email To unsubscribe or change your options, please go to: http://v2.listbox.com/member/?list_id=303
Re: [agi] Betting and multiple-component truth values
On Wed, 07 Feb 2007 10:57:04 -0500, Ben Goertzel [EMAIL PROTECTED] wrote: The dramatic probabilistic incoherency of humans is demonstrated by human behavior in casinos. You mean something more stringent than me by the word incoherency, then. Human betting behavior in casinos is stupid but it is not incoherent in the De Finetti sense as I understand it. It's easy to prove incoherence: one need only show how a dutch book can be made against the allegedly incoherent person. Vulnerability to dutch books is how incoherence is defined under the theory. Casino gamblers are stupid in so much as they place bets with unfavorable odds, but they do not by virtue of those stupid bets make themselves vulnerable to dutch books. One sometimes wins against unfavorable odds but it is never possible to beat a dutch book. In fact casinos do not even offer such betting situations. -gts - This list is sponsored by AGIRI: http://www.agiri.org/email To unsubscribe or change your options, please go to: http://v2.listbox.com/member/?list_id=303
Re: [agi] Betting and multiple-component truth values
Ben, Of course the world is an enormously complex relation of interdependencies between many causes and effects. I do not dispute that fact. I question however whether this should really be an important consideration in developing AGI. One's probabilistic judgements should always be justified, yes? And when a probabilistic judgement P(A) is justified only by one or more other probabilistic judgements [P(Q), P(R), and P(S), say] then one is not justified in assuming P(A) should have a value greater than [P(Q) * P(R) * P(S)]. Yes? If that coherency condition is not true for an AGI then I might have trouble trusting its probabilistic judgements. I do not much care in this case whether our AGI is correct in its probabilistic judgement about A (it may be ignorant about many facts of the world including many facts related to judgements about Q, R and S) but I do care whether our AGI is *justified* in its appraisal of P(A). Note that dutch books cannot be made against an AGI that does not claim to have knowledge it does not have. -gts - This list is sponsored by AGIRI: http://www.agiri.org/email To unsubscribe or change your options, please go to: http://v2.listbox.com/member/?list_id=303
Re: [agi] Betting and multiple-component truth values
On Wed, 07 Feb 2007 16:07:13 -0500, Ben Goertzel [EMAIL PROTECTED] wrote: only under an independence assumption. True, I did not make the independence assumption explicit. Note that dutch books cannot be made against an AGI that does not claim to have knowledge it does not have. That is true and important, and is why Pei and I and others use multiple-component truth values in our systems -- we explicitly track the weight of evidence associated with uncertainty estimates. I don't see how multiple-component truth values might block a fully developed Novamente from being vulnerable to dutch books, if that is what you are saying here. In fact I have been thinking about how one might attempt a dutch book against Novamente involving your multiple component values, but I do not yet fully understand b. My impression at the moment is that b is similar to 'power' in conventional statistics -- a real number from 0 to 1 that roughly speaking acts as a measure of the robustness of the analysis. Fair comparison? -gts - This list is sponsored by AGIRI: http://www.agiri.org/email To unsubscribe or change your options, please go to: http://v2.listbox.com/member/?list_id=303
Re: [agi] Betting and multiple-component truth values
Russell, I'm not suggesting that an omniscient player would not win over time as a result of its superior knowledge. I am suggesting that a non-omniscient player need not necessarily be bilked in the sense meant by De Finetti; that is, it needn't be forced to lose automatically due to dutch books made against it. To illustrate a dutch book: Say you believe in life on Mars with p=.1 and in intelligent life on Mars with p=.01. To De Finetti (and Ramsey), this is the same as saying you would pay 10 cents for a ticket worth $1 if there is life on Mars, and 1 cent for a ticket worth $1 if there is intelligent life on Mars. Also you believe these are fair bets such that you would be willing to take either side of either transaction. You are coherent here in the De Finetti sense no matter how right or wrong you may be about the probabilities of life on Mars. No dutch books can be made against you. No bookie can bilk you. Would you consider instead valuing the tickets such that the first is worth 1 cent and the second is worth 10 cents? No, you would not, because in that case you would be incoherent: someone (an omniscient bookie or otherwise) could exploit your incoherency by buying from you the first ticket and selling you the second, locking in a profit of at least 9 cents at your expense no matter what is true about life on Mars. That is a dutch book. Can entities with limited knowledge and resources be coherent in the sense described, thus avoiding being bilked by omniscient bookies seeking to make dutch books? I don't see why not. -gts - This list is sponsored by AGIRI: http://www.agiri.org/email To unsubscribe or change your options, please go to: http://v2.listbox.com/member/?list_id=303
