tisdag 20 augusti 2013

Artificial intelligence and Solomonoff induction: what to read?

I am deeply concerned about the future of humanity.1 There are several factors that may have a huge impact (for good and/or for bad) but are mostly overlooked in mainstream public discourse, and one of my ambitions is to try to understand these, as best as I can - with the ultimate goal of suggesting courses of action that might improve the odds of a favorable future. One of these factors is the possibility of a breakthrough in AGI - Artificial General Intelligence.2 Hence it is important to me to try to stay informed about major developments in AGI (notwithstanding the extreme difficulty even for experts to forecast in this area).

One concept that is sometimes claimed to be of central importance in contemporary AGI research is the so-called AIXI formalism. I came across a recent presentation named Uncertainty & Induction in AGI where AIXI's inventor Marcus Hutter presents its basic ingredients, which, very briefly, are as follows:

An AIXI agent is (like all of us) situated in an environment which is partly unknown to it, and wishes to act in such a way as to maximize some pre-defined utility. Based on what it observes it updates its beliefs about the environment, and unlike most of us, it does so using Bayesian updating. Before it has made any observations, the agents starts from a particular prior distribution known as Solomonoff's universal prior, defined by assigning to every possible environment H probability 2-K(H), where K(H) is the Kolmogorov complexity of H, which in turned is defined as the length of the shortest program that produces H on a given universal Turing machine. The AIXI also acts, and it does so based by plugging its utility function and its Bayesian beliefs about the environment into the Bellman equations of standard decision theory, in order to maximize its futire utility (with some finite time horizon).

In his presentation, Hutter advices, in fairly strong terms, AGI researchers to take AIXI seriously. It provides, he says, "the best conceptual solutions of the induction/AGI problem so far". The approach he advocates outperforms all earlier attempts at theories of uncertainty and induction, including "Popper’s denial of induction, frequentist statistics, much of statistical learning theory, subjective Bayesianism, Carnap’s confirmation theory, the data paradigm, eliminative induction, pluralism, and deductive and other approaches". Moreover, "AIXI is the most intelligent environmental independent, i.e. universally optimal, agent possible". If nothing else, this last word "possible" is a bit of an exaggeration, because AIXI is in fact not possible to implement, as Kolmogorov complexity is not computable). Hutter's tone then moves on from immodest to outright condescending, when he writes that "cranks who have not understood the giants and try to reinvent the wheel from scratch can safely be ignored".3

Concluding from Hutter's boasting tone that Solomonoff induction and AIXI are of no interest would, of course, be a non sequitur. But Uncertainty & Induction in AGI is just a PowerPoint presentation and can only give so much detail, so I felt I needed to look further. In the presentation, Hutter advices us to consult his book Universal Artificial Intelligence. Before embarking on that, however, I decided to try one of the two papers that he also directs us to in the presentation, namely his A philosophical treatise of universal induction, coauthored with Samuel Rathmanner and published in the journal Entropy in 2011. After reading the paper, I have moved the reading of Hutter's book far down my list of priorities, because gerneralizing from the paper leads me to suspect that the book is not so good.

I find the paper bad. There is nothing wrong with the ambition - to sketch various approaches to induction from Epicurus and onwards, and to try to argue how it all culminates in the concept of Solomonoff induction. There is much to agree with in the paper, such as the untenability of relying on uniform priors and the limited interest of the so-called No Free Lunch Theorems (points I've actually made myself in a different setting). The authors' emphasis on the difficulty of defending induction without resorting to circularity (see the well-known anti-induction joke for a drastic illustration) is laudable. And it's a nice perspective to view Solomonoff's prior as a kind of compromise between Epicurus and Ockham, but does this particular point need to be made in quite so many words? Judging from the style of the paper, the word "philosophical" in the title seems to mean something like "characterized by lack of rigor and general verbosity".4 Here are some examples of my more specific complaints:
  • The definition of Kolmogorov complexity depends on the choice of universal Turing machine, of which there are infinitely many. The same ambiguity is therefore true of Solomonoff induction, and the authors admit this. So far, so good. But how, then, should we choose which particular machine to use?

