Inductive Amnesia

Inductive AmnesiaInductive Amnesia

The Reliability The Reliability

of of

Iterated Belief RevisionIterated Belief Revision

A Table of OppositesA Table of Opposites

EvenEven OddOdd StraightStraight CrookedCrooked ReliabilityReliability “Confirmation”“Confirmation” Performance Performance “Primitive norms”“Primitive norms” CorrectnessCorrectness “Coherence”“Coherence” Classical statistics Classical statistics BayesianismBayesianism Learning theoryLearning theory Belief Revision TheoryBelief Revision Theory

The IdeaThe Idea

Belief revision is inductive reasoningBelief revision is inductive reasoning A restrictive norm prevents us from finding truths A restrictive norm prevents us from finding truths

we could have found by other meanswe could have found by other means Some proposed belief revision methods are Some proposed belief revision methods are

restrictiverestrictive The restrictiveness is expressed as The restrictiveness is expressed as inductive inductive

amnesiaamnesia

Inductive AmnesiaInductive Amnesia

No restriction on memory...No restriction on memory... No restriction on predictive power...No restriction on predictive power... But prediction causes memory loss...But prediction causes memory loss... And perfect memory precludes prediction!And perfect memory precludes prediction! Fundamental dilemmaFundamental dilemma

OutlineOutline

I. I. Seven belief revision methodsSeven belief revision methods II. II. Belief revision as learningBelief revision as learning III. III. Properties of the methodsProperties of the methods IV. IV. The Goodman hierarchyThe Goodman hierarchy V. V. Negative resultsNegative results VI. VI. Positive resultsPositive results VII. VII. DiscussionDiscussion

Points of InterestPoints of Interest

Strong negative and positive resultsStrong negative and positive results Short run advice from limiting analysisShort run advice from limiting analysis 2 is magic for reliable belief revision2 is magic for reliable belief revision Learning as cube rotationLearning as cube rotation GrueGrue

Part IPart I


Bayesian (Vanilla) UpdatingBayesian (Vanilla) Updating

B

Propositions are sets of “possible worlds”


B

E newevidence


Perfect memoryPerfect memory No inductive leapsNo inductive leaps

BB’

E

B’ = B *E = B E

Epistemic HellEpistemic Hell

B


B

E

Epistemic hell

Surprise!


Scientific revolutionsScientific revolutions Suppositional reasoningSuppositional reasoning Conditional pragmaticsConditional pragmatics Decision theoryDecision theory Game theoryGame theory Data basesData bases

B

E

Epistemic hell

Ordinal EntrenchmentOrdinal EntrenchmentSpohn 88Spohn 88

Epistemic state Epistemic state SS maps worlds to ordinals maps worlds to ordinals Belief state of Belief state of SS = = bb ( (S S ) = ) = S S -1-1(0)(0) Determines “centrality” of beliefsDetermines “centrality” of beliefs Model: orders of infinitesimal probabilityModel: orders of infinitesimal probability

S

B = b (S)

2

1

0

+ 1

Belief Revision MethodsBelief Revision Methods

*

S’

S

E

* takes an epistemic state and a proposition to an epistemic state

b(S)

b (S *E )

Spohn Conditioning *Spohn Conditioning *CCSpohn 88Spohn 88

b (S )

S


E

S

E newevidencecontradicting b (S )

b (S )


E

B’

S

E

*C

b (S )

S *C E


Conditions an entire Conditions an entire entrenchment orderingentrenchment ordering

Perfect memoryPerfect memory Inductive leapsInductive leaps No epistemic hell on consistent No epistemic hell on consistent

sequencessequences Epistemic hell on inconsistent Epistemic hell on inconsistent

sequencessequences

E

B’

S

E

S *C E

*C

b (S )

Lexicographic Updating *Lexicographic Updating *LLSpohn 88, Nayak 94Spohn 88, Nayak 94

S


S

E


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

Perfect memory on consistent Perfect memory on consistent sequencessequences

Inductive leapsInductive leaps No epistemic hellNo epistemic hell

B’

S S *L E

E

*L

Minimal or “Natural” Updating *Minimal or “Natural” Updating *MMSpohn 88, Boutilier 93Spohn 88, Boutilier 93

B

S


B

E

S


Drop the lowest Drop the lowest possibilities consistent possibilities consistent with the data to the bottom with the data to the bottom and raise everything else and raise everything else up one notchup one notch

inductive leapsinductive leaps No epistemic hellNo epistemic hell But...But...

