1
Belief Revision and Truth-Finding

• Kevin T. Kelly
• Department of Philosophy
• Carnegie Mellon University
• kk3n_at_andrew.cmu.edu

2
Further 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

3
The Idea

• Belief revision theory... rational belief
  change
• Learning theory...............reliable belief
  change
• Conflict?

Truth
4
Part I

• Iterated Belief Revision

5
Bayesian (Vanilla) Updating
Propositional epistemic state

B
6
Bayesian (Vanilla) Updating

• New belief is intersection
• Perfect memory
• No inductive leaps

E
new evidence
B
7
Bayesian (Vanilla) Updating

• New belief is intersection
• Perfect memory
• No inductive leaps

E
B
B
8
Bayesian (Vanilla) Updating

• New belief is intersection
• Perfect memory
• No inductive leaps

E
B
B
9
Epistemic Hell (a.k.a. Nirvana)
B
10
Epistemic Hell (a.k.a. Nirvana)
E
Surprise!
B
11
Epistemic Hell (a.k.a. Nirvana)

• Scientific revolutions
• Suppositional reasoning
• Conditional pragmatics
• Decision theory
• Game theory
• Data bases

E
B
Epistemic hell
12
Ordinal Epistemic StatesSpohn 88

• Ordinal-valued degrees of implausibility
• Belief state is bottom level

w 1
w
2
1
b (S)
0
S
13
Iterated Belief Revision
epistemic state trajectory
initial state
S0
input propositions
E0
E1
E2
14
Iterated Belief Revision
epistemic state trajectory
S1
S0
input propositions
E1
E2
15
Iterated Belief Revision
epistemic state trajectory
S1
S0
input propositions
E1
E2
16
Iterated Belief Revision
epistemic state trajectory
S2
input proposition
E2
17
Iterated Belief Revision
epistemic state trajectory
S2
input proposition
E2
18
Iterated Belief Revision
epistemic state trajectory
S3
19
Iterated Belief Revision
epistemic state trajectory
S3
b (S0)
b (S1)
b (S2)
b (S3)
belief state trajectory
20
Generalized Conditioning CSpohn 88
S
21
Generalized Conditioning CSpohn 88

• Condition entire epistemic state

E
S
22
Generalized Conditioning CSpohn 88

• Condition entire epistemic state

E
C
S
S C E
23
Generalized 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
24
Lexicographic Updating LSpohn 88, Nayak 94
S
25
Lexicographic Updating LSpohn 88, Nayak 94

• Lift refuted possibilities above non-refuted
  possibilities preserving order.

S
26
Lexicographic Updating LSpohn 88, Nayak 94

• Lift refuted possibilities above non-refuted
  possibilities preserving order.

L
S
S L E
27
Lexicographic 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
28
Minimal or Natural Updating MSpohn 88,
Boutilier 93
S
29
Minimal 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
30
Minimal 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
31
Minimal 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
32
The Flush-to-a Method F,a Goldszmidt and
Pearl 94
S
33
The 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
34
The 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
35
The 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
36
Ordinal Jeffrey Conditioning J,a Spohn 88
S
37
Ordinal Jeffrey Conditioning J,a Spohn 88
38
Ordinal 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
39
Ordinal 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

40
Ordinal 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
41
Empirical Backsliding

• Ordinal Jeffrey conditioning can increase the
  plausibility of a refuted possibility

E
a
42
The Ratchet Method R,a Darwiche and Pearl 97
S
43
The 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
44
The 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
45
The 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
46
Part II

• Properties of the Methods

47
Timidity and Stubbornness

• Timidity no inductive leaps without refutation.
• Stubbornness no retractions without refutation
• Examples all the above
• Nutty!

B
B
48
Timidity and Stubbornness

• Timidity no inductive leaps without refutation.
• Stubbornness no retractions without refutation
• Examples all the above
• Nutty!

B
B
49
Timidity and Stubbornness

• Timidity no inductive leaps without refutation.
• Stubbornness no retractions without refutation
• Examples all the above
• Nutty!

B
B
50
Local Consistency

• Local consistency new belief must be
  consistent with the current consistent datum
• Examples all the above

51
Positive Order-invariance

• Positive order-invariance preserve original
  ranking inside conjunction of data
• Examples
• C, L, R, a, J, a.

52
Data-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
53
Enumerate 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
54
Part III

• Belief Revision as Learning

55
A Very Simple Learning Paradigm
data trajectory
mysterious system
56
A Very Simple Learning Paradigm
data trajectory
mysterious system
57
A Very Simple Learning Paradigm
data trajectory
mysterious system
58
A Very Simple Learning Paradigm
data trajectory
mysterious system
59
Possible Outcome Trajectories
possible data trajectories
e
en
60
Finding the Truth

• (, S0) identifies e Û
• for all but finitely many n,
• b(S0 (0, e(0), ... , n, e(n)))
  e

61
Finding the Truth

• (, S0) identifies e Û
• for all but finitely many n,
• b(S0 (0, e(0), ... , n, e(n)) e

truth
62
Finding the Truth

• (, S0) identifies e Û
• for all but finitely many n,
• b(S0 (0, e(0), ... , n, e(n)) e

truth
63
Finding the Truth

• (, S0) identifies e Û
• for all but finitely many n,
• b(S0 (0, e(0), ... , n, e(n)) e

truth
64
Finding the Truth

• (, S0) identifies e Û
• for all but finitely many n,
• b(S0 (0, e(0), ... , n, e(n)) e

truth
65
Finding the Truth

• (, S0) identifies e Û
• for all but finitely many n,
• b(S0 (0, e(0), ... , n, e(n)) e

completely true belief
66
Reliability 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.

