Ben,
Thanks for the references. I do not have any particularly good reason
for trying to do this, but it is a fun exercise and I find myself
making the attempt every so often :).
I haven't read the PLN book yet (though I downloaded a copy, thanks!),
but at present I don't see why term probabilities are needed... unless
inheritance relations "A inh B" are interpreted as conditional
probabilities "A given B". I am not interpreting them that way-- I am
just treating inheritance as a reflexive and transitive relation that
(for some reason) we want to reason about probabilistically. As such,
it is easy to set up probabilistic treatments-- the challenge is to
get them to behave in a way that resembles NARS.
Another way of putting this is that I am not worrying too much about
the semantics, I'm just trying to get the formal manipulations to
match up.
And the definition 3.7 that you mentioned *does* match up, perfectly,
when the {w+, w} truth-value is interpreted as a way of representing
the likelihood density function of the prob_inh. Easy! The challenge
is section 4.4 in the paper you reference: syllogisms. The way
evidence is spread around there doesn't match with definition 3.7, not
without further probabilistic assumptions.
--Abram
On Sat, Sep 20, 2008 at 4:13 PM, Ben Goertzel <[EMAIL PROTECTED]> wrote:
>
> Abram,
>
> I think the best place to start, in exploring the relation between NARS
> and probablity theory, is with Definition 3.7 in the paper
>
> From Inheritance Relation to Non-Axiomatic Logic
> [International Journal of Approximate Reasoning, 11(4), 281-319, 1994]
>
> which is downloadable from
>
> http://nars.wang.googlepages.com/nars%3Apublication
>
> It is instructive to look at specific situations, and see how this
> definition
> leads one to model situations differently from the way one traditionally
> uses
> probability theory to model such situations.
>
> The next place to look, in exploring this relation, is at the semantics that
> 3.7 implies for the induction and abduction rules. Note that unlike in PLN
> there are no term (node) probabilities in NARS, so that induction and
> abduction cannot rely on Bayes rule or any close analogue of it. They must
> be justified on quite different grounds. If you can formulate a
> probabilistic
> justification of NARS induction and abduction truth value formulas, I'll be
> quite interested. I'm not saying it's impossible, just that it's not
> obvious ...
> one has to grapple with 3.7 and the fact that the NARS relative frequency
> w+/w is combining intension and extension in a manner that is unusual
> relative to ordinary probabilistic treatments.
>
> The math here is simple enough that one does not need to do hand-wavy
> philosophizing ;-) ... it's just elementary algebra. The subtle part is
> really
> the semantics, i.e. the way the math is used to model situations.
>
> -- Ben G
>
>
>
> On Sat, Sep 20, 2008 at 2:22 PM, Abram Demski <[EMAIL PROTECTED]> wrote:
>>
>> It has been mentioned several times on this list that NARS has no
>> proper probabilistic interpretation. But, I think I have found one
>> that works OK. Not perfectly. There are some differences, but the
>> similarity is striking (at least to me).
>>
>> I imagine that what I have come up with is not too different from what
>> Ben Goertzel and Pei Wang have already hashed out in their attempts to
>> reconcile the two, but we'll see. The general idea is to treat NARS as
>> probability plus a good number of regularity assumptions that justify
>> the inference steps of NARS. However, since I make so many
>> assumptions, it is very possible that some of them conflict. This
>> would show that NARS couldn't fit into probability theory after all,
>> but it is still interesting even if that's the case...
>>
>> So, here's an outline. We start with the primitive inheritance
>> relation, A inh B; this could be called "definite inheritance",
>> because it means that A inherits all of B's properties, and B inherits
>> all of A's instances. B is a superset of A. The truth value is 1 or 0.
>> Then, we define "probabilistic inheritance", which carries a
>> probability that a given property of B will be inherited by A and that
>> a given instance of A will be inherited by B. Probabilistic
>> inheritance behaves somewhat like the full NARS inheritance: if we
>> reason about likelihoods (the probability of the data assuming (A
>> prob_inh B) = x), the math is actually the same EXCEPT we can only use
>> primitive inheritance as evidence, so we can't spread evidence around
>> the network by (1) treating prob_inh with high evidence as if it were
>> primitive inh or (2) attempting to use deduction to accumulate
>> evidence as we might want to, so that evidence for "A prob_inh B" and
>> evidence for "B prob_inh C" gets combined to evidence for "A prob_inh
>> C".
>>
>> So, we can define a second-order-probabilistic-inheritance "prob_inh2"
>> that is for prob_inh what prob_inh is for inh. We can define a
>> third-order over the second-order, a fourth over the second, and so
>> on. In fact, each of these are generalizations: simple inheritance can
>> be seen as a special case of prob_inh (where the probability is 1),
>> prob_inh is a special case of prob_inh2, and so on. This means we can
>> define an infinite-order probabilistic inheritance, prob_inh_inf,
>> which is a generalization of any given level. The truth value of
>> prob_inh_inf will be very complicated (since each prob_inhN has a more
>> complicated truth value than the last, and prob_inh_inf will include
>> the truth values from each level).
>>
>> My proposal is to add 2 regularity assumptions to this structure.
>> First, we assume that the prior over probability values for prob_inh
>> is even. This givens us some permission to act like the probability
>> and the likelihood are the same thing, which brings the math closer to
>> NARS. Second, assume that a "high" truth value on one level strongly
>> implies a high one on the next value, and similarly that low implies
>> low. They will already weakly imply eachother, but I think the math
>> could be brought closer to NARS with a stronger assumption. I don't
>> have any precise suggestions however. The idea here is to allow
>> evidence that properly should only be counted for prob_inh2 to cound
>> for prob_inh as well, which is the case in NARS. This is point (1)
>> above. More generally, it justifies the NARSian practice of using the
>> simple prob_inh likelihood as if it were a likelihood for
>> prob_inh_inf, so that it recursively acts on other instances of itself
>> rather than only on simple inh.
>>
>> Of course, since I have not given precise definitions, this solution
>> is difficult to evaluate. But, I thought it would be of interest.
>>
>> --Abram Demski
>>
>>
>> -------------------------------------------
>> agi
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>
>
>
> --
> Ben Goertzel, PhD
> CEO, Novamente LLC and Biomind LLC
> Director of Research, SIAI
> [EMAIL PROTECTED]
>
> "Nothing will ever be attempted if all possible objections must be first
> overcome " - Dr Samuel Johnson
>
>
> ________________________________
> agi | Archives | Modify Your Subscription
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