Re: [agi] Betting and multiple-component truth values
On Tue, 06 Feb 2007 09:56:10 -0500, Russell Wallace [EMAIL PROTECTED] wrote: I'm not talking about dutch book, I'm talking about the following quoted from Ben's original post, emphasis added): I think Ben is talking about dutch books, at least implicitly. I think he wants to show that multiple-component truth values are consistent with a De Finetti-like subjectivist interpretation of probability. Dutch book considerations are central to that interpretation. -gts - This list is sponsored by AGIRI: http://www.agiri.org/email To unsubscribe or change your options, please go to: http://v2.listbox.com/member/?list_id=303
Re: [agi] Betting and multiple-component truth values
On Tue, 06 Feb 2007 11:18:09 -0500, Ben Goertzel [EMAIL PROTECTED] wrote: The scenario I described was in fact a dutch book scenario... The next step might then be to show how Novamente is constrained from allowing dutch books to be made against it. This would prove Novamente's probabilistic reasoning to be coherent in the sense meant by De Finetti. -gts - This list is sponsored by AGIRI: http://www.agiri.org/email To unsubscribe or change your options, please go to: http://v2.listbox.com/member/?list_id=303
Re: [agi] Betting and multiple-component truth values
On Tue, 06 Feb 2007 16:27:22 -0500, Jef Allbright [EMAIL PROTECTED] wrote: You would have to assume that statement 2 is *entirely* contingent on statement 1. I don't believe so. If statement S is only partially contingent on some other statement, or contingent on any number of other statements, then simple coherency demands only that we assign the p of S to be less than the p of any of those other statements on which S is contingent. It makes no difference for the sake of coherency how many of those other statements are known or in memory, nor does it matter whether our assigned probabilities match reality. I think coherency is probably a necessary but not a sufficient condition for intelligence. I hope it is not really outside the range of what is possible in AI. -gts - This list is sponsored by AGIRI: http://www.agiri.org/email To unsubscribe or change your options, please go to: http://v2.listbox.com/member/?list_id=303
Re: [agi] Probabilistic consistency
Inconsistency, though annoying, is a major driving force for learning and creativity. Along these lines I was reading an old research paper about subjective notions of randomness a few weeks back (sorry I don't have a reference). It seems back in the 30's, a radio station sponsored a series of experiments in ESP in which someone at the station would attempt to mentally broadcast to the listening audience a random number from 0 to 9. The audience then wrote to the station with their mental impressions of the random numbers. Their votes were tabulated. This experiment was repeated numerous times. The experiment failed -- no correlation was found to support the ESP hypothesis. So these other researchers used the data to analyze the subjective meaning of 'random number'. As it turned out, the number 7 was predicted about twice as often as any other number. The data was statistically significant with n something like 1800. From this one can infer that the typical human mind regards the number 7 as the 'most random' digit. These researchers posited a theory to explain the human penchant for 7 as most random (it was not clear if the theory was ad hoc or not, but I think it's interesting regardless): According to the theory, the numbers 2, 4, 6 and 8 are multiples of 2, which one might say makes them less random than 7 which is not a multiple of any other digit. 0 and 9 are endpoints on the 0-9 scale, which also makes them less random than 7. The number 5 is in the middle, which is non-random, etc. It seems that less can be said about 7 than about any other digit, and that the human mind considers this to be evidence that 7 is the most random. These considerations may not be exactly rational but apparently the human mind sees them as rational at some unconscious level. One might ask what this means in terms of AGI. Should an AGI also regard 7 as about twice as random as any other digit? Or would that be irrational and inconsistent with probability theory? I would suppose little considerations like these would make the difference between 'robot-like' and 'human-like'... -gts - This list is sponsored by AGIRI: http://www.agiri.org/email To unsubscribe or change your options, please go to: http://v2.listbox.com/member/?list_id=303
