When working on your new proposal, remember that in NARS all measurements must be based on what the system has --- limited evidence and resources. I don't allow any "objective probability" that only exists in a Platonic world or the infinite future.
Pei On Sun, Sep 21, 2008 at 1:53 PM, Abram Demski <[EMAIL PROTECTED]> wrote: > Hmm... I didn't mean infinite evidence, only infinite time and space > with which to compute the consequences of evidence. But that is > interesting too. > > The higher-order probabilities I'm talking about introducing do not > reflect inaccuracy at all. :) > This may seem odd, but it seems to me to follow from your development > of NARS... so the difficulty for me is to account for why you can > exclude it in your system. Of course, this need only arises from > interpreting your definitions probabilistically. > > I think I have come up with a more specific proposal. I will try to > write it up properly and see if it works. > > --Abram > > On Sat, Sep 20, 2008 at 11:28 PM, Pei Wang <[EMAIL PROTECTED]> wrote: >> On Sat, Sep 20, 2008 at 11:02 PM, Abram Demski <[EMAIL PROTECTED]> wrote: >>> You are right in what you say about (1). The truth is, my analysis is >>> meant to apply to NARS operating with unrestricted time and memory >>> resources (which of course is not the point of NARS!). So, the >>> question is whether NARS approaches a probability calculation as it is >>> given more time to use all its data. >> >> That is an interesting question. When the weight of evidence w goes to >> infinite, so does confidence, and frequency converge to the limit of >> positive evidence among all evidence, so it becomes probability, under >> a certain interpretation. Therefore, as far as a single truth value is >> concerned, probability theory is an extreme case of NARS. >> >> However, to take all truth values in the system into account, it is >> not necessarily true, because the two theories specify the relations >> among statements/propositions differently. For example, probability >> theory has conditional B|A, while NARS uses implication A==>B, which >> are similar, but not the same. Of course, there are some overlaps, >> such as disjunction and conjunction, where NARS converges to >> probability theory in the extreme case (infinite evidence). >> >>> As for higher values... NARS and PLN may be using them for the purpose >>> you mention, but that is not the purpose I am giving them in my >>> analysis! In my analysis, I am simply trying to justify the deductions >>> allowed in NARS in a probabilistic way. Higher-order probabilities are >>> potentially useful here because of the way you sum evidence. Simply >>> put, it is as if NARS purposefully ignores the distinction between >>> different probability levels, so that a NARS frequency is also a >>> frequency-of-frequencies and frequency-of-frequency-of frequencies and >>> so on, all the way up. >> >> I see what you mean, but as it is currently defined, in NARS there is >> no need to introduce higher-order probabilities --- frequency is not >> an estimation of a "true probability". It is uncertain because the >> influence of new evidence, not because it is inaccurate. >> >>> The simple way of dealing with this is to say that it is wrong, and >>> results from a confusion of similar-looking mathematical entities. >>> But, to some extent, it is intuitive: I should not care too much in >>> normal reasoning which "level" of inheritance I'm using when I say >>> that a truck is a type of vehicle. So the question is, can this be >>> justified probabilistically? I think I can give a very tentative >>> "yes". >> >> Hopefully we'll know better about that when you explore further. ;-) >> >> Pei >> >>> --Abram >>> >>> On Sat, Sep 20, 2008 at 9:38 PM, Pei Wang <[EMAIL PROTECTED]> wrote: >>>> On Sat, Sep 20, 2008 at 9:09 PM, Abram Demski <[EMAIL PROTECTED]> wrote: >>>>>> >>>>>> (1) In probability theory, an event E has a constant probability P(E) >>>>>> (which can be unknown). Given the assumption of insufficient knowledge >>>>>> and resources, in NARS P(A-->B) would change over time, when more and >>>>>> more evidence is taken into account. This process cannot be treated as >>>>>> conditioning, because, among other things, the system can neither >>>>>> explicitly list all evidence as condition, nor update the probability >>>>>> of all statements in the system for each piece of new evidence (so as >>>>>> to treat all background knowledge as a default condition). >>>>>> Consequently, at any moment P(A-->B) and P(B-->C) may be based on >>>>>> different, though unspecified, data, so it is invalid to use them in a >>>>>> rule to calculate the "probability" of A-->C --- probability theory >>>>>> does not allow cross-distribution probability calculation. >>>>> >>>>> This is not a problem the way I set things up. The likelihood of a >>>>> statement is welcome to change over time, as the evidence changes. >>>> >>>> If each of them is changed independently, you don't have a single >>>> probability distribution anymore, but a bunch of them. In the above >>>> case, you don't really have P(A-->B) and P(B-->C), but P_307(A-->B) >>>> and P_409(B-->C). How can you use two probability values together if >>>> they come from different distributions? >>>> >>>>>> (2) For the same reason, in NARS a statement might get different >>>>>> "probability" attached, when derived from different evidence. >>>>>> Probability theory does not have a general rule to handle >>>>>> inconsistency within a probability distribution. >>>>> >>>>> The same statement holds for PLN, right? >>>> >>>> Yes. Ben proposed a solution, which I won't comment until I see all >>>> the details in the PLN book. >>>> >>>>>> The first half is fine, but the second isn't. As the previous example >>>>>> shows, in NARS a high Confidence does implies that the Frequency value >>>>>> is a good summary of evidence, but a low Confidence does implies that >>>>>> the Frequency is bad, just that it is not very stable. >>>>> >>>>> But I'm not talking about confidence when I say "higher". I'm talking >>>>> about the system of levels I defined, for which it is perfectly OK. >>>> >>>> Yes, but the whole purpose of adding another value is to handle >>>> inconsistency and belief revision. Higher-order probability is >>>> mathematically sound, but won't do this work. >>>> >>>> Think about a concrete example: if from one source the system gets >>>> P(A-->B) = 0.9, and P(P(A-->B) = 0.9) = 0.5, while from another source >>>> P(A-->B) = 0.2, and P(P(A-->B) = 0.2) = 0.7, then what will be the >>>> conclusion when the two sources are considered together? >>>> >>>> Pei >>>> >>>> >>>> ------------------------------------------- >>>> agi >>>> Archives: https://www.listbox.com/member/archive/303/=now >>>> RSS Feed: https://www.listbox.com/member/archive/rss/303/ >>>> Modify Your Subscription: https://www.listbox.com/member/?&; >>>> Powered by Listbox: http://www.listbox.com >>>> >>> >>> >>> ------------------------------------------- >>> agi >>> Archives: https://www.listbox.com/member/archive/303/=now >>> RSS Feed: https://www.listbox.com/member/archive/rss/303/ >>> Modify Your Subscription: https://www.listbox.com/member/?&; >>> Powered by Listbox: http://www.listbox.com >>> >> >> >> ------------------------------------------- >> agi >> Archives: https://www.listbox.com/member/archive/303/=now >> RSS Feed: https://www.listbox.com/member/archive/rss/303/ >> Modify Your Subscription: https://www.listbox.com/member/?&; >> Powered by Listbox: http://www.listbox.com >> > > > ------------------------------------------- > agi > Archives: https://www.listbox.com/member/archive/303/=now > RSS Feed: https://www.listbox.com/member/archive/rss/303/ > Modify Your Subscription: https://www.listbox.com/member/?&; > Powered by Listbox: http://www.listbox.com > ------------------------------------------- agi Archives: https://www.listbox.com/member/archive/303/=now RSS Feed: https://www.listbox.com/member/archive/rss/303/ Modify Your Subscription: https://www.listbox.com/member/?member_id=8660244&id_secret=114414975-3c8e69 Powered by Listbox: http://www.listbox.com
