We introduce a ``rarity of defeat'' principle, valid in cases of deference to an expert, to address intuitions involved in a puzzle of Nissan-Rozen concerning epistemic deference and evidential screening-off

McCutcheon, Randall

English

Philsci-Archive

ON A PUZZLE ABOUT EXPERTS, SCREENING-OFF
AND THE RARITY OF DEFEAT
Abstract. We introduce a \rarity of defeat" principle, valid in
cases of deference to an expert, to address intuitions involved in
a puzzle of Nissan-Rozen concerning epistemic deference and evi-
dential screening-o.
Ittay Nissan-Rozen (2018) invites us to consider the following proposi-
tions.
H { John is a physics professor at Harvard
T { John has published at least 3 papers in physics journals
X = X(x) { Jane believes H to degree x1
Nissan-Rozen stipulates that we know that Jane knows whether or
not T is true, and that, moreover, \you are certain that Jane has
much more information about John (who you have never met and know
almost nothing about) than you and you are certain that she is more
epistemically competent than you in evaluating this evidence...."
Nissan-Rozen claims (he cites Hall 2004, Elga 2007 and Joyce 2007
in support of this claim) that there is \widespread agreement" that
in cases like this we should defer to the more knowledgeable party,
i.e. adopt her credences as our own, should we learn them. That is,
our credence function c ought to have the property that c(HjX) = x;
we should treat Jane as an \expert", exhibiting what Nissan-Rozen
calls Epistemic Deference. But if that's the case, our credence function
ought to satisfy both c(HjTX) = x and c(Hj:TX) = x as well; Nissan-
Rozen calls this Strong Epistemic Deference. For even if we were to
learn that T is the case (say), Jane would still be more knowledgeable
and competent than we are, and X says that Jane's credence in H is
x.2 So we should still defer, and adopt Jane's credence in H as our own.
That is, conditional on X we ought to view T and H as independent.
1We follow Nissan-Rozen in writing simply \X" rather than \X(x)", though the
reader should bear the dependence of X on x continuously in mind.
2A referee wrote \(Nissan-Rozen) seems to think that if Jane is an expert about
H for us then she should remain one even after we learn the truth of T . This is
wrong. Experts need not remain experts conditional on new information." But they
do remain experts conditional on information that's not new to them. (Provided we
know that it's not, as is the case here.) So this is an apparent misunderstanding.
1
ON A PUZZLE ABOUT EXPERTS 2
That independence, however, does not sit well with Nissan-Rozen. For
note that, since c(HjTX) = c(Hj:TX) = c(HjX) is equivalent to
c(T jHX) = c(T j:HX) = c(T jX) and the former holds, the latter also
holds. But imagine now that we learn the truth of X for some low
value of x. Since our only interesting evidence for or against T is that
Jane has the low credence x in H, it seems reasonable that we should
come to have low credence in T as well, i.e. c(T jX) should be low. By
independence, though, c(T jHX) = c(T jX), so c(T jHX) should be low.
Nissan-Rozen rejects this conclusion out of hand, writing \However,
c(T jHX) is very high (if John is a physics professor at Harvard he
probably has published at least 3 papers in physics journals)".
Though the argument postures as a reductio of Epistemic Deference,
there are reasons not to grant it this status. First there is some con-
fusion caused by an inconsistency. It is stated at the beginning of the
thought experiment that one can assume of the expert Jane \any level
of epistemic competence that you would like". That can't be right,
though, for that wouldn't preclude Jane being ideally rational, and
that one ought to defer to ideally rationally agents who know more
than we do is wholly unassailable. Indeed, there is cause to suspect
that a referee for the paper called attention to this fact. Consider the
following passage from footnote 1:
My discussion is...limited to cases of deference to human ex-
perts. As an anonymous referee pointed to me [sic], when
considering (Epistemic Deference) in cases in which the ex-
pert is ideally rational...our intuitions regarding formally
equivalent examples dramatically change.
What the referee called attention to, we suspect, is that while the
intuition that \c(T jHX) is very high" might be appropriate to a case
in which Jane is human (and therefore an imperfect reasoner), it isn't
appropriate to a case in which Jane is ideally rational. If that's right
then it was an oversight to fail to remove the earlier claim that one
can assume of Jane \any level of epistemic competence that you would
like". For in order for this argument to work, we have to believe that
Jane might be imperfect.
That clears up one confusion, but it creates a new problem. For al-
though there is surely \widespread agreement" that one ought to defer
completely to the credences of a better-informed party if one is certain
that she is ideally rational, there's no such agreement to the eect that
one ought to defer completely to the credences of a better-informed