Re: [agi] Betting and multiple-component truth values
On Mon, 05 Feb 2007 02:03:21 -0500, Ben Goertzel [EMAIL PROTECTED] wrote: I thought of a way to define two-component truth values in terms of betting strategies (vaguely in the spirit of de Finetti). I think your thought-experiment here is ingenious! I'm not yet totally sure whether I agree with your set-up or your conclusions (I need to think about this more) but in general I applaud your effort to make subjectivist sense of multiple-component truth values. Kudos. :) -gts - This list is sponsored by AGIRI: http://www.agiri.org/email To unsubscribe or change your options, please go to: http://v2.listbox.com/member/?list_id=303
Re: [agi] Optimality of probabilistic consistency
On Sun, 04 Feb 2007 07:52:02 -0500, Pei Wang [EMAIL PROTECTED] wrote: However, the axioms of probability theory and interpretations of probability (frequentist, logical, subjective) all take a consistent probability distribution as precondition. Also I think the meaning of 'probabilistic consistency' might change according to the interpretation of probability. For example two subjectivist-like AGI's might arrive at different conclusions, at least early in the learning process, without probabilistic inconsistency. Such apparent inconsistencies are however prohibited under the logical interpretation. This I think may also go to the question of resources. I'm thinking a subjectivist (De Finetti-Ramsey inspired) AGI should require a different amount of resources than a logical (Keynes-Jaynes-Cox inspired) AGI. At the moment my conjecture is that implementations of the logical interpretation would require the greater resources in that it imposes more restraints, but I can also see some possible rationale for the converse. Ben, this is also why I was wondering why your hypothesis is framed in terms of both Cox and De Finetti. Unless I misunderstand Cox, their interpretations are in some ways diametrically opposed. De Finetti was a radical subjectivist while Cox is (epistemically) an ardent logical/objectivist (or so I gather). Apparently you see their ideas as complementary rather than mutually exclusive, which is interesting... is it because De Finetti's subjective interpretation gives a theoretical foundation to your use of [U,L] ranges in your quadruples? Another question on my mind is if and how it might be possible to design an AGI based entirely on the subjectivist ideas of De Finetti, an idea that I find very attractive. However I am at the moment stumped on that question; it may be true that no matter the philosophy of the programmer, he must for practical reasons implement something like a logical/objective interpretation of bayes' rule. Comments? -gts - This list is sponsored by AGIRI: http://www.agiri.org/email To unsubscribe or change your options, please go to: http://v2.listbox.com/member/?list_id=303
Re: [agi] Optimality of probabilistic consistency
A mathematical test for objectivity/subjectivity might be whether Novamente (or any AGI) could allow, in principle, for the possibility of different posterior probabilities on bayes rule as can happen under subjectivism. My thought is that a programmer is essentially forced for practical reasons to disallow that sort of inconsistency -- that he must implement an objective interpretation. The definition of 'probabilistic consistency' that I was using comes from ET Jaynes' book _Probability Theory - The Logic of Science_, page 114. These are Jaynes' three 'consistency desiderata' for a probabilistic robot: 1. If a conclusion can be reasoned out in more than one way, then every possible way must lead to the same result. 2. The robot takes into account all information relevant to the question. 3. The robot always represents equivalent states of information with equivalent plausibility assignments. Seems to me that strict enforcement of these desiderata (especially #3) would make the robot an objective bayesian as opposed to a subjective bayesian in the De Finetti sense. -gts - This list is sponsored by AGIRI: http://www.agiri.org/email To unsubscribe or change your options, please go to: http://v2.listbox.com/member/?list_id=303
Re: [agi] Optimality of probabilistic consistency