    From p 1111 and onwards, the authors talk about choosing a machine that is "unbiased", i.e. it doesn't favor certain objects blatantly more than other universal Turing machines do. It is not at all clear (at least not to the present reader) that such a notion of unbiasedness can be made sense of. On p 1112, the vague notion of natural universal Turing machines is introduced, meaning machines that do not exhibit such biases, and the authors offers this:

      One possible criterion is that a reference machine is natural if there is a short interpreter/compiler for it on some predetermined and universally agreed upon reference machine. If a machine did have an in-built bias for any complex strings then there could not exist a short interpreter/compiler. If there is no bias then we assume it is always possible to find a short compiler.
    This does not make any progress towards solving the ambiguity problem, due to the obvious circularity: We should choose a machine that is unbiased in comparison to a "universally agreed upon reference machine", but how should that machine be chosen? Perhaps it should be chosen in such a way as to be unbiased in comparison to some absolutely honest-to-God universally agreed upon reference machine? And so on.5

  • Solomonoff induction involves the model assumption that the environment is computable. On p 1118, the authors address this assumption, and we find the following passage:
      John von Neumann once stated “If you will tell me precisely what it is that a machine cannot do, then I can always make a machine which will do just that”. This is because, given any “precise” description of a task we can design an algorithm to complete this task and therefore the task is computable. Although slightly tongue in cheek and not quite mathematically correct, this statement nicely captures the universality of the concept of computability.
    All right, thank you for the "not quite mathematically correct", but I still think this is downplaying the problem in a misleading way. It's not the case that von Neumann's statement could be turned into something similar in spirit but correct - the problem is that the statement is downright false. It would have been easy for the authors at this point to provide a counterexample by sketching Turing's diagonalization argument for the halting problem.

    In the same paragraph, they say that "according to the Church-Turing thesis, the class of all computable measures includes essentially any conceivable natural environment". Here it would have been suitable for the authors to modestly admit that the procedure they advocate (Solomonoff prediction) is in fact not computable and therefore impossible in "any conceivable natural environment". (Such a discussion comes much later in the paper.)

    Of course it is OK and meaningful to discuss, as most of this paper does, the hypothetical scenario of having a machine that performs Solomonoff induction. But to insist, in this scenario, that the environment must be computable, is to introduce an outright contradiction and thus to enter the realm of the meaningless.

  • The black raven paradox is a major issue that any theory of induction ought to address. To accept induction is to accept that the observation of a black raven will (usually) strengthen our belief in the hypothesis H that all ravens are black. Note, however, that "black" and "raven" are just arbitrarily chosen examples of properties here, and might as well be replaced by, e.g., "non-raven" and "non-black", so that accepting induction equally well entails accepting that the observation of a non-black non-raven will (usually) strengthen our belief in the hypothesis H* that all non-black items are non-ravens. But H* and H are logically equivalent, and surely logically equivalent hypoteses must be supported by the same evidence, so we seem to be forced to accept that the observation of a non-black non-raven (such as a yellow banana) will (usually) strengthen our belief in the hypothesis H that all ravens are black. But this seems paradoxical: most people will intuitively resist the idea that the observation of a yellow banana should have any evidential impact on the credibility of a hypothesis concerning the color of ravens.

    Rathmanner and Hutter suggest that Solomonoff induction makes progress on the black raven paradox, but their case is very weak, for the following reasons:

    • Nowhere in their discussion about Solomonoff induction and the black raven paradox on p 1122-1124 of their paper is there any hint of any asymmetry in how Solomonoff induction would treat black ravens versus non-black non-ravens as regards the the all-ravens-are-black hypothesis. No such hint means no argument that Solomonoff induction helps to solve the paradox.
    • The highlight, supposedly, of the authors' argument on Solomonoff induction and the black raven paradox is the displayed formula near the top of p 1124, which states that when observing objects, one after another, the conditional probability of never ever seeing a black raven given what has been seen so far after n observations tends to 1 as n→∞ almost surely on the event that no black raven is ever seen. This is a nice property, but has little or nothing to do with Solomonoff induction, because the same is true for any Bayesian model incorporating such an infinite sequence of observations.6
    • There is a difference between the statements "all ravens are black" and "all ravens ever appearing in our sequence of observations are black". The statement we're originally interested in is the former, whereas the aforementioned highlight on p 1124 is about the latter. The preceding displayed formula, near the bottom of p 1123, bridges this difference. That formula, however, is not about Solomonoff induction but only about a certain simple class of i.i.d. mixtures.
    To summarize, there is nothing in the paper to back up its penultimate sentence (on p 1134) saying that that the authors have shown "how Solomonoff induction deals with the black raven paradox and argued that it should give the desired result".