party simply because one is \certain that she is more epistemically
ON A PUZZLE ABOUT EXPERTS 3
competent". Indeed, without some reworking this criterion is wholly
implausible, for it admits extreme cases in which we plainly should not
defer, such as when we and the better-informed party exhibit dierent
positive degrees of anti-expertise. (And, of course, the miniscule prob-
ability of :TH relative to TH makes c(T jHX) extremely sensitive to
gratuitous reports of credence x on Jane's part in TH scenarios, so not
even small deviations from \ideal" behavior can be safely ignored.)
In particular, we see little sympathy in Nissan-Rozen's sources for
complete deference to non-ideal agents. Hall (2004) speaks of agents
whose conditional probabilities are deferred to (he calls these \analyst-
experts") being perceived as having \astonishing powers of evidential
reasoning". Elga (2007) does write in one spot that \When it comes
to the weather, I completely defer to the opinions of my local weather
forecaster", but later he concedes that this isn't really true: \But upon
nding out that my forecaster is condent that it will rain eggplants
tomorrow, I will not follow suit. I will conclude that my forecaster is
crazy." Thus sobered, he concludes that \only in highly idealized cases
is it appropriate to treat someone as an expert..." Joyce (2007) mean-
while describes a range of partial deference scenarios: \C might see q's
values as a better guide to A's truth-value than her own, but still assign
her own views some weight, so that C(Ajq(A) = x) falls between x and
1
2
C(A) + 1
2
x:" So even if one concedes that Nissan-Rozen's argument
works in suciently non-ideal cases, it isn't clear that it would thereby
refute a position that anyone subscribes to.
Nevertheless it's still possible appreciate the puzzle for the creative
manner in which it challenges naive intuition (even in the case where
Jane is taken to be ideally rational{an assumption we will make going
forward). Nissan-Rozen writes of his paper that it \emerged from an
exercise in Bayesian epistemology" he wrote for a course at the Hebrew
University, and he thanked his students for \challenging me to nd
interesting ways to teach epistemology and by doing so to gain a better
understanding of several key issues in epistemology." We think that
the puzzle serves this end well, and that we have a good line on the
issues arising in it. Follows now our attempt to communicate it.
Nissan-Rozen entertains then rejects the idea that c(T jHX) should be
low in the following passage:
ON A PUZZLE ABOUT EXPERTS 4
The thought might be that since Jane knows whether T is
true (i.e. she knows whether John has published at least 3
papers in physics journals) the fact that she believes H to
a given low degree is (for some low degrees) indicative for
the falsity of T . This is so because she might believe H to
a low degree because (or partly because) she knows that T
is false (i.e. she might believe to a high degree that John is
not a physics professor at Harvard because she knows he has
not published at least 3 papers in physics journals). While I
agree that this line of reasoning can support the claim that
c(T jHX) might be lower than c(T jH), I do not see how it
can plausibly support the claim that c(T jHX) might be low
in absolute terms (say lower than 0.5). Jane might believe
H to a low degree for many reasons. Jane might know that
John lives in London or she might know that he once took
an oath never to teach in Harvard, or she might know that
he spends most of his waking hours playing soccer.
As noted, denying lowness of c(T jHX) when x is low requires one to
renounce Epistemic Deference, a radical and plainly unwarranted move
in the case we are interested in (where Jane is ideally rational). Still,
the intuition that c(T jHX) ought to be high is stubborn, even in this
case. It's wrong of course{but how can one (more perspicuously than
does the independence argument) defuse it?
We're sure there are many ways, but we think we've found an inter-
esting one. Namely, via a \rarity of defeat" principle (of independent
interest) that, to our minds, wholly and cleanly undercuts the stub-
born intuition. The principle will be established using two features
(the rst entails the second) that apply whenever an agent C with cre-
dence function c defers to an expert J having credence function j and
A is a measurable event. Denoting C's expectation operator by E, they
are:
(i) E(1Ajj(A)  x)  x; and
(ii) E(j(A)) = c(A).
Here now is our formulation of the principle.
Theorem 1. (Rarity of Defeat) Let A be a measurable event. Then
for any x < 1 with 0 < x  c(A),
c(j(A)  xjA) 
x(1  c(A))
(1  x)c(A)
: (1)
ON A PUZZLE ABOUT EXPERTS 5
Proof. Denote by X the event j(x)  x. (ii) says that
c(A) = E(j(A))  xc(j(x)  x) + 1  c(j(x) > x);
i.e. c(A)  xc(X) + 1  c(X), from which it follows that
c(X) 
1  c(A)
1  x
: (2)
(i) meanwhile says that c(AjX)  x. Together with (2) this yields
c(AX)  xc(X) 
x(1  c(A))
1  x
:
Dividing both sides of this equation by c(A) gives (1). qed
Theorem 1's estimate is sharp; if J has conditioned on the partition
n
A ^