67
Completeness

• is complete Û
• for each identifiable K
• there is an S0 such that,
• K is identifiable by (, S0).
• Else is restrictive.

68
Completeness

• Proposition If enumerates and tests, is
  complete.

69
Completeness

• Proposition If enumerates and tests, is
  complete.

• Enumerate K
• Choose arbitrary e in K

e
70
Completeness

• Proposition If enumerates and tests, is
  complete.

71
Completeness

• Proposition If enumerates and tests, is
  complete.

data precedence
positive invariance
72
Completeness

• Proposition If enumerates and tests, is
  complete.

73
Completeness

• Proposition If enumerates and tests, is
  complete.

data precedence
positive invariance
74
Completeness

• Proposition If enumerates and tests, is
  complete.

75
Completeness

• Proposition If enumerates and tests, is
  complete.

data precedence
local consistency
convergence
76
Amnesia

• Without data precedence, memory can fail
• Same example, using J,1.

77
Amnesia

• Without data precedence, memory can fail
• Same example, using J,1.

78
Amnesia

• Without data precedence, memory can fail
• Same example, using J,1.

79
Amnesia

• Without data precedence, memory can fail
• Same example, using J,1.

80
Amnesia

• Without data precedence, memory can fail
• Same example, using J,1.

81
Amnesia

• Without data precedence, memory can fail
• Same example, using J,1.

82
Amnesia

• Without data precedence, memory can fail
• Same example, using J,1.

83
Amnesia

• Without data precedence, memory can fail
• Same example, using J,1.

E
E is forgotten
84
Duality
conjectures and refutations
tabula rasa
remembers doesnt predict
predicts may forget
85
Rationally Imposed Tension
compression for memory
Can both be accommodated?
rarefaction for inductive leaps
86
Inductive Amnesia
compression for memory
Restrictiveness No possible initial state
resolves the pressure
Bang!
rarefaction for inductive leaps
87
Question

• Which methods are guilty?
• Are some worse than others?

88
Part IV

• The Goodman Hierarchy

89
The Grue OperationNelson Goodman
n
e
e n
90
Grue 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)
91
Classification 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
92
Classification 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
93
Hamming Algebra

• a H b mod e Û
• a differs from e only where b does.

Hamming
94
R,1 ,J,1 can identify Gweven(e)
a
Example
e
a
Learning as rigid hypercube rotation
e
95
R,1 ,J,1 can identify Gweven(e)
Learning as rigid hypercube rotation
a
e
96
R,1 ,J,1 can identify Gweven(e)
Learning as rigid hypercube rotation
e
a
97
R,1 ,J,1 can identify Gweven(e)
e
Learning as rigid hypercube rotation
a
convergence
98
Classification 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
99
Classification 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
100
Classification 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
101
R,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
102
R,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
103
Classification 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
104
Wrench In the Works

• Suppose J,2 succeeds with Hamming rank.
• Feed e until it is uniquely at the bottom.

k
e
By convergent success
105
Wrench In the Works

• So for some later n,
  

k
n
a
Hamming rank and positive invariance.
b
If empty, things go even worse!
e
Still alone since timid and stubborn
106
Wrench 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
107
Wrench 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
108
Hamming 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
109
Epistemic States as Boolean Ranks
Goodman
Hamming
Gwodd (e)
Gw(e)
Gweven (e)
e
e
110
J,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.

111
Classification 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
112
Methods J,1 M Fail on G1(e)

• Proof Suppose otherwise
• Feed e until e is uniquely at the bottom

e
data so far
113
Methods J,1 M Fail on G1(e)

• By the well-ordering condition,

...else infinite descending chain
e
data so far
114
Methods 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
115
Methods 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
116
Classification 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
117
Method R,1 Fails on G2(e)with Oliver Schulte

• Proof Suppose otherwise
• Bring e uniquely to the bottom, say at stage k

k
e
118
Method R,1 Fails on G2(e)with Oliver Schulte

• Start feeding a e k

k
e
a
119
Method 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
120
Method 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
121
Method 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
122
Method R,1 Fails on G2(e)with Oliver Schulte
k
k
j

• So c(j - 1) is not a(j - 1)

c
a
123
Method 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
124
Method R,1 Fails on G2(e)with Oliver Schulte

• Show c agrees with e after j.

k
k
j
d
c
1
a
125
Method 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
126
Method 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
127
Method 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
128
Method 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
129
Without 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
130
Summary

• 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