Two Approaches to Belief Revision

Setup
EUT Revision
AGM Revision
Comparison
Future Work
References
Extras
Setup
EUT Revision
AGM Revision
Comparison
Future Work
References
Extras
Today, we’ll sketch a new approach to (qualitative) belief
revision based on epistemic utility theory (EUT) and contrast
it with the traditional AGM theory of belief revision.
Two Approaches to Belief Revision
Ted Shear1
The EUT approach is based on a (normative) Lockean thesis.
It’s well known (lottery and preface paradoxes, etc. [2]) that
Lockean approaches to full belief fail to satisfy Cogency.
Branden Fitelson2
1 Philosophy
Cogency. An agent’s belief set B should (at any given time)
be deductively consistent and closed under logic.
@ UC Davis
Our main focus will be on divergences between EUT & AGM
that are orthogonal to the classic debates about Cogency.
[email protected]
2 Philosophy
& RuCCS @ Rutgers
As we will see, there is a precise sense in which AGM is
more demanding on the agent’s commitments than EUT.
[email protected]
The upshot of this will be that — from an EUT perspective
— AGM is an epistemically risk-seeking strategy for revising
one’s qualitiative beliefs. We begin with some setup.
Shear & Fitelson
Setup
1
Two Approaches to Belief Revision
EUT Revision
AGM Revision
Comparison
Future Work
References
Extras
Our agents possess both numerical credence functions, b(·),
and qualitative belief sets, B. When p ∈ B, we write B(p).
EUT Revision
AGM Revision
Comparison
Future Work
References
Extras
We take a veritistic (i.e., accuracy-centered) approach to
epistemic utility according to which the only feature of
epistemic attitudes that matters is their accuracy.
(1) b(·) is a classical (Kolmogorov) probability function.
(2) given a prior b(·), the posterior b0 (·) is generated via
conditionalizing b(·) on E — i.e., b0 (·) = b(· | E).
More precisely, we will adopt the following naïve,
accuracy-centered epistemic utility function for belief.

r
if p is true at w
u(B(p), w) Ö
−w if p is false at w
On the belief side, we’ll assume our agent entertains
(classical, possible world) propositions in some finite
agenda A, generated by a classical sentential language.
(3) B is the set of all and only those members of A which our
agent believes; and
The only constraint we will impose on r and w is
(4) given a prior belief set B, the posterior belief set B0 is
generated by revising the prior by E — i.e., B0 = B ? E
(where ? is some qualitative belief revision operator).
Two Approaches to Belief Revision
Setup
2
Two Approaches to Belief Revision
The basic idea tenet underlying EUT is that an agent’s belief
set should (at any given time) maximize expected epistemic
utility from the point of view of her credence function.
We will be talking about agents who revise both their
credence functions and their belief sets, upon learning
(exactly) some proposition E. On the credence side:
Shear & Fitelson
Shear & Fitelson
(†)
1 ≥ w > r ≥ 0.
The (Lockean) rationale for (†) will be explained shortly.
3
Shear & Fitelson
Two Approaches to Belief Revision
4
Setup
EUT Revision
AGM Revision
Comparison
Future Work
References
Extras
Setup
The expected epistemic utility (EEU) of a belief B(p), from
the point of view of a credence function b(·), is given by
X
EEU (B(p), b) Ö
b(w) · u(B(p), w)
Setup
w
.
r+w
Two Approaches to Belief Revision
EUT Revision
AGM Revision
Comparison
Future Work
References
Formally, B0 = B ÷ E, where
n B ÷ E Ö p b(p | E) >
5
Extras
Shear & Fitelson
Setup
w
r+w
o
.
6
Two Approaches to Belief Revision
EUT Revision
AGM Revision
Comparison
Future Work
References
Extras
w
w
and b(q) > r+w
(i.e., that
Proposition. Suppose b(p) > r+w
w
1
our EUT agent believes both p and q) and /2 < r+w
≤ 1.
(P2) If an agent learns something that she already believes, then
her belief set should remain unchanged.
Then, b(p | q) >
0
[More formally, X ∈ B ⇒ B = B ? X = B.]
w−r
w ,
and b(p | q) − b(p) <
r2
rw+w2 .
And, in the limit as an agent’s credences in p and q
approach 1, (P2) will be satisfied by EUT revision.