y 
x(1  c(A))
(1  x)c(A)

;:A _

y >
x(1  c(A))
(1  x)c(A)
o
;
where y is independently and uniformly distributed on (0; 1), there is
equality in (1).
The rarity of defeat principle gets its name from cases where c(A) is
near 1 and one seeks lower bounds on the condence one should have
that one knows that A. Suppose an ideal C's current credence in A
is .999, and we subscribe to the formalism that KA if and only if A
is true and C's credence in A will justiably reach 1 eventually while
never falling below .9. Taking J to be the future time slice of C where
her credence in A is at its global future minimum (assuming that J 's
credence function j always has the form j() = c(jP) for a measurable
partition P), Theorem 1 says:
c(:KA) = c(:A)+c(A)c(j(A)  :9jA)  :001+(:999)
:9(1  :999)
(1  :9):999
= :01:
Therefore C has at least 99% condence that she knows that A.
The point at which the argument we are critiquing falls apart in the
case where Jane is ideally rational can be isolated from the following
excerpts:
1. \However, c(T jHX) is very high (if John is a physics professor
at Harvard he probably has published at least 3 papers in physics
journals)..."
2. \Jane might believe H to a low degree for many reasons. Jane might
know that John lives in London or..."
Taken by itself, 1. is a clear non-sequitur; all that follows from the
parenthetical observation is that c(T jH) is very high. 2. looks like
ON A PUZZLE ABOUT EXPERTS 6
an attempt to explain this away by minimizing the impact of having
observed X. We can check to see if this works when x is low by letting
X denote the event j(H)  x = 2c(Hj:T ) and writing:
c(T jHX) = c(T jH)
c(XjTH)
c(XjH)

= c(T jH)
 c(XjTH)
c(:T jH)c(Xj:TH) + c(T jH)c(XjTH)

:
By the intuitions we are scrutinizing this too is \very high". Of course
we concede that c(T jH) is very high, so the second summand in the
denominator of the rightmost expression is essentially equal to the nu-
merator. So this reasoning requires that the rst summand should be
negligible relative to the second. In particular, it requires:
Key Premise: c(:T jH)c(Xj:TH) is negligible relative to c(XjTH).
We doubt that anyone would nd the key premise appealing because
they think that c(Xj:TH) is small. It shouldn't be that surprising that
Jane would have credence in H less than or equal to twice c(Hj:T )
conditional on :TH, because in such a case :T would be part of her
evidence. (She could have additional evidence raising her credence inH
above 2c(Hj:T ), but not necessarily.) Rather, we think that what gives
the above premise its rst blush appeal is the intuition that c(:T jH)
should be negligible relative to c(XjTH). Indeed, we think that is the
point excerpt 2. is getting at. \Jane might believe H to a low degree
for many reasons" looks to be an imprecise way of conveying that we
should not be that surprised should Jane turn out to have such a low
degree of belief, even in a case where both T and H are true.
According to the rarity of defeat principle, however, c(:T jH) can't be
negligible relative c(XjTH) if Jane is an ideal rational agent. This is
because Jane's credence function j is expert with respect to k() =
c(jT ), so Theorem 1 says that
c(XjTH) = k

j(H)  xjH


x(1  k(H))
(1  x)k(H)
=
2c(Hj:T )(1  c(HjT ))
(1  2c(Hj:T ))c(HjT )
=
2c(:TH)(c(T )  c(TH))
c(H)c(:T )
 c(H)
(1  2c(Hj:T ))c(TH)

The second factor in the nal expression is  1, and presumably c(T ) 
1
4
(this is generous; we know essentially nothing about John, and far
ON A PUZZLE ABOUT EXPERTS 7
less than a quarter of people in the population have published papers
in physics journals), so
c(XjTH) < 3c(:T jH)
c(T )
c(:T )
 c(:T jH):
So it isn't just that c(:T jH) isn't negligible relative to c(XjTH)...it's
in fact always larger than c(XjTH). Indeed, in most scenarios it's
probably much larger, because even near equality holds in Theorem 1
only in very special circumstances.
This analysis also shows how proneness to irrationally extreme cre-
dences on Jane's part might make room for a scenario in which c(T jHX)
is in fact high. For what Theorem 1 shows to be rare is \justied de-
feat" or \defeat by the evidence"; defeat by one's own irrationality
needn't be rare at all. So, if it's not that unlikely that Jane's credences
in H have been decimated by evidence that is less supportive of :H
than she believes (John lives in Rhode Island, say, or spends a fair
number of weekend hours playing soccer), the key premise could turn
out to be true. Under such an assumption, one would, upon learning
that T , H and X were all true, come to believe it likely that Jane had
adopted a gratuitously low credence in H.
References
Elga, A. 2007. Reection and disagreement. Nous 41:478-502.
Hall, N. 2004. Two mistakes about credence and chance. Australasian
Journal of Philosophy 82:93-111.
Joyce, M.J. 2007. Epistemic deference: the case of chance. Proceedings
of the Aristotelian Society 107:187-206.
Nissan-Rozen, I. 2018. A puzzle about experts, evidential screening-o
and conditionalization. Episteme. Forthcoming.


On a Puzzle About Experts, Screening-Off and the Rarity of Defeat

http://philsci-archive.pitt.edu/16025/1/gigoD2.pdf

On a Puzzle About Experts, Screening-Off and the Rarity of Defeat

Abstract

Similar works

Full text

Available Versions

Philsci-Archive