E

S S *M E

*M

AmnesiaAmnesia

What goes up can come downWhat goes up can come down Belief no longer entails past dataBelief no longer entails past data

E

AmnesiaAmnesia


E E’

*M

AmnesiaAmnesia


E E’

*M *M

The Flush-to-The Flush-to- Method * Method *F,F,Goldszmidt and Pearl 94Goldszmidt and Pearl 94

B

S


E

S

EE


Send non-Send non-EE worlds to a worlds to a fixed level fixed level and drop and drop E E --worlds rigidly to the worlds rigidly to the bottombottom

Perfect memory on Perfect memory on sequentially consistent sequentially consistent data data ifif is high enough is high enough


E

S S *F, E

EE

*F,

Ordinal Jeffrey Conditioning *Ordinal Jeffrey Conditioning *J,J,Spohn 88 Spohn 88

S


E

S

EE


E

S

EE


Drop Drop E E worlds to the worlds to the bottom. Drop non-bottom. Drop non-EE worlds worlds to the bottom and then jack to the bottom and then jack them up to level them up to level

Perfect memory on Perfect memory on consistent sequences if consistent sequences if is is large enoughlarge enough

No epistemic hellNo epistemic hell ReversibleReversible But...But...

B

E

S S *J, E

B’

EE

*J,

Empirical BackslidingEmpirical Backsliding


E


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

E

The Ratchet Method *The Ratchet Method *R,R,Darwiche and Pearl 97Darwiche and Pearl 97

S


S

E


Like ordinal Jeffrey Like ordinal Jeffrey conditioning except conditioning except refuted possibilities move refuted possibilities move up by up by from their current from their current positionspositions

Perfect memory if Perfect memory if is is large enoughlarge enough


S

S *R, E

E

B

B’

*R,

Part IIPart II

Belief Revision as LearningBelief Revision as Learning


*

S0 S1 S2

b (S2)

S0

((SS00 * ()) = * ()) = SS00

((SS00 * ( * (EE00, ..., , ..., EEnn, , EEn+n+11)) = ()) = (SS00 * ( * (EE00, ..., , ..., EEnn, )) * , )) * EEn+n+11

E0 E1

b (S1)b (S0)

A Very Simple Learning ParadigmA Very Simple Learning Paradigm

mysterious system

outcomesequence

0 0 1 0 0

e

e|n

possible infinite trajectories

n

Empirical PropositionsEmpirical Propositions

[] = the proposition

that has occurred[k, n] = theproposition thatk occurs at stage n

n

Empirical propositions are sets of possible trajectoriesEmpirical propositions are sets of possible trajectories Some special cases:Some special cases:

e

{e} = the proposition that the future trajectory is exactly e

“fan”k

Trajectory IdentificationTrajectory Identification

((*, S*, S00) ) identifies eidentifies e for all but finitely many for all but finitely many nn, ,

bb((SS00 * ([0, * ([0, ee(0)], ..., [(0)], ..., [nn, , ee((nn)]) = {)]) = {ee}}



bb((SS00 * ([0, * ([0, ee(0)], ..., [(0)], ..., [nn, , ee((nn)]) = {)]) = {ee}}

e

possible trajectories



bb((SS00 * ([0, * ([0, ee(0)], ..., [(0)], ..., [nn, , ee((nn)]) = {)]) = {ee}}

b (S 0)