Informally, the reason ÷ violates (P2) is that — even if an
agent already believes q — learning q can lower her
credence in some other p proposition she also believes.
This goes some way toward explaining why (P2) may seem
like a plausible diachronic constraint on full belief, since it
is “approximately” true if full beliefs have sufficiently high
credence (and it is exactly true in the extremal case).
Indeed, learning something one already believes (q) can
drop one’s credence in another proposition one also
believes (p) below one’s EUT Lockean threshold.
More generally, extremal EUT agents (i.e., agents such that
w = 1 and r = 0, who will have Lockean thresholds equal to
one) will always satisfy all AGM principles.
However, there is a precise upper-bound on the “degree” to
which (P2) can fail from the perspective of EUT.
Two Approaches to Belief Revision
Extras
The following proposition provides a bound on how much
an EUT agent’s credence in one of her beliefs can be lowered
by learning something else that she already believes.
To get a feel for how EUT Revision works, it is instructive to
note that ÷ does not generally satisfy the following (AGM)
principle, which has recently been employed by Leitgeb [6].
Shear & Fitelson
References
EUT Revision. If an agent with a prior belief set B learns
(exactly) E, then her posterior B0 should maximize EEU
relative to her conditional credence function b(· | E).
Theorem (Dorst [3], Easwaran [4]) A belief set B (on A)
maximizes EEU relative to b if and only if, for every p ∈ B
Shear & Fitelson
Future Work
Our diachronic requirement will be that — upon learning E
via conditionaliation — our agent beleives exactly those
propositions that are sufficiently probable, a posteriori.
p∈B
EEU-maximization entails a normative Lockean thesis, with
w
threshold r+w
. This explains (†), since w ≤ r would permit
B(p) when b(p) ≤ 1/2, which is not in the Lockean spirit.
Comparison
Since our agents are (Bayesian) conditionalizers, this
immediately suggests a natural diachronic requirement.
The overall EEU of an agent’s belief set B, from the point of
view of her credence function b(·) is defined as
X
EEU (B, b) Ö
EEU (B(p), b)
+
AGM Revision
We just explained how EUT implies a synchronic coherence
constraint — the normative Lockean thesis.
w∈W
b(p) >
EUT Revision
7
Shear & Fitelson
Two Approaches to Belief Revision
8
Setup
EUT Revision
AGM Revision
Comparison
b(p | q)
Future Work
References
Extras
Setup
EUT Revision
Future Work
References
Extras
AGM’s underlying principle is the principle of
Conservativity (also sometimes called the principle of
informational economy, or minimal mutilation).
0.8
0.6
Conservativity. When an agent learns E, she should revise
to a posterior belief set B0 such that (a) B0 accommodates E,
(b) B0 is deductively cogent, and (c) B0 constitutes the
minimal change to B which satisfies (a) and (b).
0.4
0.2
0.6
0.7
0.8
0.9
1.0
Figure: The “degree” to which EUT belief updating can violate
Leitgeb’s (P2), as a function of the EUT Lockean threshold t =
Shear & Fitelson
EUT Revision
t=

+
Comparison
The AGM revision operator can be characterized
axiomatically. Next, we look at the AGM axioms.
w
r+w .
9
Two Approaches to Belief Revision
AGM Revision
AGM’s core synchronic requirement is Cogency (which
needs to be assumed as a standing constraint) and its
revisions are generally “logical” in nature.
Future Work
References
Extras
Shear & Fitelson
Setup
10
Two Approaches to Belief Revision
EUT Revision
AGM Revision
Comparison
Future Work
References
Extras
Given the other AGM axioms, Superexpansion and
Subexpansion imply Inclusion and Vacuity, respectively,
assuming only the following weak additional postulate.
Basic AGM postulates:
(*1) B ∗ E = Cn(B ∗ E)
Closure
(*2) E ∈ B ∗ E
Idempotence. B ∗ > = B.