On Sun, 04 Feb 2007 11:10:57 -0500, Pei Wang [EMAIL PROTECTED] wrote: I don't think any intelligent system (human or machine) can achieve any of the three desiderata, except in trivial cases. I have no doubt you and Ben are correct on this point. Enormous resources would be required for an ideal version of Jaynes' objective bayesian model of the probabilistic robot, which is one reason why I think it might be important to consider which philosophical interpretation to emulate. Personally I would be inclined to allow exceptions to Jaynes' second and third desiderata. The reason for compromising the second is easy enough to see: it is simply not always feasible to have and consider all the relevant information before making a decision. Any compromise of the third desiderata (that our AGI must by some supposed force of objective logic always represent equivalent states of information with equivalent plausibility assignments) is more controversial. People of Keynesian/logical persuasion might cry heresy, but I would respond that all is not lost; that these apparent sacrifices still leave us with the perfectly reasonable and coherent subjectivist account of De Finetti. The question then would be how to go about implementing it. I'm a bit skeptical that it can be done, but, unlike you and Ben, I am by no means an expert in the field of AI. Is it possible to program AGI without forcing it to abide by the tenets of objective/logical bayesianism? Subjectivists like De Finetti and Ramsey define probability as degree of belief but unlike the objective/logical bayesians they measure it according to an agent's *willingness to act* on said degrees of belief, (as opposed to some supposed calculable mental barometer of rationally determined belief separate from the will). Even though I might support the subjectivist programme philosophically, I'm not sure if or how a programmer might get a handle on this subjective 'willingness to act', as distinct from the logical restraints that objective bayesians would already seek to impose. -gts - This list is sponsored by AGIRI: http://www.agiri.org/email To unsubscribe or change your options, please go to: http://v2.listbox.com/member/?list_id=303
Re: [agi] Optimality of probabilistic consistency
On Sun, 04 Feb 2007 13:15:27 -0500, Pei Wang [EMAIL PROTECTED] wrote: none of the existing AGI project is designed [according to the tenets of objective/logical bayesianism] Hmm. My impression is that to whatever extent AGI projects use bayesian reasoning, they usually do so in a way that satisfies the tenets of objective/logical bayesianism. I hope you understand I mean objective in the epistemic and not the physical sense. I see objective/logical bayesianism embodied in Jaynes' third desiderata of probabilistic consistency, a principle that I doubt all AGI projects reject, assuming any do. Those projects which do allow for any compromise of that principle, if they exist, would I think be better described as implementations of subjective rather than objective bayesianism. Of course this is only according to my understanding of these two schools of bayesian thought and their differences, which may be different from yours. -gts - This list is sponsored by AGIRI: http://www.agiri.org/email To unsubscribe or change your options, please go to: http://v2.listbox.com/member/?list_id=303
Re: [agi] Relevance of Probability
On Sun, 04 Feb 2007 12:46:06 -0500, Richard Loosemore [EMAIL PROTECTED] wrote: If we knew for sure that the human mind was using something like a formalized system (and not the messy nonlinear stuff I described), then we could quite comfortably say Hey, let's do the same, but simpler and maybe even better. My problem is, of course, that the human mind may well not be doing it that way... I'm somewhat sympathetic to that point of view, Richard, in case it's any consolation to you. :) Your words remind me of the criticism that the subjectivist theorist F.P. Ramsey had of the logical theories of J.M. Keynes, which I mentioned here yesterday or the day before and which I find very persuasive. Keynes argued for the existence of something he called probability relations. These relationships were supposed to be perceivable by the human mind in the same manner in which it sees logical relationships. For Keynes, probability theory was in fact a sort of extension of deductive logic in which probable conclusions were partially entailed by their premises. The degree of partial entailment was supposed to be equal to the probability. So for example on Keynes' view the statement Ten black ravens exist partially entails the statement All ravens are black and the degree of entailment = P(All ravens are black). On this view all rational minds should assign exactly the same value to: P(All ravens are black|Ten black ravens exist) Keynes was influenced heavily by Bertrand Russell and Alfred North Whitehead who had together attempted to do something similar with their *Principia Mathematica*. It's doubtful that Russell and Whitehead succeeded, and I think the same can be said of Keynes. Ramsey's most pointed criticism was that these Keynesian probability relationships, if they exist, certainly are not perceived by the mind as Keynes claimed. And who here can argue with Ramsey's criticism? If these probability relationships were perceivable in the same way as ordinary logical relations then there would be hardly any question about the correct way to do probabilistic reasoning in AGI -- we'd all immediately recognize the correct algorithms and agree. -gts - This list is sponsored by AGIRI: http://www.agiri.org/email To unsubscribe or change your options, please go to: http://v2.listbox.com/member/?list_id=303