e



bb((SS00 * ([0, * ([0, ee(0)], ..., [(0)], ..., [nn, , ee((nn)]) = {)]) = {ee}}

b (S 1)



bb((SS00 * ([0, * ([0, ee(0)], ..., [(0)], ..., [nn, , ee((nn)]) = {)]) = {ee}}

b (S 2)



bb((SS00 * ([0, * ([0, ee(0)], ..., [(0)], ..., [nn, , ee((nn)]) = {)]) = {ee}}

b (S 3)



bb((SS00 * ([0, * ([0, ee(0)], ..., [(0)], ..., [nn, , ee((nn)]) = {)]) = {ee}}

convergence to {e }

b (S 4)



bb((SS00 * ([0, * ([0, ee(0)], ..., [(0)], ..., [nn, , ee((nn)]) = {)]) = {ee}}

b (S 5)

etc...

ReliabilityReliability

Let Let K K be a set of possible outcome be a set of possible outcome trajectoriestrajectories

((*, S*, S00) ) identifies Kidentifies K ((*, S*, S00) identifies each ) identifies each

ee in in KK

Identifiability CharacterizedIdentifiability Characterized

Proposition: Proposition: KK is identifiable just in case is identifiable just in case KK is countable is countable

Completeness and RestrictivenessCompleteness and Restrictiveness

* is * is completecomplete each ientifiable each ientifiable KK is is identifiable by (identifiable by (*, S*, S00), for some choice of ), for some choice of SS00..

Else * is Else * is restrictiverestrictive. .

Part IIIPart III

Properties of the MethodsProperties of the Methods

Timidity and StubbornnessTimidity and Stubbornness

timidity: no inductive leaps timidity: no inductive leaps without refutationwithout refutation

stubbornness: no retractions stubbornness: no retractions without refutationwithout refutation

BB’




BB’


““Belief is Bayesian in the non-Belief is Bayesian in the non-problematic case”problematic case”

All the proposed methods are All the proposed methods are timid and stubborntimid and stubborn

Vestige of the dogma that Vestige of the dogma that probability rules inductionprobability rules induction

BB’

Local ConsistencyLocal Consistency

Local consistency: The Local consistency: The updated belief must updated belief must always be consistent with always be consistent with the current datumthe current datum

All the methods under All the methods under consideration are designed consideration are designed to be locally consistentto be locally consistent




BB’

Positive Order-invariancePositive Order-invariance

Positive order-invariance: Positive order-invariance: ranking among worlds ranking among worlds satisfying all the data so satisfying all the data so far are preservedfar are preserved

All the methods considered All the methods considered are positively order-are positively order-invariantinvariant

Data-RetentivenessData-Retentiveness

Data-retentiveness: Each Data-retentiveness: Each world satisfying all the data is world satisfying all the data is placed above each world placed above each world failing to satisfy some datumfailing to satisfy some datum

Data-retentiveness is sufficient Data-retentiveness is sufficient but not necessary for perfect but not necessary for perfect memorymemory

**CC, *, *LL are data-retentive are data-retentive

**R,R,,, * *J,J, are data-retentive if are data-retentive if is above the top of is above the top of SS. .

S

Enumerate and TestEnumerate and Test

A method enumerates and A method enumerates and tests just in case it is: tests just in case it is:

locally consistent,locally consistent, positively order-invariant,positively order-invariant, data-retentivedata-retentive Enumerate and test Enumerate and test

methods: *methods: *CC, *, *LL

The methods with The methods with parameter parameter if a is above if a is above the top of the top of SS 00. .

preserved entrenchmentordering on live possibilities

epistemicdump forrefutedpossibilities

CompletenessCompleteness

Proposition: Proposition: If * enumerates and tests, then * is If * enumerates and tests, then * is completecomplete

Proof: Let Proof: Let SS00 be an enumeration of be an enumeration of KK


Proposition: Proposition: If * enumerates and tests, then * is If * enumerates and tests, then * is completecomplete

Proof: Let Proof: Let SS00 be an enumeration of be an enumeration of KK

Let Let e e be in be in KK

e


Feed successive data along Feed successive data along ee:: [0, [0, ee(0)], [1, (0)], [1, ee(1)], ..., [(1)], ..., [nn, , ee((nn)], ...)], ...