  • A bit further down on p 1124, in Section 8.1, the authors suggest that a certain property of Solomonoff's univeral prior M implies "fast convergence in practice". I resent the term "in practice" here, because there is no practice: Solomonoff's prior is not computable. (The paper is full of similar slips.)

  • In the introductory section, on p 1078, we learn that Solomonoff induction means that "it can be argued that the problem of formalizing optimal inductive inference is solved", and from then on, we hear about this optimality again and again. But I have trouble understanding the concept of e procedure that is both optimal and (if we accept the aforementioned Church-Turing thesis) impossible. Here's another induction procedure which is equally impossible, and arguably even more optimal: whenever we wish to know something about the world, ask an infallible oracle!

    Presumably, what the authors mean by Solomonoff induction being "optimal" is that no real-world system can perform better at the given task. But there are many such "optimal" levels, as my oracle example shows, so a better term than "optimal" would be "upper bound on performance". The upper bound on performance provided by Solomonoff induction is obviously tighter than mine, and therefore better. But how tight is it, compared to what is actually achievable? Is there room for better bounds? There'd better not be, in order for the authors' statement about "the problem of formalizing optimal inductive inference [being] solved" not to look silly, but on this issue they are silent.

  • In Section 10.4 entitled "Conclusions", the authors claim on p 1134 that Solomonoff induction "gives us new insights into the mechanics of our own thinking". Well, isn't that nice? Even nicer would have been if the paper had contained at least one example of such an insight that can plausibly be attributed to Solomonoff induction.
I still consider it plausible to think that Kolmogorov complexity and Solomonoff induction are relevant to AGI7 (as well as to statistical inference and the theory of science), but the experience of reading Uncertainty & Induction in AGI and A philosophical treatise of universal induction strongly suggests that Hutter's writings are not the place for me to go in order to learn more about this. But where, then? Can the readers of this blog offer any advice?


1) For the benefit of Samuel Rathmanner and Marcus Hutter, and to give them a chance to comment, I write this blog post in English.

2) What such a breakthrough might entail is extremely difficult to predict, but I have discussed various scenarios en earlier blog posts, e.g., here, here and here.

3) By "giants", he means "Bayes, Shannon, Turing, Kolmogorov, Solomonoff, Wallace, Rissanen [and] Bellman".

4) I have great respect for the subject of philosophy. Naturally, then, I disapprove of Rathmanner's and Hutter's useage of the term "philosophical".

5) There is a structural similarity between, on one hand, this failure to make progress by introducing yet another reference machine, and, on the other hand, the postulation that everything that exists must be created and thus have a creator, and pretending to solve the problem by introducing God Almighty. (Who created God Almighty?)

6) One interesting strengthening concering Solomonoff induction would be if the "almost surely" qualifier might be dropped in the asymptotic result considered here. This would be less trivial, because the corresponding statement in the generality of Bayesian models incorporating an infinite sequence of observations would be false.

7) I am not a computer scientist, so the following should perhaps be taken with a grain of salt. While I do think that computability and concepts derived from it such as Kolmogorov complexity may be relevant to AGI, I have the feeling that the somewhat more down-to-earth issue of computability in polynomial time is even more likely to be of crucial importance.

23 kommentarer:

  1. I think it is highly improbable that AGI will arise from anything based on some theoretical results from information theory. The basic reason for me saying this is empirical. Most machine learning researchers use information theory and Bayesian updating in the algorithms they develop, yet they can't get even the slightest bit of intelligence out of a computer. They are just simple search algorithms. It seems daft then to investigate the formal basis of the reasoning with a hope of getting to the bottom of the key components of what artificial intelligence looks like.