Re: [agi] foundations of probability theory
On Fri, 02 Feb 2007 22:01:34 -0500, Ben Goertzel [EMAIL PROTECTED] wrote: In Novamente, we use entities called indefinite probabilities, which are described in a paper to appear in the AGIRI Workshop Proceedings later this year... Roughly speaking an indefinite probability is a quadruple (L,U,b,N) with interpretation The probability is b that after I make N more observations, my estimated mean for the probability distribution attached to statement S will be in the interval (L,U) Where statement S might be some general hypothesis, e.g., All ravens are black, is that right? And then b increases as N increases -- as Novamente sees more black ravens. Yes? Does the confidence interval also change? -gts - This list is sponsored by AGIRI: http://www.agiri.org/email To unsubscribe or change your options, please go to: http://v2.listbox.com/member/?list_id=303
Re: [agi] Optimality of using probability
On Sat, 03 Feb 2007 07:29:26 -0500, Ben Goertzel [EMAIL PROTECTED] wrote: Can't we just speak about this in terms of optimization? ... The subtlety comes in the definition of what it means to use an approximation to probability theory. The cleanest definition would be: To act in such a way that its behaviors are approximately consistent with probability theory Now, how can we define this? It seems to me you're just offering up a definition of decision theory which might be defined as the science of acting in such a way that one's goal-seeking behaviors are optimized and approximately consistent with probability theory. Decision theory is the hand-maiden of probability theory and of course there is already a huge body of knowledge on the subject. Or do you mean something that a decision theorist would not consider part of his domain? -gts - This list is sponsored by AGIRI: http://www.agiri.org/email To unsubscribe or change your options, please go to: http://v2.listbox.com/member/?list_id=303
Re: [agi] foundations of probability theory
On Thu, 01 Feb 2007 14:00:06 -0500, Ben Goertzel [EMAIL PROTECTED] wrote: Discussing Cox's work is on-topic for this list... Okay, I'll get a copy and read it. Let me tell you one research project that interests me re Cox and subjective probability: Justifying Probability Theory as a Foundation for Cognition. Cox's axioms and de Finetti's subjective probability approach, developed in the first part of the last century, give mathematical arguments as to why probability theory is the optimal way to reason under conditions of uncertainty. What are you quoting here, if I may ask? I'm surprised to see Cox mentioned this way in the same sentence with de Finetti, because it's my impression that Cox's views are similar to those of Jaynes, who was a pretty sharp critic of de Finetti. I was under the impression that Cox, like Jaynes, rejected the extreme subjectivist views of de Finetti in favor of a more objective/logical interpretation. But this is admittedly based only on my very scant knowledge of Cox. I don't know of any work explicitly addressing this sort of issue, do you? No, none that address Cox and AI directly, but I suspect one is forthcoming perhaps from you. Yes? :) The only work I know of that addresses both AI and probability theory is one currently on my reading list by Professor Donald Gillies of King's College, London (not to be confused with some Canadian character named Donald B. Gillies, whose name comes up in a google search). Gillies earned his Phd under your own favorite Lakatos, with a dissertation in probability theory (I think) and wrote a book about AI and the scientific method which I believe also deals with at least tangentially with probability theory. Maybe you've already read it. It was published a while ago and you probably stay on the leading of edge of AI. Artificial Intelligence and Scientific Method (Paperback) http://www.amazon.com/Artificial-Intelligence-Scientific-Method-Gillies/dp/0198751591/sr=8-2/qid=1170441700/ref=sr_1_2/103-6974055-7831844?ie=UTF8s=books I should mention here that although I am certified with Microsoft as a C++ application developer, I clam no special knowledge of AI programming techniques. I expect this may change soon, however. -gts - This list is sponsored by AGIRI: http://www.agiri.org/email To unsubscribe or change your options, please go to: http://v2.listbox.com/member/?list_id=303
Re: [agi] foundations of probability theory
