Title:
1Belief Revision and Truth-Finding
- Kevin T. Kelly
- Department of Philosophy
- Carnegie Mellon University
- kk3n_at_andrew.cmu.edu
2Further Reading
- (with O. Schulte and V. Hendricks) Reliable
Belief Revision, in Logic and Scientific
Methods, Dordrecht Kluwer, 1997. - The Learning Power of Iterated Belief Revision,
in Proceedings of the Seventh TARK Conference,
1998. - Iterated Belief Revision, Reliability, and
Inductive Amnesia, Erkenntnis, 50 1998
3The Idea
- Belief revision theory... rational belief
change - Learning theory...............reliable belief
change - Conflict?
Truth
4Part I
5Bayesian (Vanilla) Updating
Propositional epistemic state
B
6Bayesian (Vanilla) Updating
- New belief is intersection
- Perfect memory
- No inductive leaps
E
new evidence
B
7Bayesian (Vanilla) Updating
- New belief is intersection
- Perfect memory
- No inductive leaps
E
B
B
8Bayesian (Vanilla) Updating
- New belief is intersection
- Perfect memory
- No inductive leaps
E
B
B
9Epistemic Hell (a.k.a. Nirvana)
B
10Epistemic Hell (a.k.a. Nirvana)
E
Surprise!
B
11Epistemic Hell (a.k.a. Nirvana)
- Scientific revolutions
- Suppositional reasoning
- Conditional pragmatics
- Decision theory
- Game theory
- Data bases
E
B
Epistemic hell
12Ordinal Epistemic StatesSpohn 88
- Ordinal-valued degrees of implausibility
- Belief state is bottom level
w 1
w
2
1
b (S)
0
S
13Iterated Belief Revision
epistemic state trajectory
initial state
S0
input propositions
E0
E1
E2
14Iterated Belief Revision
epistemic state trajectory
S1
S0
input propositions
E1
E2
15Iterated Belief Revision
epistemic state trajectory
S1
S0
input propositions
E1
E2
16Iterated Belief Revision
epistemic state trajectory
S2
input proposition
E2
17Iterated Belief Revision
epistemic state trajectory
S2
input proposition
E2
18Iterated Belief Revision
epistemic state trajectory
S3
19Iterated Belief Revision
epistemic state trajectory
S3
b (S0)
b (S1)
b (S2)
b (S3)
belief state trajectory
20Generalized Conditioning CSpohn 88
S
21Generalized Conditioning CSpohn 88
- Condition entire epistemic state
E
S
22Generalized Conditioning CSpohn 88
- Condition entire epistemic state
E
C
S
S C E
23Generalized Conditioning CSpohn 88
- Condition entire epistemic state
- Perfect memory
- Inductive leaps
- No epistemic hell if evidence sequence is
consistent
E
C
S
S C E
24Lexicographic Updating LSpohn 88, Nayak 94
S
25Lexicographic Updating LSpohn 88, Nayak 94
- Lift refuted possibilities above non-refuted
possibilities preserving order.
S
26Lexicographic Updating LSpohn 88, Nayak 94
- Lift refuted possibilities above non-refuted
possibilities preserving order.