Success
(*3) B ∗ E ⊆ Cn(B ∪ {E})
Inclusion
(*4) If E is consistent with B, then B ∗ E ⊇ Cn(B ∪ {E})
Vacuity
(*5) If E is not a contradiction, then B ∗ E is consistent Consistency
(*6) If X a Y , then B ∗ X = B ∗ Y
Extensionality
Supplementary AGM postulates:
(*7) B ∗ (X ∧ Y ) ⊆ Cn((B ∗ X) ∪ {Y })
(*8) If Y is consistent with Cn(B ∗ X), then
B ∗ (X ∧ Y ) ⊇ Cn((B ∗ X) ∪ {Y })
Shear & Fitelson
Comparison
The AGM theory of belief revision (Alchourron, Gärdenfors
& Makinson [1]) is the most widely investigated and
influential account of qualitative belief revision.
b(p) = b(q) = t
1.0
Setup
AGM Revision
Two Approaches to Belief Revision
Superexpansion
Subexpansion
11
1.
2.
3.
4.
5.
6.
Y ∈B∗X
B ∗ X = B ∗ (> ∧ X)
Y ∈ B ∗ (> ∧ X)
Y ∈ Cn((B ∗ >) ∪ {X})
Cn((B ∗ >) ∪ {X}) = Cn(B ∪ {X})
Y ∈ Cn(B ∪ {X})
Assumption
(1), Extensionality
(1), (2), Logic
(3), Logic, Superexpansion
Idempotence, Logic
(4), (5), Logic
1.
2.
3.
4.
5.
X
Y
Y
Y
Y
Assumption
Assumption
(2), Idempotence, Logic
(3), Subexpansion
(4), Extensionality Shear & Fitelson
is consistent with B
∈ Cn(B ∪ {X})
∈ Cn((B ∗ >) ∪ {X})
∈ B ∗ (> ∧ X)
∈B∗X
Two Approaches to Belief Revision
12
Setup
EUT Revision
AGM Revision
Comparison
Future Work
References
Extras
Setup
EUT Revision
AGM Revision
Comparison
Future Work
References
(*4) If E is consistent with B, then B ∗ E ⊇ Cn(B ∪ {E})
Proof: We just need an example in which E is consistent
with B, but B ÷ E É Cn(B ∪ {E}). To this end, consider the
credence function generated by the following assignments:
Closure
(*2) E ∈ B ∗ E
Success
(*5) If E is not a contradiction, then B ∗ E is consistent Consistency
(*6) If X a Y , then B ∗ X = B ∗ Y
Extensionality
(*7) B ∗ (X ∧ Y ) ⊆ Cn((B ∗ X) ∪ {Y })
Superexpansion
(*8) If Y is consistent with Cn(B ∗ X), then
B ∗ (X ∧ Y ) ⊇ Cn((B ∗ X) ∪ {Y })
(*9) B ∗ > = B
Vacuity
Proposition. ÷ does not satisfy Vacuity (or Subexpansion).
Alternative Axiomatization of AGM, using Idempotence.
(*1) B ∗ E = Cn(B ∗ E)
Extras
Subexpansion
p
b(p)
E∧X
E ∧ ¬X
¬E ∧ X
¬E ∧ ¬X
.3
.05
.325
.325
Given a Lockean threshold of t = 0.9, the prior belief set is
Idempotence
B = {E ⊃ X}.
And, the (EUT) posterior belief set is
B0 = B ÷ E = {E, E ∨ X, E ∨ ¬X}.
Shear & Fitelson
Setup
EUT Revision
13
Two Approaches to Belief Revision
AGM Revision
Comparison
Future Work
(*3) B ∗ E ⊆ Cn(B ∪ {E})
References
Extras
EUT Revision
AGM Revision
Comparison
Future Work
References
Extras
Because EUT satisfies Inclusion, it will never require the
agent to gain more new beliefs than AGM. However, EUT
may require the agent to lose more beliefs than AGM.
Proposition. B ÷ E ⊆ Cn(B ∪ {E})
This feature (in conjunction with the fact that AGM may
require the agent to gain more new beliefs than EUT) shows
that EUT is less demanding of an agent (commitment-wise).
Proof: Suppose X ∈ B ÷ E. Then, b(X | E) > t. And, it is a
theorem of probability calculus that Pr(E ⊃ X) ≥ Pr(X | E).
Therefore, b(E ⊃ X) > t. So, E ⊃ X ∈ B. Hence, by modus
ponens (for material implication), X ∈ Cn(B ∪ {E}).
In this sense, AGM is more risk-seeking in its diachronic
requirements. Pettigrew [8] has independently argued (via
thw use of an epistemic Hurwicz Criterion) that the
insistance on Cogency is epistemically risk-seeking.