On Fri, 02 Feb 2007 15:57:24 -0500, Ben Goertzel [EMAIL PROTECTED] wrote: Interpretation-wise, Cox followed Keynes pretty closely. Keynes had his own eccentric view of probability... Although I don't yet know much about Cox, (Amazon is shipping his book to me), I have studied a bit about Keynes and yes, eccentric is my view an understatement! I assume you are familiar with F.P. Ramsey? (If not, he was one of the founders/discoverers of the subjective theory along with de Finneti, but separately.) I read Ramsey's classic paper Truth and Probability and found his arguments very convincing, including his criticisms of Keynes. For example: But let us now return to a more fundamental criticism of Mr Keynes' views, which is the obvious one that there really do not seem to be any such things as the probability relations he describes. He supposes that, at any rate in certain cases, they can be perceived; but speaking for myself I feel confident that this is not true. I do not perceive them, and if I am to be persuaded that they exist it must be by argument; moreover I shrewdly suspect that others do not perceive them either, because they are able to come to so very little agreement as to which of them relates any two given propositions. [1] I agree with Ramsey that Keynes' supposed probability relations do not seem to exist and that in any case they cannot be perceived in the way Keynes claimed. I echo Ramsey here in saying, I do not perceive them, and if I am to be persuaded that they exist it must be by argument. I suspect that if Ramsey were alive today, he would shudder at the thought of programming Keynesian-like probability relations in AGI. Are you attempting something like this in Novamente? (Please forgive my ignorance of your Novamente project. I'm still learning about it.) -gts 1. Truth and Probability by Frank P. Ramsey cepa.newschool.edu/het/texts/ramsey/ramsess.pdf - This list is sponsored by AGIRI: http://www.agiri.org/email To unsubscribe or change your options, please go to: http://v2.listbox.com/member/?list_id=303
Re: [agi] foundations of probability theory
Hi Ben, Well, Jaynes showed that the PI can be derived from another assumption, right?: That equivalent states of information yield equivalent probabilities Yes, as I understand it the principle of indifference is a special case of Jaynes' principle of maximum entropy. I have no problem with the principle of maximum entropy except in this special case in which we have *zero* information relevant to the true probabilities of outcomes. The ramifications of mathematical concepts can sometimes change radically when considering zero quantities, and I strongly suspect this is an example. It seems to me that in this special case of the maximum entropy principle, known as the indifference principle, we are not actually considering 'equivalent states of information' which might in theory yield equivalent probabilities. We are not at all considering information. We have no pieces of information to analyze, test, compare, or otherwise consider. Instead we are considering non-information (whatever that means). How can we justify probabilistic inferences from non-information? How can we justify a decision to infer something from nothing? I use the word justify here in the formal epistemological/logical sense. This goes to the question of whether the principle of indifference is truly a valid logical principle, in the formal sense of that word, as is maintained by certain people loyal to certain logical interpretations of probability theory. As you know I believe the PI falls short of that definition -- that it is instead merely a heuristic device -- a bit of semi-religious quasi-logic left over from the essentially defunct classical theory of Laplace. Perhaps someone can convince me otherwise (you came very close, when you answered the wine/water paradox!) This seems to also be dealt with at the end of Cox's book... Interesting. I'm tempted to read Cox's book so that you and I can discuss his ideas in more detail here on your list. (I worry that my enthusiasm for this subject is only annoying people on that other discussion list.) Is that something you would like to do? Please let me know! I'm copying Jef and Stu here, as this is not the first time the principle of maximum entropy has come up in the dialogue. (I don't want you guys to think your thoughts on this subject went ignored or unanswered.) -gts - This list is sponsored by AGIRI: http://www.agiri.org/email To unsubscribe or change your options, please go to: http://v2.listbox.com/member/?list_id=303
Re: [agi] Chaitin randomness