[0, e (0)]

e


[0, e (0)]

e local consistencypositive invariance

data retentiveness

e


[0, e (0)]

e

e

[1, e (1)]


[0, e (0)]

e

e

e

[1, e (1)]

local consistencypositive invariance

data retentiveness


[0, e (0)]

e

e

e

[1, e (1)][2, e (2)]


[0, e (0)]

e

e

e

[1, e (1)][2, e (2)]

e

local consistpositive invar

data retentiveness

Convergence

QuestionQuestion

What about the methods that aren’t data What about the methods that aren’t data retentive?retentive?

Are they complete?Are they complete? If not, can they be objectively compared?If not, can they be objectively compared?

Part IV:Part IV:

The Goodman HierarchyThe Goodman Hierarchy

The Grue OperationThe Grue OperationNelson GoodmanNelson Goodman

A way to generate inductive problems A way to generate inductive problems of ever higher difficulty of ever higher difficulty

ee ‡ ‡ nn = ( = (ee||nn)¬()¬(nn||ee))

nnee

nnkk

ee ‡ ‡ nnee



ee ‡ ‡ nn = ( = (ee||nn)¬()¬(nn||ee))



ee ‡ ‡ nn = ( = (ee||nn)¬()¬(nn||ee))

nn

mmkk

ee ‡ ‡ nnee

((ee ‡ ‡ nn) ‡ ) ‡ mm



ee ‡ ‡ nn = ( = (ee||nn)¬()¬(nn||ee))

nn

mmkk

ee

((((ee ‡ ‡ nn) ‡ ) ‡ mm) ‡ ) ‡ kk

nn

mmkk

ee ‡ ‡ nn((ee ‡ ‡ nn) ‡ ) ‡ mm

The Goodman HierarchyThe Goodman Hierarchy

GGnn((ee) = the set of all trajectories you can get by gruing ) = the set of all trajectories you can get by gruing ee up to up to nn positions positions

GGnneveneven((ee) = the set of all trajectories you can get by ) = the set of all trajectories you can get by

gruing gruing ee an even number of distinct positions up to 2 an even number of distinct positions up to 2nn

GG00((ee))GG11((ee))


GG00eveneven((ee))

GG11eveneven

((ee))

nn

mmkk

The Goodman LimitThe Goodman Limit

GG((ee) = ) = nnGGnn((ee))

GGeveneven((ee) = ) = nnGGnn

eveneven ((ee))

Proposition: Proposition: GGeveneven((ee) = the set of all finite ) = the set of all finite

variants of variants of ee

The Goodman SpectrumThe Goodman Spectrum


GG22((ee))

GG33((ee))

GG((ee))

Min Flush Jeffrey Ratch Lex Cond

= 2

= 2

= 2 yes

yes

yes

yes yes

yes

yes

yes

= 2

= 2

no

no

no

no = = 2 = 2

= n +1

= 3

= 2

yes yes

= 0 = 0 = 0yes

= 1

The Even Goodman SpectrumThe Even Goodman Spectrum

GG00eveneven

((ee))GG11

eveneven ((ee))

GG22eveneven

((ee))

GGnneveneven

((ee))

GGeveneven

((ee))

= 1


= 0

= 2

= 3

= n +1

= 0 = 0

= 1

= 1

= 1

= 1

yes

yes

yes

yes

yes yes

yes

yes

yes

yes

= 1

= 1

yes

no

no

no

no =

= 1

Part V:Part V:

Negative ResultsNegative Results

Epistemic DualityEpistemic Duality

. . .