A similar argument shows that EUT revision satisfies the
more general principle Superexpansion. Suppose
P ∈ B ÷ (X ∧ Y ). Then, b(P | X ∧ Y ) > t. It is a theorem of
probability calculus that Pr((X ∧ Y ) ⊃ P ) ≥ Pr(P | X ∧ Y ).
Therefore, b((X ∧ Y ) ⊃ P ) > t. So, (X ∧ Y ) ⊃ P ∈ B. So, by
Success and modus ponens, P ∈ Cn((B ÷ X) ∪ {Y }).
Two Approaches to Belief Revision
Setup
14
Two Approaches to Belief Revision
The crucial difference between AGM and EUT (aside, of
course, from Cogency) is Vacuity/Subexpansion.
Inclusion
Intuitively, Inclusion requires that a revision does not
include any more than the logical closure of the union of
the original beliefs with the learned proposition.
Shear & Fitelson
Shear & Fitelson
We close with a final theorem, which illuminates the tight
connection between EUT’s violations of Vacuity (and
Subexpansion) and its “risk aversion” (vs AGM revision).
15
Shear & Fitelson
Two Approaches to Belief Revision
16
Setup
EUT Revision
AGM Revision
Comparison
Future Work
References
Extras
Setup
EUT Revision
EUT violates Vacuity (wrt B, E) a E is consistent with B and
Proof.
Future Work
Longer term, we also want to investigate iterated revision
(EUT vs AGM), and also theories of belief update (in contrast
to revision). We’re planning a series of four papers . . .
17
Two Approaches to Belief Revision
Comparison
Extras
Jan van Eijck & Bryan Renne [5] recently provided a modal
logic for belief given a Lockean threshold of 1/2. A near-term
task is to investigate how their modal logic may be used to
define a system of belief revision for a threshold of 1/2.
(⇐) Suppose E is consistent with B and B ÷ E ⊂ B ∗ E. Then, there
exists an X such that X ∈ B ∗ E but X ∉ B ÷ E. Because E is consistent
with B, Vacuity and Inclusion imply that B ∗ E = Cn(B ∪ {E}).
Therefore, X ∈ Cn(B ∪ {E}); but, X ∉ B ÷ E.
AGM Revision
References
It would be nice to have a purely qualitative
characterization/axiomatization of EUT Revision. Ideally,
we’d like to have one for arbitrary Lockean thresholds.
(⇒) Suppose EUT violates Vacuity (wrt B and E). Then, (a) E is
consistent with B; and, (b) B ÷ E É Cn(B ∪ {E}). By (b), there exists an
X such that X ∈ Cn(B ∪ {E}) but X ∉ B ÷ E. It follows from (a), Vacuity
and Inclusion that Cn(B ∪ {E}) = B ∗ E. Therefore, X ∈ B ∗ E and
X ∉ B ÷ E. And, by Inclusion, B ÷ E ⊆ Cn(B ∪ {E}) = B ∗ E.
EUT Revision
Future Work
It would be useful to investigate EUT Revision from a
“non-classical probability” perspective (e.g., two-place
Popper functions, imprecise probability functions, etc.).
B ÷ E ⊂ B ∗ E.
Setup
Comparison
While we now have a clearer understanding of the
(interesting) ways in which EUT and AGM revision diverge,
there is much more work to be done in this project.
Theorem
Shear & Fitelson
AGM Revision
References
Extras
Shear & Fitelson
Setup
EUT Revision
18
Two Approaches to Belief Revision
AGM Revision
Comparison
Future Work
References
Extras
[1] Carlos Alchourron, Peter Gärdenfors, and David Makinson. On the logic of
theory change: Partial meet contraction and revision functions. The Journal of
Symbolic Logic, 50(2): 510–530, June 1985.
[2] D. Christensen, Putting Logic in its Place, OUP, 2007.
1.
2.
3.
4.
5.
6.
7.
8.
9.
[3] Kevin Dorst. An Epistemic Utility Argument for the Threshold View of Outright
Belief Manuscript, 2014.
[4] Kenny Easwaran. Dr. Truthlove, or how I learned to stop worrying and love
Bayesian probabilities. Manuscript, 2014.
[5] Jan van Eijck and Bryan Renne. Belief as Willingness to Bet. Manuscript, 2014.
