OK, here is attempt 2 to interpret NARS probabilistically. (Actually it has a higher number than that, but my previous attempts have not been successful enough to count.) I do not guarantee the consistency of the probabilistic assumptions.

The account is in terms of infinite-order probabilities, an entity that I heard about by reading Ben Goertzel's blog :). It does not work otherwise, which is interesting.

As I am not greatly familiar with that entity, I must apologize to him for any abuses that follow.

Pei Wang suggested that I should make no reference to objective probabilities from an external world. But, this does not fit with my goal-- I'm trying to give a probabilistic interpretation, after all. However, the system never manipulates the external probabilities. They are referenced only to indicate the semantics.

The basic definition:

A=>B represents the probability distribution of A=>C or C=>B given B=>C or C=>A. (More formally, the distribution is the average of p(A=>C | B=>C) and p(C=>B | C=>A), each with randomly chosen C.)

The probabilistic assumptions:

1. All of these distributions are density functions of the form p(z) = y*(x^z + (1-x)^(1-z)) + (1-y)*(prior), where x is a parameter defining the distribution A=>B, y is the value of that parameter for the given (B=>C or C=>A), and z is the value of that parameter for the distribution being conditioned (A=>C or C=>B). The parameter ranges between 0 and 1, but it is not a probability; it is a way of describing a distribution. Let's call the parameter 'par', just to give it a name.

2. The prior is even over values of par.

The system is supposed to work with likelihoods rather than probabilities, reflecting the experienced-based semantics of NARS. (The even prior means this doesn't matter *too* much, but it will be more intuitive to explain it in terms of likelihoods I think.)

The likelihood of a statement given experience is the probability of experience given that statement. Since the probabilities are density functions over par, the likelihoods will also be density functions over par (except they aren't required to sum to 1). The nature of the likelihoods happens to lend itself to an easy representation: we can record the likelihood as a pair <f, w>, with f ranging from 0 to 1 and w from 0 to infinity. The following calculation defines the likelihood of par based on f and w:

par^(w*f) + (1-par)^(w*(1-f))

If 'par' represented the probability of a coin coming up heads, 'f' recorded the experimental frequency, and 'w' recorded the number of trials, then the above calculation would still be the likelihood of 'par'. That analogy makes it very tempting to call 'f' the frequency and 'w' the weight of evidence, as in NARS. However, there are no coin flips; in practice, few or no inheritance relations will be perfect. The next calculation tells us how to accumulate evidence despite this, matching the induction and abduction calculations of NARS. Also note that as evidence accumulates, f will approach par. This provides some justification for using f in a calculation when the unknown par is what we should really use, a practice that seems to follow from the NARS calculations.

The current evidence for a statement A=>B can be collected into pairs of statements of one of the two forms, (A=>C, B=>C) and (C=>B, C=>A). (Again, C is any term.) All evidence statements have their current <f, w> attached. Call the values in a pair (<f1,w1>,<f2,w2>). The 'f' of A=>B should be the weighted average of all the f1 values, as weighted by f2*w1*w2. The 'w' value is the sum of all these weights. This follows from the definition of A=>B along with some purposeful confusion between par and f. (At least, it seems so to me. A more careful derivation would be valuable. Feel free to correct!)

As with NARS, we need to avoid double-counting evidence. The math would more complicated when we need to worry about non-treelike information flow, of course... I won't worry about that for now...

Aside from ignoring that, I have also ignored the deduction rule for now. Maybe someone else will comment on that, or maybe I'll get to it later.

--Abram Demski