    I heard Josh Tennenbaum (http://web.mit.edu/cocosci/josh.html) give an excellent talk recently where first he explained what they had achieved in showing how people reason using Bayesian updating about the likilhood which of their various models of the world. He then spent the second half of his talk saying why this answered only a small set of problems. His latest research is looking at whether the brain runs 'physics simulators' to make predictions about future states. This is in my opinion is pushing the limits of what can be done and is where you should start. Start with the actual science.

    I appreciate the honesty in your post. But I feel forced to point out that you are picking soft targets. Clearly this guy was some sort of crackpot, and in a way it is bad taste to attack him.

    1. There are lots of crackpots in the world and they cluster. Move to where the science is done instead.

    2. It may well be, David, that Solomonoff induction and AIXI are not the most fertile grounds for further inquiry, in which case you are right that I ouught to move on to more promising areas. But even if this should be the case, I don't see how it would be "bad taste" to offer feedback on my way out. I daresay all research areas - healthy or not - stand to benefit from criticism from the outside.

      Thanks, by the way, for the link to Tenenbaum. Will have a closer look.

    3. You're right. Its actually me that's being snobby about his work. It isn't below the quality threshold where criticism is no longer allowed. But I just feel its better to keep quiet. You never know, you might end up increasing his citation index!

    4. David Sumpter, can you elaborate on the bit about "can't get even the slightest bit of intelligence out of a computer"? For example, Hutter claims good performance on test problems, such as Kuhn Poker and a version of Pacman, for his practical algorithm. What's the main reason this doesn't constitute artificial intelligence, in your view?


    5. The point in AGI is to pass the Turing test or even to produce an entity that can reason in some simple but general way about a range of problems. There are lots of games for which we can develop good algorithms and computers can do well, but this does not qualify as 'general' intelligence. General Intelligence is when I can recognise the difference between Pac man and Poker and start to think what would be best to do in each situation.

      For example, just now I looked up Kuhn Poker on google, because I had never heard of it before. I read two lines about what the creator was trying to achieve and understood what is all about and roughly what the rules should be. I thought about why there should be three cards not two. I don't know what the optimal strategy is, but I know why it was created, what purpose it serves and why it is interesting. I could program a computer to work out the optimal strategy too. This is general intelligence.

      There is no reason to believe that algorithms which are good at finding optimal game strategies will have any relation to an algorithm which produces general intelligence.

    6. David Sumpter, that's mostly fair enough, I think. I don't have a clear notion of what it means for an algorithm to "reason", but on some intuitive level I can agree that what this algorithm does is different. I just want to point out regarding "There are lots of games for which we can develop good algorithms and computers can do well" that Hutter's practical algorithm was developed neither for Kuhn Poker nor for Pacman. It was developed to be very general, Kuhn Poker and Pacman are just test cases. Natural language processing and the Turing test are in principle other conceivable tests, though I'd be surprised if this is presently practically feasible within reasonable CPU time.


  2. You might like the stuff at http://syntience.com/links
    and especially http://syntience.com/rch.pdf that provides a more bioplausible view of a dual process cognitive system that focuses on Reduction rather than Induction. Induction is "too logical" to be bioplausible.

    The video "A New Direction in AI Research" contains the main points but later results and more domain-specific examples are available in the other videos.

  3. Praktiska algoritmer inspirerade av Solomonoff-induktion:


    1. Hmm, more stuff by Hutter... not sure I'm sufficiently motivated to read this.

    2. Active LZ-algoritmen som de jämför med är också inspirerad av Solomonoff-induktion: http://arxiv.org/abs/0707.3087


  4. I notified Marcus Hutter about my blog post, and received the following response. I find it thoroughly unsatisfactory:

    "Hi Olle,

    Thanks for your interest in my work. In general I don't have time to engage in blog discussions, especially if they are non-constructive. I am happy to leave such discussions to others. Just two brief comments:

    (1) If you don't like the verbose style of [RH11], you should consult my book and/or technical papers at http://www.hutter1.net/official/bib.htm

    (2) "Hutter's writings are not the place for me to go in order to learn more about this. But where, then?" Your question implies that [RH11] contains "the best conceptual solutions of the induction/AGI problem" you are aware of. In case you find or develop something better, i.e. which better withstands comparable scrutiny, I would be interested to hear about it.