[6] Hannes Leitgeb. The review paradox: On the diachronic costs of not closing
rational belief under conjunction. Noûs, 78(4): 781–793, 2013.
[7] David Makinson and James Hawthorne. Lossy Inference Rules and their Bounds:
A Brief Review. The Road to Universal Logic: Festschrift for 50th Birthday of
Jean-Yves Béziau, Vol. 1, pages 385–407, Springer, 2015.
[8] Richard Pettigrew. Accuracy and Risk: a Jamesian investigation in formal
epistemology. Manuscript, 2015.
B is consistent.
B is closed, i.e., B = Cn(B).
X ∈ B.
X is consistent with B.
B ∗ X = Cn(B ∪ {X}).
B ∗ X = Cn(B).
B ∗ X = Cn(B ∗ X)
Cn(B ∗ X) = Cn(B)
B∗X =B
Assumption
Assumption
Assumption
(1), (3), Logic
(4), Vacuity, Inclusion
(5), (3), Logic
Closure
(6), (7), Logic
(7), (8), (2), Logic
Figure: Derivation of (P2) from Closure, Inclusion, and Vacuity
1
[9] Alexander Pruss. The badness of being certain of a falsehood is at least log(4)−1
times greater than the value of being certain of a truth. Logos & Episteme, 3(2):
229–238, 2012.
[10] Florian Steinberger. Explosion and the normativity of logic. Mind, to appear.
Shear & Fitelson
Two Approaches to Belief Revision
19
Shear & Fitelson
Two Approaches to Belief Revision
20
Setup
EUT Revision
1.
2.
3.
4.
5.
6.
AGM Revision
Comparison
Y ∈B∗X
B ∗ X = B ∗ (> ∧ X)
Y ∈ B ∗ (> ∧ X)
Y ∈ Cn((B ∗ >) ∪ {X})
Cn((B ∗ >) ∪ {X}) = Cn(B ∪ {X})
Y ∈ Cn(B ∪ {X})
Future Work
References
Extras
Assumption
(1), Extensionality
(1), (2), Logic
(3), Logic, Superexpansion
Idempotence, Logic
(4), (5), Logic
Figure: Derivation of Inclusion from Superexpansion, Extensionality,
and Idempotence
Setup
EUT Revision
AGM Revision
p
b(p)
E∧X
E ∧ ¬X
¬E ∧ X
¬E ∧ ¬X
E
X
E≡X
E ≡ ¬X
¬E
¬X
E∨X
E ∨ ¬X
¬E ∨ X
¬E ∨ ¬X
.3
.05
.325
.325
.35
.625
.625
.375
.65
.375
.675
.675
.95
.7
Comparison
b(p | E)
6/7
1/7
0
0
1
6/7
6/7
1/7
0
1/7
1
1
6/7
1/7
Future Work
p ∈ Cn(B ∪ {E})?
Yes
No
No
No
Yes
Yes
Yes
No
No
No
Yes
Yes
Yes
No
References
Extras
p ∈ B ÷ E?
No
No
No
No
Yes
No
No
No
No
No
Yes
Yes
No
No
Table: Full counterexample to Vacuity for EUT Revision
Shear & Fitelson
Setup
21
Two Approaches to Belief Revision
EUT Revision
AGM Revision
p
b(p)
E∧X
E ∧ ¬X
¬E ∧ X
¬E ∧ ¬X
E
X
E≡X
E ≡ ¬X
¬E
¬X
E∨X
E ∨ ¬X
¬E ∨ X
¬E ∨ ¬X
.3
.05
.325
.325
.35
.625
.625
.375
.65
.375
.675
.675
.95
.7
Comparison
b(p | E)
6/7
1/7
0
0
1
6/7
6/7
1/7
0
1/7
1
1
6/7
1/7
p ∈ B?
No
No
No
No
No
No
No
No
No
No
No
No
Yes
No
Future Work
p ∈ B ÷ E \ B?
No
No
No
No
Yes
No
No
No
No
No
Yes
Yes
No
No
References
p ∈ B ∗ E \ B?
Table: Demonstration that B ÷ E ⊂ B ∗ EB in our example
Shear & Fitelson
Two Approaches to Belief Revision
Extras
Yes
No
No
No
Yes
Yes
Yes
No
No
No
Yes
Yes
No
No
23
Shear & Fitelson
Two Approaches to Belief Revision
22