L
S
S L E
27Lexicographic Updating LSpohn 88, Nayak 94
- Lift refuted possibilities above non-refuted
possibilities preserving order. - Perfect memory on consistent data sequences
- Inductive leaps
- No epistemic hell
L
S
S L E
28Minimal or Natural Updating MSpohn 88,
Boutilier 93
S
29Minimal or Natural Updating MSpohn 88,
Boutilier 93
- Drop the lowest possibilities consistent with the
data to the bottom and raise everything else up
one notch
E
S
30Minimal or Natural Updating MSpohn 88,
Boutilier 93
- Drop the lowest possibilities consistent with the
data to the bottom and raise everything else up
one notch
E
M
S
S M E
31Minimal or Natural Updating MSpohn 88,
Boutilier 93
- Drop the lowest possibilities consistent with the
data to the bottom and raise everything else up
one notch - inductive leaps
- No epistemic hell
E
M
S
S M E
32The Flush-to-a Method F,a Goldszmidt and
Pearl 94
S
33The Flush-to-a Method F,a Goldszmidt and
Pearl 94
- Send non-E worlds to a and drop E -worlds
rigidly to the bottom
boost parameter
a
E
E
S
S
34The Flush-to-a Method F,a Goldszmidt and
Pearl 94
- Send non-E worlds to a and drop E -worlds
rigidly to the bottom
a
E
E
F,a
S
S F,a E
S
35The Flush-to-a Method F,a Goldszmidt and
Pearl 94
- Send non-E worlds to a and drop E -worlds
rigidly to the bottom - Perfect memory on sequentially consistent data if
a is high enough - Inductive leaps
- No epistemic hell
a
E
E
F,a
S
S F,a E
36Ordinal Jeffrey Conditioning J,a Spohn 88
S
37Ordinal Jeffrey Conditioning J,a Spohn 88
38Ordinal Jeffrey Conditioning J,a Spohn 88
- Drop E worlds to the bottom. Drop non-E worlds
to the bottom and then jack them up to level a
E
E
E
a
S
39Ordinal Jeffrey Conditioning J,a Spohn 88
- Drop E worlds to the bottom. Drop non-E worlds
to the bottom and then jack them up to level a
40Ordinal Jeffrey Conditioning J,a Spohn 88
- Drop E worlds to the bottom. Drop non-E worlds
to the bottom and then jack them up to level a - Perfect memory on consistent sequences if a is
large enough - No epistemic hell
- But...
J,a
E
E
E
a
S
S J,a E
41Empirical Backsliding
- Ordinal Jeffrey conditioning can increase the
plausibility of a refuted possibility
E
a
42The Ratchet Method R,a Darwiche and Pearl 97
S
43The Ratchet Method R,a Darwiche and Pearl 97
- Like ordinal Jeffrey conditioning except refuted
possibilities move up by a from their current
positions
b a
b
E
S
44The Ratchet Method R,a Darwiche and Pearl 97
- Like ordinal Jeffrey conditioning except refuted
possibilities move up by a from their current
positions
b a
b
E
R,a
B
B
S
S R,a E
45The Ratchet Method R,a Darwiche and Pearl 97
- Like ordinal Jeffrey conditioning except refuted
possibilities move up by a from their current
positions - Perfect memory if a is large enough
- Inductive leaps
- No epistemic hell
b a
b
E
R,a
B
B
S
S R,a E
46Part II
- Properties of the Methods
47Timidity and Stubbornness
- Timidity no inductive leaps without refutation.
- Stubbornness no retractions without refutation
- Examples all the above
- Nutty!
B
B
48Timidity and Stubbornness
- Timidity no inductive leaps without refutation.
- Stubbornness no retractions without refutation
- Examples all the above
- Nutty!
B
B
49Timidity and Stubbornness
- Timidity no inductive leaps without refutation.
- Stubbornness no retractions without refutation
- Examples all the above
- Nutty!
B
B
50Local Consistency
- Local consistency new belief must be
consistent with the current consistent datum - Examples all the above
51Positive Order-invariance
- Positive order-invariance preserve original
ranking inside conjunction of data - Examples
- C, L, R, a, J, a.
52Data-Precedence
- Data-precedence Each world satisfying all the
data is placed above each world failing to
satisfy some datum. - Examples
- C, L
- R, a, J, a, if a is above S .
S
53Enumerate and Test
- Enumerate-and-test
- locally consistent,
- positively invariant
- data-precedent
- Examples
- C, L
- R, a, J, a, if a is above S .