    Best regards,


    Hutter begins with declaring that he declines to participate in scientific discussions unless they take place in traditional formats (i.e., not blogs). For someone who primarily values scientific progress (as opposed to superficial things like career-boosting) such format issues really ought to be secondary compared to the actual scientific content.

    He then criticizes me for being "non-constructive". But when Hutter claims to make progress on problems like the choice of reference machine or the black raven paradox, and I point out the hollowness of his claims, it's more than a little disingenious to shrug off my criticism by noting that I haven't provided alternative approaches to these deep and difficult unsolved problems.

    In his comment (1), Hutter pretends that my main criticism concerns style rather than content.

    In his comment (2), he draws a conclusion about what my question "...to learn more about this. But where, then?" implies. This conclusion is an obvious non sequitur, even if we should ignore that he pretends that my word "this" refers to the general "induction/AGI problem" rather than (as is entirely clear from my blog post's final paragraph, from which he quoted my question) the specific approach of using Solomonoff induction.

  5. Angående svarta korpar:

    * Rathmanner & Hutter diskuterar detta exempel på flera ställen i sin artikel. Speciellt avsnittet omkring s. 1106 där de diskuterar Mahers text ger intrycket att problemet de främst menar sig lösa är att på ett naturligt sätt ge hypoteser som "alla korpar är svarta" en ändlig och icke-försumbar sannolikhet. Om problemformuleringen är sådan att antalet korpar ska betraktas som oändligt, så ger de mest direkta angreppssätten att hypotesen "alla korpar är svarta" antingen alltid har sannolikhet 0 eller särbehandlas från andra hypoteser.

    * Bland bayesianer är den vanligasta uppfattningen att en icke-svart icke-korp faktiskt ger pyttelite empiriskt stöd till hypotesen att alla korpar är svarta. Med Rathmanner & Hutter egna ord är asymmetrin mellan observation av en svart korp resp. en icke-svart icke-korp:

    "The solution to this problem lies in the relative degree of confirmation. The above result only states that the belief in the hypothesis must increase after observing either a black raven or a white sock, it says nothing about the size of this increase. If the size of the increase is inversely proportional to the proportion of this object type in the relevant object population then the result can become quite consistent and intuitive." (s. 1109)

    "This also solves the problem of our intuition regarding black ravens. Black ravens make up a vanishingly small proportion off all possible objects, so the observation of a black raven gives an enormously greater degree of confirmation to “all ravens are black” than a non-black non-raven. So much so that the observation of a non-black non-raven has a negligible affect on our belief in the statement.
    Unfortunately no formal inductive system has been shown to formally give this desired result so far. It is believed that Solomonoff Induction may be able to achieve this result but is has not been shown rigorously. Later we will argue the case for Solomonoff induction." (s. 1109)

    * Ditt påstående "Nowhere in their discussion about Solomonoff induction and the black raven paradox on p 1122-1124 of their paper is there any hint of any asymmetry..." är falskt. Det finns visst antydningar om asymmetri. Men framför allt så skriver de mycket explicit om detta på andra ställen i artikeln. I kommentaren ovan skriver du att Hutters e-mail svar är "more than a little disingenious". Om något så ser ditt selektiva refererande av Hutters diskussion av "svarta korpar"-exemplet mer ohederligt ut än Hutters e-mail svar till dig. Men framförallt så borde man inte slänga ur sig sådana anklagelser på lösa grunder.


    1. Varken diskussionen på sidan 1109 eller den sista meningen på sidan 1122 ("Since non-black non-ravens make up the vast majority of objects in the real world, the correct hypothesis corresponds...") involverar Solomonoffinduktion. Ingen av dessa passager kan anföras som exempel på att författarna har några argument för att Solomonoffinduktion skulle lösa svartkorpsparadoxen. Jag står fast vid att "nowhere in their discussion about Solomonoff induction and the black raven paradox [...] is there any hint of any asymmetry in how Solomonoff induction would treat black ravens versus non-black non-ravens".