On Sat, 20 Jan 2007 00:32:18 -0500, Benjamin Goertzel [EMAIL PROTECTED] wrote: I'm not sure exchangeability implies Chaitin randomness. Yeah, you're right, this statement needs qualification -- it wasn;t quite right as stated. You're right that a binary series formed by tossing a weighted coin is exchangeable but not Chaitin random. Okay, then this observation leads me back to the same puzzlement I expressed on extropy-chat, which I will re-state here in slightly different language: Very improbable-appearing subsequences can and inevitably do appear in long random sequences. Flip a fair coin a few thousand times, for example, and there is a very good chance you'll see some extraordinarily long runs of heads and tails along with other very non-random-appearing subsequences. In binary terms, you'll see many runs like 111 and 101010101010101. We can imagine ourselves parsing the sequence, dividing it into two groups: 1) complex/disorderly subsequences not amenable to simple algorithmic derivation and 2) simple/orderly subsequences such as those above that are so amenable. Now, if I understand Chaitin's information-theoretic compressibility definition of randomness correctly (and I very likely do not), the simple/orderly subsequences in group 2) are compressible and so would count against the larger sequence in any compressibility measure of its randomness. If that is so then a maximally random sequence might be best considered as one that is at least slightly compressible. But this definition would be contrary to Chaitin's idea that maximally random sequences are incompressible! I have to conclude that either a) my understanding of the information-theoretic incompressibility definition of randomness is deficient, or b) incompressible Chaitin-random numbers are in some sense 'artificial'. Probably a) is true, but at the moment I don't see why. -gts - This list is sponsored by AGIRI: http://www.agiri.org/email To unsubscribe or change your options, please go to: http://v2.listbox.com/member/?list_id=303
Re: [agi] Chaitin randomness
This author makes a distinction, similar to the one in my mind, between algorithmic and intuitive randomness. === We can say that a sequence is algorithmically random if it has an amount of algorithmic information approximately equal to its length. Note that this is related to, but not exactly the same as our intuitive conception of randomness. Intuitively, we apply the term to processes (like coin tossing) rather than the results of such processes (like the resulting sequence). We would naturally call the process random even if it (freakishly) ended up producing a long string of heads. http://www.amirrorclear.net/academic/research-topics/algorithmic-randomness.html === -gts - This list is sponsored by AGIRI: http://www.agiri.org/email To unsubscribe or change your options, please go to: http://v2.listbox.com/member/?list_id=303
Re: [agi] Chaitin randomness
On Sat, 20 Jan 2007 20:41:55 -0500, Matt Mahoney [EMAIL PROTECTED] wrote: Any information you save by compressing the compressible bits of a random sequence is lost because you also have to specify the location of those bits. (You can use the counting argument to prove this). Ah, yes... Thank you. Your (and other's) mention of the counting argument reminded me of something I once considered in the past, and led me to this comp.compression FAQ which explains it all very nicely: Compression of random data (WEB, Gilbert and others) http://www.faqs.org/faqs/compression-faq/part1/section-8.html -gts - This list is sponsored by AGIRI: http://www.agiri.org/email To unsubscribe or change your options, please go to: http://v2.listbox.com/member/?list_id=303
[agi] Chaitin randomness
On Fri, 19 Jan 2007 18:32:54 -0500, Benjamin Goertzel [EMAIL PROTECTED] wrote: I think this topic is more appropriate for agi@v2.listbox.com Sorry, I thought that was where I was! :) Sending there now... Anyway, to respond to your point: Yep, I agree that exchangeability is different from, but closely related to Chaitin randomness, in the sense that for finite series it seems to be the case that * Chaitin randomness almost always implies exchangeability * Exchangeability almost always implies Chaitin randomness I'm not sure exchangeability implies Chaitin randomness. Exchangeability is the subjective correlate to independence and it's my understanding that independence does not imply Chaitin randomness. Consider for example a finite sequence of independent trials of a heavily weighted coin that turns up heads 99% of the time. Am I wrong to think this sequence would be highly compressible and thus not Chaitin-random? My thinking here is that the number bits required to encode the sequence would be much fewer than the bits in the sequence, and that following Chaitin, a series of numbers is random in the Chaitin sense iff the smallest algorithm capable of specifying it has about the same number of bits of information as the series itself. (This is my understanding of Chatin randomness gleaned from http://www.cs.auckland.ac.nz/CDMTCS/chaitin/sciamer.html) -gts - This list is sponsored by AGIRI: http://www.agiri.org/email To unsubscribe or change your options, please go to: http://v2.listbox.com/member/?list_id=303