epistemic dump for refuted possibilities
preserved implausibility structure
54Part III
- Belief Revision as Learning
55A Very Simple Learning Paradigm
data trajectory
mysterious system
56A Very Simple Learning Paradigm
data trajectory
mysterious system
57A Very Simple Learning Paradigm
data trajectory
mysterious system
58A Very Simple Learning Paradigm
data trajectory
mysterious system
59Possible Outcome Trajectories
possible data trajectories
e
en
60Finding the Truth
- (, S0) identifies e Û
- for all but finitely many n,
- b(S0 (0, e(0), ... , n, e(n)))
e
61Finding the Truth
- (, S0) identifies e Û
- for all but finitely many n,
- b(S0 (0, e(0), ... , n, e(n)) e
truth
62Finding the Truth
- (, S0) identifies e Û
- for all but finitely many n,
- b(S0 (0, e(0), ... , n, e(n)) e
truth
63Finding the Truth
- (, S0) identifies e Û
- for all but finitely many n,
- b(S0 (0, e(0), ... , n, e(n)) e
truth
64Finding the Truth
- (, S0) identifies e Û
- for all but finitely many n,
- b(S0 (0, e(0), ... , n, e(n)) e
truth
65Finding the Truth
- (, S0) identifies e Û
- for all but finitely many n,
- b(S0 (0, e(0), ... , n, e(n)) e
completely true belief
66Reliability is No Accident
- Let K be a range of possible outcome trajectories
- (, S0) identifies K Û (, S0) identifies each e
in K. - Fact K is identifiable Û K is countable.
67Completeness
- is complete Û
- for each identifiable K
- there is an S0 such that,
- K is identifiable by (, S0).
- Else is restrictive.
68Completeness
- Proposition If enumerates and tests, is
complete.
69Completeness
- Proposition If enumerates and tests, is
complete.
- Enumerate K
- Choose arbitrary e in K
e
70Completeness
- Proposition If enumerates and tests, is
complete.
71Completeness
- Proposition If enumerates and tests, is
complete.
data precedence
positive invariance
72Completeness
- Proposition If enumerates and tests, is
complete.
73Completeness
- Proposition If enumerates and tests, is
complete.
data precedence
positive invariance
74Completeness
- Proposition If enumerates and tests, is
complete.
75Completeness
- Proposition If enumerates and tests, is
complete.
data precedence
local consistency
convergence
76Amnesia
- Without data precedence, memory can fail
- Same example, using J,1.
77Amnesia
- Without data precedence, memory can fail
- Same example, using J,1.
78Amnesia
- Without data precedence, memory can fail
- Same example, using J,1.
79Amnesia
- Without data precedence, memory can fail
- Same example, using J,1.
80Amnesia
- Without data precedence, memory can fail
- Same example, using J,1.
81Amnesia
- Without data precedence, memory can fail
- Same example, using J,1.
82Amnesia
- Without data precedence, memory can fail
- Same example, using J,1.
83Amnesia
- Without data precedence, memory can fail
- Same example, using J,1.
E
E is forgotten
84Duality
conjectures and refutations
tabula rasa
remembers doesnt predict
predicts may forget
85Rationally Imposed Tension
compression for memory
Can both be accommodated?
rarefaction for inductive leaps
86Inductive Amnesia
compression for memory
Restrictiveness No possible initial state
resolves the pressure
Bang!
rarefaction for inductive leaps
87Question
- Which methods are guilty?
- Are some worse than others?
88Part IV
89The Grue OperationNelson Goodman
n
e
e n
90Grue Complexity Hierarchy
Gw(e)
Gweven (e)
finite variants of e
finite variants of e ,e
G4(e)
G2even (e)
G3(e)
G2(e)
G1even (e)
G1(e)
G0(e)
G0even(e)
91Classification even grues
Min
Flush
Jeffrey
Ratch
Lex
Cond
Gweven (e)
a 1
yes
yes
a 1
no
a w
Gneven (e)
a n 1
a 1
yes
yes
a 1
no
G2even (e)
a 3
a 1
yes
yes
no
a 1
G1even (e)
a 1
a 2
a 1
yes
yes
no
G0even (e)
a 0
a 0
a 0
yes
yes
yes
92Classification even grues
Min
Flush
Jeffrey
Ratch
Lex
Cond
Gweven (e)
a 1
yes
yes
a 1
no
a w
Gneven (e)
a 1
a n 1
a 1
yes
yes
no
G2even (e)
a 3
a 1
yes
yes
no
a 1
G1even (e)
a 1
a 2
a 1
yes
yes
no
G0even (e)
a 0
a 0
a 0
yes
yes
yes
93Hamming Algebra
- a H b mod e Û
- a differs from e only where b does.