    2. Det vore helt legitimt att konstruera en lösning som kombinerar ingredienser specifika till Solomonoff-induktion med andra ingredienser som också är tillgängliga i konkurrerande formalismer. De gör uppenbarligen inga anspråk på originalitet när det gäller tanken att svarta korpar är mycket mindre vanliga än icke-svarta icke-korpar, och att detta är en ingrediens i den totala lösningen. Det finns ingen anledning att förvänta sig att de ska presentera något argument för asymmetri som endast är tillgängligt via Solomonoff-induktion. Och du har ingen grund för att blankt ignorera det de faktiskt skriver om asymmetrin, låtsas som deras påstående om asymmetri var något annat, samt kritisera dem för att de inte diskuterat frågan på det märkliga, Solomonoff-specifika sätt du förväntade.

      Oavsett i vilken grad de faktiskt gör framsteg mot en lösning på "svarta korpar"-problemet, så är just din läsning och kritik inte så träffande.


    3. Men snälla Erik, har du läst vad Rathmanner och Hutter skriver i Avsnitt 10.3, fjärde punkten?

      "Solomonoff solves many persistent philosophical problems such as the zero prior and confirmation problem for universal hypotheses. It also deals with the problem of old evidence and we argue that it should solve the black ravens paradox."

      Jag repeterar: "we argue that it [Solomonoffinduktion alltså] should solve the black ravens paradox".

      Ett minimikrav för att deras påstående "we argue that it should solve the black ravens paradox" skall vara sant är att det någonstans finns ett argument (bra eller dåligt) som inbegriper både Solomonoffinduktion och asymmetrin mellan svarta korpar och icke-svarta icke-korpar. Något sådant står emellertid inte att finna i uppsatsen. De pratar visserligen en hel del om Solomonoffinduktion, och de nämner dessutom sagda asymmetri, men ingenstans för de samman dessa saker. Ett resonemang måste hänga ihop. För att troliggöra att det finns ett samband mellan ett begrepp A och ett annat begrepp B räcker det inte att först pladdra lite grand om A och senare i uppsatsen pladdra lite grand om B.

      Rathmanner och Hutter saknar alltså täckning för sitt påstående "we argue that it should solve the black ravens paradox", och för detta förtjänar de kritik.

    4. 1. "...should solve..." är inte detsamma som "...does solve..." eller "...provably solves...". Det är en smula oklart för mig vad den avsedda tolkningen av "should" är här, men eftersom de på andra ställen...:

      (a) skriver om mängden svarta korpar vs. icke-svarta icke-korpar i generella Bayesianska termer, och
      (b) skriver saker som att "It is believed that Solomonoff Induction may be able to achieve this result [=asymmetry] but is has not been shown rigorously." (s. 1109) och "What remains to be seen is whether M also gets the absolute and relative degree of confirmation right." (s. 1124)

      ...så tolkar jag "should" som en förväntning att det går att lösa problemet inom ramverket för Solomonoff-induktion.

      2. Enligt författarna så bidrar Solomonoff-induktion systematisk med hantering av universella hypoteser ("alla x med egenskapen F har också egenskapen G"), där en nollskild sannolikhet och bayesiansk uppdatering är möjlig i en (uppräkneligt) oändlig domän utan särbehandling. Det är detta de framhäver som ett lovande resultat på väg mot en lösning av "svarta korp"-problemet.

      3. Tanken att asymmetrin kommer från att svarta korpar är ovanligare än icke-svarta icke-korpar är inte specifik till Solomonoff-induktion. So what?? De påstår inget annat och de skriver uttryckligen att de inte har lyckats visa hur asymmetrin hanteras specifikt inom deras formalism.