“tabula rasa”Bayesian

“conjectures and refutations”Popperian

Epistemic ExtremesEpistemic Extremes

. . .

perfect memoryno projections

projects the futuremay forget

*J,2

Opposing Epistemic PressuresOpposing Epistemic Pressures


rarefaction for inductive leaps


Identification requires Identification requires bothboth Is there a critical value of Is there a critical value of for which for which

they can be balanced for a given they can be balanced for a given problem problem KK??

compression for memory

rarefaction for inductive leaps

Methods *Methods *S,1S,1; *; *MM Fail on Fail on GG11((ee))


GG22((ee))

GG33((ee))

GG((ee))


= 2

= 2

= 2 yes

yes

yes

yes yes

yes

yes

yes

= 2

= 2

no

no

no

no = = 2

= n +1

= 3

= 2

yes yes

= 0 = 0 = 0yes

= 1

= 2


Proof: Proof: Suppose otherwiseSuppose otherwise Feed Feed ee until until ee is uniquely at the bottom is uniquely at the bottom


Proof: Proof: Suppose otherwiseSuppose otherwise Feed Feed ee until until ee is uniquely at the bottom is uniquely at the bottom

e

data so far

?


By the well-ordering condition, By the well-ordering condition,

e

data so far

else...

?


Now feed Now feed e’ e’ foreverforever By stage By stage n, n, the picture is the samethe picture is the same

e

?

e’

e’’

e’ n

positive order invariance

timidity and stubbornness


At stage At stage nn +1, +1, ee stays at the stays at the bottom (timid and stubborn).bottom (timid and stubborn).

So So e’ e’ can’t travel down can’t travel down (definitions of the rules)(definitions of the rules)

e’’ e’’ doesn’t rise (definitions doesn’t rise (definitions of the rules)of the rules)

Now Now e’’ e’’ makes it to the makes it to the bottom at least as soon as bottom at least as soon as e’e’

e

?

e’

e’’

e’ n

Method *Method *R,1R,1 Fails on Fails on GG22((ee))


GG22((ee))

GG33((ee))

GG((ee))


= 2

= 2

= 2 yes

yes

yes

yes yes

yes

yes

yes

= 2

= 2

no

no

no

no = = 2 = 2

= n +1

= 3

= 2

yes yes

= 0 = 0 = 0yes

= 1

Method *Method *R,1R,1 Fails on Fails on GG22((ee))with Oliver Schultewith Oliver Schulte

ProofProof: Suppose otherwise: Suppose otherwise Bring Bring ee uniquely to the bottom, say at stage uniquely to the bottom, say at stage kk

e

k


Start feeding Start feeding a a = = ee ‡ ‡ kk

e

k

a


By some stage By some stage k’k’, , a a is uniquely downis uniquely down So between So between k + k + 1 and 1 and k’k’, there is a first stage , there is a first stage j j when no finite variant of when no finite variant of ee is at is at

the bottomthe bottom

a

k k’

a


Let Let c c in in GG22((ee) be a finite variant ) be a finite variant of of e e that rises to level 1 at that rises to level 1 at jj

k’

a

k j

c


Let Let c c in in GG22((ee) be a finite variant ) be a finite variant of of e e that rises to level 1 at that rises to level 1 at jj

k’

a

k j

c


SoSo c c((j j - 1) - 1) aa((j j - 1)- 1)

k’

a

k j

c


Let Let dd be be aa up to up to jj and and ee thereafterthereafter

So is in So is in GG22((ee)) Since Since dd differs from differs from ee, , dd is at is at

least as high as level 1 at least as high as level 1 at j j k’

a

k j

c

d

1


Show: Show: cc agrees with agrees with ee after after jj..

