Hi,
When reading this nice survey paper of Eliezer's
"
_Cognitive biases potentially affecting judgment of global risks_
http://singinst.org/Biases.pdf
"
I was reminded of some of the heuristics and biases that exist in the
Novamente system right now.
For instance, consider the case of conjunctions, as exemplified by problems like
"
Example: Linda is 31, outspoken, and very bright. She majored in
philosophy in college. As a student, she was deeply concerned with
discrimination and other social issues, and she participated in
antinuclear demonstrations. Which statement is more likely to be
true?
a. Linda is a bank teller
b. Linda is a bank teller and active in the feminist movement.
"
Here we have a problem of the form
a.
P( bank teller | dataset)
b.
P( bank teller AND active feminist | dataset)
Humans tend to rank b as more likely than a, which is wrong...
A study of how Novamente handles this problem using different
heuristic inference approaches is interesting in terms of exemplifying
the ways Novamente is and is not irrational....
Basic probability theory says
P( bank teller AND active feminist | dataset) =
P(bank teller | dataset) P(active feminist | bank teller AND dataset)
<= P(bank teller)
The first, preliminary version of Novamente's heuristic/probabilistic
reasoning system would have simply said, in the absence of any further
information,
P( bank teller AND active feminist | dataset) =
P(bank teller | dataset) P(active feminist | dataset)
i.e. assuming independence of "bank teller" and "active feminist"
given the dataset.
However, we found that this gave a systematic *conjunction
underestimation* bias.
When applied to appropriately generated random mathematical sets, it
is of course completely unbiased, due to its foundation on an
independence assumption.
However, in practice, the concepts being combined within the Novamente
system had a certain bias:
-- they tended to be "cohesive", in the sense that the elements a
concept gathers together often tend to be approximately clustered in
"feature vector space"
It turns out that if you are estimating
P(A & B)
where it is strongly suspected that there is at least one element in
the intersection, and that A and B are cohesive, then other heuristic
formulas give less error than the standard independence assumption
P(A&B) = P(A) P(B)
As a heuristic, we use the theorem
***
DEFINITION
LET f(x,y) = expected (probability) measure of the intersection of a
set with measure x and a set with measure y, assuming that the two
sets have nonzero intersection, and assuming a measure space with
total measure 1.
THEOREM
Suppose one has sets that are "k-spheres" on the surface of a
(k+1)-sphere. A special case of this is where the sets are intervals
on the boundary of the unit circle. Let V(d) denote the volume of the
k-sphere with diameter d, assuming a measure in which the volume of
the whole surface of the (k+1)-sphere is 1. Given a number r, let
diam(r) denote the diameter of the k-sphere with measure r.
Then, we have f(x, y) = x y / V(diam(x)+diam(y))
***
As a special case for k=1 we get the heuristic
f(r,q) = r * q / min (1, r+q)
So assuming k=1 we get the heuristic
P(A&B) = P(A) P(B) / min(1, P(A) + P(B) )
Note that this "concept geometry" approach will tend to underestimate
less than the standard heuristic independence assumption, given the
stated assumptions (A and B have nonzero intersection and tend to be
round).
But it's still a crude heuristic that may lead to errors, of course....
For instance if (to choose some plausible numbers)
P(bank teller) = .01
P(active in the feminist movement) = .1
then it yields
P(bank teller AND active feminist) = .01 * .1 / .11 = .009
as opposed to the
P(bank teller AND active feminist) = .01 * .1 / .11 = .001
value that the naive independence assumption would give.
Similarly we could estimate
P(bank teller & active feminist | dataset) =
P(bank teller | dataset) P(active feminist | dataset) * c
where
c = P(dataset) / min(1, P(active feminist & dataset) + P(bank teller & dataset))
So given
P(dataset) = .001
P(active feminist & dataset) = .0005
P(bank teller & dataset) = .000001
[knowledge embodying prior information that the dataset about Linda is
more conducive to the active feminist hypothesis than the bank teller
hypothesis]
thus
P(active feminist | dataset) = .5
P(bank teller | dataset) = .001
we find an estimate of
P(bank teller & active feminist | dataset) = .0005
using an independence assumption naively but using the 1-d concept
geometry heuristic we find
c = .001 /(.0005+.000001) = 1.996...
yielding an estimate of
P(bank teller & active feminist | dataset) = .000998...
Note that we still have
P(bank teller & active feminist | dataset) < P(bank teller | dataset)
as required, but this is weak, it just means
.000998 < .001
So, using a plausible cognitive heuristic for overcoming the
conjunction UNDERestimation bias implicit in naive probabilistic
independence assumptions, one arrives at the conclusion that
P(bank teller | dataset)
and
P(bank teller & active feminist | dataset)
are about the same...
Note that the kind of heuristic I've described here does NOT yield the
kind of error that is seen by humans carrying out this sort of
inference.
This illustrates a difference between the way Novamente works and the
way the human mind works, regarding estimation of uncertainties.
Novamente uses various heuristics along with probabilistic rules, and
some of these heuristics are cruder than others. However, the
heuristics are applied within a probabilistic framework so that the
kinds of really gross errors we see in naive human commonsense
reasoning don't happen.
-- ben
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