      4. Ditt "minimikrav" är bara ett symptom på din slarviga, alternativt tententiösa, läsning. Det är näppeligen ett "minimikrav" att hålla med dig om problemformuleringen, och dina tolkningar är väl tillrättalagda för din kritik. Om det, enligt deras uppfattning, är signifikant att Solomonoff-induktion systematiskt kan hantera universella påstående som "alla korpar är svarta", så räcker det gott till deras diskussion av "svarta korpar"-problemet". Hanteringen av universella påståenden är ju något de menar är en stor brist i t.ex. Mahers analys av problemet. Eventuellt tycker du att asymmetrin är den enda aspekten av "svarta korp"-exemplet som är relevant eller där det är möjligt att göra lovande framsteg, men Rathmanner och Hutter är inte skyldiga att hålla med om detta.

      5. De förtjänar antagligen kritik. Jag tycker själv inte språk- och Turingmaskin-beroendet hanteras så övertygande. Men de förtjänar inte dina feltolkningar. De förtjänar inte heller anklagelser om ohederlighet pga något så otroligt trivialt som att Hutter inte vill diskutera på din blog eller att han tolkade delar av ditt e-mail annorlunda än det var avsett.


    5. Det du skriver i punkt 4 tyder på att du inte förstått vad svarta korpar-paradoxen är. För att råda bot på det rekommenderar jag Wikipedias koncisa formulering.

    6. Nej du, problemet är och förblir att du läser slarvigt eller tendentiöst. Återigen, det har betydelse för paradoxen huruvida det ens är möjligt att systematiskt hantera universella påståenden som "alla korpar är svarta", tilldela dem nollskilda sannolikheter som kan uppdateras, etc.


    7. Ett visst mått av välvilja när man läser andras arbeten är förstås bra, men den överdrivna välvilja med vilken du läser Rathmanner och Hutter slår över i vad som i mina öron låter som ett avskaffande av kravet på att vetenskapliga arbeten skall vara välargumenterade.

      Givetvis finns ett samband mellan (a) "huruvida det ens är möjligt att systematiskt hantera universella påståenden som 'alla korpar är svarta', tilldela dem nollskilda sannolikheter som kan uppdateras" och (b) svarta korpar-paradoxen. Båda sakerna handlar ju om induktion. Men det gör ju även exempelvis (c) Goodmans paradox. Så varför kan författarnas påpekanden om (a) väntas utgöra viktiga ledtrådar om hur Solomonoffinduktion skulle kunna lösa (b), snarare än t.ex. (c)? Här behövs ett argument, men författarna ger inget sådant, varför deras påstående "we argue that it should solve the black ravens paradox" saknar täckning.

    8. När det gäller *argument* så verkar du kräva ytterst lite av dig själv och massor av andra.

      Ett sista försök: Här är en formulering av "svarta korpar"-paradoxen fritt översatt och nedkortat från s. 2 av Fitelson 2006 (http://fitelson.org/ravens.pdf):

      1. NC: För alla a, och vilka som helst predikat R och B, så bekräftar Ra & Ba hypotesen H = "för alla x, om Rx så Bx".

      2. EC: Om observation E bekräftar H1, och H2 är logiskt ekvivalent med H1, så bekräftar E även H2.

      3. PC: ~Ra & ~Ba bekräftar H' = "för alla x, om ~Bx så ~Rx" och därmed även H = "för alla x, om Rx så Bx".

      Här står PC för "Paradoxical Conclusion". NC står för "Nicod's Condition", som nog de flesta efter en stunds eftertanke förkastar som en generell princip (http://arxiv.org/abs/1307.3435). Men en specialisering av NC skulle kunna vara rimlig/korrekt när det gäller just observation av svarta korpar.

      "Paradoxen" består i att PC är kontraintuitiv för många människor. En klass "lösningar" har ambitionen förkasta argumentet och ge upprättelse till intuitionen. En annan klass "lösningar" accepterar argumentet och förkastar intuitionen. Och kanske kan man både acceptera argumentet och rädda intuitionen genom att fastställa att ~Ra & ~Ba bekräftar H, men bara i pytteliten grad.