k’

a

k j

c

d

1


Case: Case: jj = = kk+1+1 Then Then cc could have been chosen could have been chosen

as as ee since since ee is uniquely at the is uniquely at the bottom at bottom at kkk’

a

k j

c

d

1


Case: Case: jj > > kk+1+1 Then Then cc wouldn’t have been at wouldn’t have been at

the bottom if it hadn’t agreed the bottom if it hadn’t agreed with with aa (disagreed with (disagreed with ee))k’

a

k j

c

d

1


Case: Case: jj > > kk+1+1 So So cc has already used up its two has already used up its two

grues against grues against ee

k’

a

k j

c

d

1


Feed c forever afterFeed c forever after By positive invariance, either By positive invariance, either

never projects never projects or or forgetsforgets the the refutation of refutation of cc at at jj-1 -1 k’ k j

c

d

1

d

The Internal Problem of InductionThe Internal Problem of Induction

Necessary condition for success by pos ord-invar methods:Necessary condition for success by pos ord-invar methods: no data stream is a no data stream is a kk-limit point of data streams as low as it -limit point of data streams as low as it

after it has been presented for after it has been presented for kk steps steps

bad

The Internal Problem of InductionThe Internal Problem of Induction

Necessary condition for success by pos ord-invar methods:Necessary condition for success by pos ord-invar methods: no data stream is a no data stream is a kk-limit point of data streams as low as it -limit point of data streams as low as it

after it has been presented for after it has been presented for kk steps steps

bad good

Corollary: Stacking LemmaCorollary: Stacking Lemma

Necessary condition for Necessary condition for identification of identification of GGnn+1+1((ee) ) by positively order-by positively order-invariant methodsinvariant methods

If If ee is at the bottom level is at the bottom level after being presented up after being presented up to stage to stage kk, then some , then some data stream data stream e’ e’ in in GGnn+1+1((ee) ) - - GGnn ((ee) agreeing with the ) agreeing with the data so far is at least at data so far is at least at level level nn+1+1



If If ee is at the bottom level is at the bottom level after being presented up after being presented up to stage to stage kk, then some , then some data stream data stream e’ e’ in in GGnn+1+1((ee) ) - - GGnn ((ee) in agreeing with ) in agreeing with the data so far is at least the data so far is at least at level at level nn+1+1

Why?



If If ee is at the bottom level is at the bottom level after being presented up after being presented up to stage to stage kk, then some , then some data stream data stream e’ e’ in in GGnn+1+1((ee) ) - - GGnn ((ee) in agreeing with ) in agreeing with the data so far is at least the data so far is at least at level at level nn+1+1

Else!

Even Stacking LemmaEven Stacking Lemma

Similarly for Similarly for GGn+n+11eveneven((ee))



Why?



Else!

Method *Method *F,F,nn Fails on Fails on GGnn((ee))


GG22((ee))

GG33((ee))

GG((ee))


= 2

= 2

= 2 yes

yes

yes

yes yes

yes

yes

yes

= 2

= 2

no

no

no

no = = 2 = 2

= n +1

= 3

= 2

yes yes

= 0 = 0 = 0yes

= 1


Proof for Proof for = 4: Suppose otherwise= 4: Suppose otherwise Bring Bring ee uniquely to the bottom uniquely to the bottom


Proof for Proof for = 4: Suppose otherwise= 4: Suppose otherwise Bring Bring ee uniquely to the bottom uniquely to the bottom

e

0

4

e’

? e’’


Apply stacking lemmaApply stacking lemma Let Let e’ e’ be in Gbe in G44((ee) - G) - G33((ee) at or ) at or

above level 4above level 4 Let Let e’’ e’’ be the same except at a be the same except at a

the first place the first place kk where where e’ e’ differs from differs from ee

Feed Feed e’e’ forever after forever after

k

e

? e’’


Timidity, stubbornness and Timidity, stubbornness and positive invariance hold the positive invariance hold the picture fixed up to picture fixed up to kk

k

0

4

e’

e

? e’’

Method *Method *F,F,nn Fails on Fails on GGnn ((ee))

Ouch! Ouch! Positive invariance, Positive invariance, timidity and stubbornness and timidity and stubbornness and = 4 = 4

k

k

= 4

0

4

e’

e 0

e’

e

Method *Method *F,F,nn Fails on Fails on GGnneveneven

((ee))