Hamming
94R,1 ,J,1 can identify Gweven(e)
a
Example
e
a
Learning as rigid hypercube rotation
e
95R,1 ,J,1 can identify Gweven(e)
Learning as rigid hypercube rotation
a
e
96R,1 ,J,1 can identify Gweven(e)
Learning as rigid hypercube rotation
e
a
97R,1 ,J,1 can identify Gweven(e)
e
Learning as rigid hypercube rotation
a
convergence
98Classification even grues
Min
Flush
Jeffrey
Ratch
Lex
Cond
Gweven (e)
a 1
yes
yes
a 1
no
a w
Gneven (e)
a n 1
a 1
yes
yes
a 1
no
G2even (e)
a 3
a 1
yes
yes
no
a 1
G1even (e)
a 1
a 2
a 1
yes
yes
no
G0even (e)
a 0
a 0
a 0
yes
yes
yes
99Classification arbitrary grues
Min
Flush
Jeffrey
Ratch
Lex
Cond
Gw(e)
no
a w
a 2
yes
a 2
yes
G3(e)
a 2
yes
yes
a 2
no
a n 1
G2(e)
a 2
yes
yes
a 2
no
a 3
G1(e)
a 2
yes
yes
no
a 2
a 1
G0(e)
yes
yes
a 0
a 0
a 0
yes
100Classification arbitrary grues
Min
Flush
Jeffrey
Ratch
Lex
Cond
Gw(e)
no
a w
a 2
a 2
yes
yes
G3(e)
a 2
yes
yes
a 2
no
a n 1
G2(e)
a 2
yes
yes
a 2
no
a 3
G1(e)
a 2
yes
yes
no
a 2
a 1
G0(e)
yes
yes
a 0
a 0
a 0
yes
101R,2 is Complete
- Impose the Hamming distance ranking on each
finite variant class - Now raise the nth Hamming ranking by n
S
C0
C1
C2
C3
C4
102R,2 is Complete
- Data streams in the same column just barely make
it because they jump by 2 for each difference
from the truth
S
1 difference from truth
2 differences from truth
C0
C1
C2
C3
C4
103Classification arbitrary grues
Min
Flush
Jeffrey
Ratch
Lex
Cond
Gw(e)
no
a w
a 2
a 2
yes
yes
Cant use Hamming rank
G3(e)
a 2
yes
yes
a 2
no
a n 1
G2(e)
a 2
yes
yes
a 2
no
a 3
G1(e)
a 2
yes
yes
no
a 2
a 1
G0(e)
yes
yes
a 0
a 0
a 0
yes
104Wrench In the Works
- Suppose J,2 succeeds with Hamming rank.
- Feed e until it is uniquely at the bottom.
k
e
By convergent success
105Wrench In the Works
k
n
a
Hamming rank and positive invariance.
b
If empty, things go even worse!
e
Still alone since timid and stubborn
106Wrench In the Works
- b moves up at most 1 step since e is still alone
(rule)
k
n
a
b
Refuted worlds touch bottom and get lifted by at
most two.
e
107Wrench In the Works
- So b never rises above a when a is true (positive
invariance) - Now a and b agree forever, so can never be
separated. - So never converges in a or forgets refutation of
b.
k
n
a
b
e
a
108Hamming vs. Goodman Algebras
- a H b mod e Û a differs from e only where b
does. - a G b mod e Û a grues e only where b
does.
Goodman
Hamming
1 0 1
1 1 0
0 1 0
1 0 0
0 1 1
0 0 1
1 1 1
0 0 0
109Epistemic States as Boolean Ranks
Goodman
Hamming
Gwodd (e)
Gw(e)
Gweven (e)
e
e
110J,2 can identify Gw (e)
- Proof Use the Goodman ranking as initial state
- Then J,2 always believes that the observed grues
are the only ones that will ever occur. - Note Ockham with respect to reversal counting
problem.