      NC tenderar att vara central i texter om "svarta korpar"-paradoxen. (Maher 2004, på s. 12, http://patrick.maher1.net/pctl.pdf, skriver t.om. att "We have already seen that (i) [=NC] is false, which suffices to resolve the paradox.", vilket kan jämföras med ditt ogrundade "minimikrav"!) Intuitivt ser NC rimlig ut och mannen-på-gatan har kanske inte någon mer förfinad förståelse av bekräftning av hypoteser än NC. Författare som analyserar paradoxen tenderar att antingen begränsa NC baserat på någon filosofisk uppfattning/begrepp (exv. "natural kinds" eller lagbunden regelbundenhet) eller ge en generell analys i termer av någon begränsning på Bayesianska sannolikheter som är relaterad till NC. Mig veterligt är Solomonoff-induktion unik i det avseendet att det är en mycket generell teori (utan speciella anpassningar till någon specifik situation), där NC inte är ett grundläggande antagande utan något som eventuellt kan härledas. Sannolikheter för universella hypoteser tilldelas och uppdateras utan någon godtycklig särbehandling (i alla inte utöver den särbehandling det innebär att välja en referens-Turingmaskin). Så vitt jag kan se är NC dessutom ofta falsk i Solomonoff-induktion pga att det Hutter kallar "M(x)" kan vara lite vad som helst för en individuell ändlig sekvens x. Om man ska formulera "svarta korpar"-paradoxen för Solomonoff-induktion behöver man nog ersätta NC med något uttalande om genomsnittlig bekräftelse av hypoteser.

      Även i väntan på att någon kommer med ett stringent bevis för att Solomonoff-induktion, i någon (genomsnittlig?) mening och för någon intressant klass av observationssekvenser, bekräftar H mycket mer när nästa symbol representerar än svart korp än när den representerar en icke-svart icke-korp, så har man här ett nytt ramverk att placera in paradoxen i.

      Solomonoff-induktion ger också ett nytt ramverk för att studera Goodmans paradox, så det Rathmanner och Hutter skriver kan mycket väl tänkas vara relevant för det också. Det är inte så att Solomonoff-induktion, eller resultat som handlar om Solomonoff-induktion, bara kan vara relevant för max en av "svarta korpar"-paradoxen och Goodmans paradox, så här finns ingen motsättning.


    9. OK, här några avslutande påpekanden. (Jag tror att både du och jag känner att det kan vara hög tid att avrunda diskussionen.)

      1. I din senaste kommentar börjar du faktiskt närma dig vad jag skulle godkänna som ett argument för att Solomonoff-induktion är lovande när det gäller att lösa svarta korpar-problemet. Men det är i så fall ditt argument, inte R&H:s, och kan därför inte anföras som stöd för R&H:s alltjämt felaktiga påstående "we argue that it should solve the black ravens paradox".

      2. Att NC är falskt är välkänt (och, när man tänker efter en smula, uppenbart). När vi väl insett det så är paradoxen i den formulering av Fitelson du refererar upplöst. Men den intressanta paradox som återstår att lösa är den som fås om vi modifierar NC (och PC) genom att peta in ordet "vanligtvis" efter "bekräftar".

      3. Givetvis är det möjligt att tänka sig ett resultat om Solomonoff-induktion som har implikationer för såväl svarta korpar-paradoxen som Goodmans paradox; jag hoppas verkligen att du inte trodde att jag är så korkad att jag hävdade motsatsen. Jag förde in Goodmans paradox i resonemanget bara för att betona att det finns många olika olösta problem inom induktion. Om man genom lite allmänna påpekanden om induktion, t.ex. rörande "huruvida det ens är möjligt att systematiskt hantera universella påståenden som 'alla korpar är svarta', tilldela dem nollskilda sannolikheter som kan uppdateras" säger sig ha funnit en lovande ingång till något specifikt av dessa olösta problem, så behöver man argumentera för att troliggöra att ens allmänna påpekanden har särskild relevans för just det problemet - en relevans som behöver vara mer konkret än blott det faktum att såväl påpekandena ifråga som problemet har med induktion att göra. Din senaste kommentar ovan är det bästa försök jag hittills sett att på detta vis koppla samman R&H:s gränsvärdesformel högst upp på sidan 1124 (och några av deras övriga påpekanden i samma avnitt) med svarta korpar-paradoxen.