Same, using the even stacking lemmaSame, using the even stacking lemma

Part VI:Part VI:

Positive ResultsPositive Results

How to Program Epistemic StatesHow to Program Epistemic States

Hamming Distance: Hamming Distance: ((ee, , e’e’) = {) = {nn: : ee((nn) °) ° e’ e’((nn)})} ((ee, , e’e’) = | ) = | ((ee, , e’e’) |) |

(e, e’ ) = 9

e

e’

Hamming AlgebraHamming Algebra

aa HH b mod e b mod e ((ee, , aa) is a subset of ) is a subset of ee, , bb))

1 1 1

1 1 0 1 0 1 0 1 1

1 0 0 0 1 0 0 0 1

0 0 0

Hamming

Epistemic States as Boolean RanksEpistemic States as Boolean Ranks

Hamming Algebra

GGeveneven

((ee))

e

Advantage of Hamming RankAdvantage of Hamming Rank

No violations of limit point condition over No violations of limit point condition over finite levels:finite levels:

nobody lower can match this

**R,1 R,1 ,*,*J,1 J,1 can identify can identify GGeveneven((ee))

GG00eveneven

((ee))GG11

eveneven ((ee))

GG22eveneven

((ee))

GGnneveneven

((ee))

GGeveneven

((ee))

= 1


= 0

= 2

= 3

= n +1

= 0 = 0

= 1

= 1

= 1

= 1

yes

yes

yes

yes

yes yes

yes

yes

yes

yes

= 1

= 1

yes

no

no

no

no =

= 1


Proof: Proof: Let Let SS be generated by be generated by finite ranks of the Hamming finite ranks of the Hamming algebra rooted at algebra rooted at ee

GGeveneven

((ee))

e

S


Let a be an arbitrary element of Let a be an arbitrary element of GG

eveneven((ee))

So So aa is at a finite level of is at a finite level of SS

e

a

S


Consider the principal ideal of Consider the principal ideal of aa These are the possibilities that These are the possibilities that

differ from differ from ee only where only where aa does does

e

a S


So these are the possibilities that So these are the possibilities that have just one difference from have just one difference from the truth for each level below the truth for each level below the truththe truth

e

a S

e

33

a


Induction = hypercube rotationInduction = hypercube rotation


a e Example


a

e

a e Example


a

e


a

e


ConvergenceConvergence

a

e


Other possibilities have more than enough Other possibilities have more than enough differences to climb above the truthdifferences to climb above the truth

**J,1 J,1 doesn’t backslide since the rotation doesn’t backslide since the rotation keeps refuted possibilities rising keeps refuted possibilities rising

**R,2R,2 is Complete is Complete


GG22((ee))

GG33((ee))

GG((ee))


= 2

= 2

= 2 yes

yes

yes

yes yes

yes

yes

yes

= 2

= 2

no

no

no

no = = 2

= n +1

= 3

= 2

yes yes

= 0 = 0 = 0yes

= 1

= 2


Proposition: Proposition: **R,2R,2 is a complete function is a complete function identifieridentifier

Proof: Proof: Let Let KK be countable be countable Partition Partition KK into finite variant classes into finite variant classes

CC00, , CC11, ..., , ..., CCn n , ..., ...


Impose the Hamming distance ranking on Impose the Hamming distance ranking on each equivalence classeach equivalence class

Now raise the Now raise the nnth Hamming ranking by th Hamming ranking by nn

CC00 CC11 CC22 CC33 CC44

SS


Else might generate horizontal limit points:Else might generate horizontal limit points:


SS


Data streams in different columns differ Data streams in different columns differ infinitely often from the truth. infinitely often from the truth.


truthzoom

!


Data streams in the same column just barely Data streams in the same column just barely make it because they jump by 2 for each make it because they jump by 2 for each difference from the truthdifference from the truth


SS


Data streams in the same column just barely Data streams in the same column just barely make it because they jump by 2 for each make it because they jump by 2 for each difference from the truthdifference from the truth


SS


Convengence at least by the stage when 2Convengence at least by the stage when 2mm differences from differences from ee have been observed for have been observed for each each eeii belowbelow e e that is not in the column of that is not in the column of ee


SSe m

e 0

e 3

e 4

How about *How about *J,2J,2??