111Classification arbitrary grues
Min
Flush
Jeffrey
Ratch
Lex
Cond
Gw(e)
no
a w
a 2
a 2
yes
yes
G3(e)
a 2
yes
yes
a 2
no
a n 1
G2(e)
a 2
yes
yes
a 2
no
a 3
G1(e)
a 2
yes
yes
no
a 2
a 1
G0(e)
yes
yes
a 0
a 0
a 0
yes
112Methods J,1 M Fail on G1(e)
- Proof Suppose otherwise
- Feed e until e is uniquely at the bottom
e
data so far
113Methods J,1 M Fail on G1(e)
- By the well-ordering condition,
...else infinite descending chain
e
data so far
114Methods J,1 M Fail on G1(e)
- Now feed e forever
- By stage n, the picture is the same
e
positive order invariance
e
e
timidity and stubbornness
e
n
115Methods J,1 M Fail on G1(e)
- At stage n 1, e stays at the bottom (timid and
stubborn). - So e cant travel down (rule)
- e doesnt rise (rule)
- Now e makes it to the bottom at least as soon
as e
e
e
e
e
n
116Classification arbitrary grues
Min
Flush
Jeffrey
Ratch
Lex
Cond
Gw(e)
no
a w
a 2
a 2
yes
yes
G3(e)
a 2
yes
yes
a 2
no
a n 1
forced backsliding
G2(e)
a 2
yes
yes
a 2
no
a 3
G1(e)
a 2
yes
yes
no
a 2
a 1
G0(e)
yes
yes
a 0
a 0
a 0
yes
117Method R,1 Fails on G2(e)with Oliver Schulte
- Proof Suppose otherwise
- Bring e uniquely to the bottom, say at stage k
k
e
118Method R,1 Fails on G2(e)with Oliver Schulte
k
e
a
119Method R,1 Fails on G2(e)with Oliver Schulte
- By some stage k, a is uniquely down
- So between k 1 and k, there is a first stage j
when no finite variant of e is at the bottom
k
k
e
a
120Method R,1 Fails on G2(e)with Oliver Schulte
- Let c in G2(e ) be a finite variant of e that
rises to level 1 at j
k
k
j
c
a
121Method R,1 Fails on G2(e)with Oliver Schulte
- Let c in G2(e ) be a finite variant of e that
rises to level 1 at j
k
k
j
c
a
122Method R,1 Fails on G2(e)with Oliver Schulte
k
k
j
- So c(j - 1) is not a(j - 1)
c
a
123Method R,1 Fails on G2(e)with Oliver Schulte
- Let d be a up to j and e thereafter
- So is in G2(e)
- Since d differs from e, d is at least as high as
level 1 at j
k
k
j
d
c
1
a
124Method R,1 Fails on G2(e)with Oliver Schulte
- Show c agrees with e after j.
k
k
j
d
c
1
a
125Method R,1 Fails on G2(e)with Oliver Schulte
- Case j k1
- Then c could have been chosen as e since e is
uniquely at the bottom at k
k
k
j
d
c
1
a
126Method R,1 Fails on G2(e)with Oliver Schulte
- Case j gt k1
- Then c wouldnt have been at the bottom if it
hadnt agreed with a (disagreed with e)
k
k
j
d
c
1
a
127Method R,1 Fails on G2(e)with Oliver Schulte
- Case j gt k1
- So c has already used up its two grues against e
k
k
j
d
c
1
a
128Method R,1 Fails on G2(e)with Oliver Schulte
- Feed c forever after
- By positive invariance, either never projects or
forgets the refutation of c at j-1
k
k
j
d
c
1
d
129Without Well-Ordering
Min
Flush
Jeffrey
Ratch
Lex
Cond
Gw(e)
no
yes
yes
G3(e)
yes
yes
no
infinite descending chains can help!
G2(e)
yes
yes
no
G1(e)
yes
yes
yes
G0(e)
yes
yes
yes
130Summary
- Belief revision constrains possible inductive
strategies - No induction without contradiction (?!!)
- Rationality weakens learning power of ideal
agents. - Prediction vs. memory
- Precise recommendations for rationalists
- boosting by 2 vs. 1
- backslide vs. ratchet
- well-ordering
- Hamming vs. Goodman rank