The same thing works, The same thing works, rightright??


SS

The Wrench in the WorksThe Wrench in the Works

Proposition: Proposition: Even *Even *J,2 J,2 can’t succeed if we can’t succeed if we add ¬ add ¬ ee to to GG

eveneven((ee) when ) when SS extends the extends the

Hamming rankingHamming ranking Proof: Proof: Suppose otherwiseSuppose otherwise


Feed ¬ Feed ¬ e e until it is uniquely at the bottom until it is uniquely at the bottom

¬¬e

k


Let Let nn exceed exceed kk and the original height of ¬ and the original height of ¬ee aa = ¬ = ¬e e ‡ ‡ n n b = b = ¬¬e e ‡ ‡ n+n+11

¬¬e

a

b

k n


By positive invariance, timidity and stubb.By positive invariance, timidity and stubb.

¬¬e

a

b

k n


By positive invariance, timidity, stubbornness and the fact that ¬By positive invariance, timidity, stubbornness and the fact that ¬ee was was alone in the basementalone in the basement

¬¬e

ab

k n

Ouch!!!

(e, e’ ) = 6

e

e’

Solution: A Different Initial StateSolution: A Different Initial State

Goodman Distance: Goodman Distance: ((ee, , e’e’) = {) = {nn: : e’e’ grues grues ee at at nn)})} gg ( (ee, , e’e’) = | ) = | ((ee, , e’e’) |) |

Hamming vs. Goodman AlgebrasHamming vs. Goodman Algebras

aa HH b mod e b mod e ((ee, , aa) is a subset of ) is a subset of ee, , bb))

aa GG b mod e b mod e ((ee, , aa) is a subset of) is a subset ofee, , bb))

Goodman1 0 1

1 1 0 0 1 0

0 1 1 0 0 1

0 0 0

1 1 1

1 1 0 1 0 1 0 1 1

1 0 0 0 1 0 0 0 1

0 0 0

Hamming

1 0 0

1 1 1


GoodmanHamming

GG((ee))GGeveneven

((ee))

GGoddodd

((ee))

e e


Goodman

GG00((ee))GG11((ee))GG22((ee))GG33((ee))

e

**J,2 J,2 can identify can identify GG ((ee))


GG22((ee))

GG33((ee))

GG((ee))


= 2

= 2

= 2 yes

yes

yes

yes yes

yes

yes

yes

= 2

= 2

no

no

no

no = = 2

= n +1

= 3

= 2

yes yes

= 0 = 0 = 0yes

= 1

= 2

**J,2 J,2 can identify can identify GG ((ee))

Proof: Proof: Use the Goodman ranking as initial stateUse the Goodman ranking as initial state Show by induction that the method projects Show by induction that the method projects ee

until a grue occurs at until a grue occurs at nn, then projects , then projects ee ‡ ‡ nn, until , until another grue occurs at another grue occurs at n’n’, etc. , etc.

Part VII:Part VII:

DiscussionDiscussion

SummarySummary

Belief revision as inductive inquiryBelief revision as inductive inquiry Reliability vs. intuitive symmetriesReliability vs. intuitive symmetries Intuitive symmetries imply reliability for large Intuitive symmetries imply reliability for large Intuitive symmetries restrict reliability for small Intuitive symmetries restrict reliability for small Sharp discriminations among proposed methodsSharp discriminations among proposed methods Isolation of fundamental epistemic dilemmaIsolation of fundamental epistemic dilemma = 2 as fundamental epistemic invariant= 2 as fundamental epistemic invariant Learning as cube rotationLearning as cube rotation Surprising relevance of tail reversalsSurprising relevance of tail reversals

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Inductive Amnesia

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