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Truth-conduciveness Without Reliability A

Skeptical Derivation of Ockhams Razor

- Kevin T. Kelly
- Department of Philosophy
- Carnegie Mellon University
- www.cmu.edu

Naivete

Lo! An apple.

Skeptical Hypothesis

Lo! An apple.

Maybe you are a brain in a vat. Everything would

look the same.

Skeptical Hypothesis

poof

Maybe you are a brain in a vat. Everything would

look the same.

Retrenchment

Thats not a serious possibility

You have the burden of proof. Its remote. Its

implausible. Its distant from the actual

world. Youre not in my community. Who cares

about the worst case?

Retrenchment

Thats not a serious possibility

You have the burden of proof. Its remote. Its

implausible. Its distant from the actual

world. Youre not in my community. Who cares

about the worst case?

Unsatisfying

- Possibilities delimited a priori circular

account. - Possibilities delimited a posteriori how do we

seek knowledge?

So there!

Zen Approach

- Dont rush to defeat the demon.

Grrrr!

Zen Approach

- Dont rush to defeat the demon.
- Get to know him extremely well.
- Justification may be located in the demons power

rather than in his weakness.

The Zen of Computation

- Algorithms are justified by efficiency.
- Efficiency means you couldnt do better.
- You couldnt do better due to a demonic argument

(the halting problem, etc).

Scientific Theory Choice

Which theory is true?

Ockham Says

Choose the Simplest!

Skeptical Hypothesis

Maybe a complex theory is true but the data are

simple

Puzzle

- An indicator must be sensitive to what it

indicates.

simple

Puzzle

- An indicator must be sensitive to what it

indicates.

complex

Puzzle

- But Ockhams razor always points at simplicity.

simple

Puzzle

- But Ockhams razor always points at simplicity.

complex

Meno

- If we know that the truth is simple, we dont

need Ockhams razor.

simple

Meno

- If we dont know that the truth is simple, what

good is Ockams razor?

complex

Some Standard Responses

Simple Theories are Virtuous

- Testable (Popper, Glymour)
- Unified (Friedman, Kitcher)
- Explanatory (Harman)
- Symmetrical (Malament)
- Compress data (Rissanen)
- Interesting (Vitanyi)

But the Truth Might Not be Virtuous

- To conclude that a theory is true because it is

virtuous is wishful thinking (van Fraassen).

Overfitting (Akaike, Sober, Forster)

- Empirical estimates based on complex models have

greater mean squared distance from the truth

Truth

Overfitting (Akaike, Sober, Forster)

- Empirical estimates based on complex models have

greater mean squared distance from the truth.

Pop! Pop! Pop! Pop!

Overfitting (Akaike, Sober, Forster)

- Empirical estimates based on complex models have

greater mean squared distance from the truth.

Truth

clamp

Overfitting (Akaike, Sober, Forster)

- Empirical estimates based on complex models have

greater mean squared distance from the truth.

Pop! Pop! Pop! Pop!

Truth

clamp

Does Not Aim at True Theory

- ...even if the simple theory is known to be false

Four eyes!

clamp

Miracle Argument (Putnam, Rosenkrantz)

- Simple data would be a miracle in a complex

world. - Simple data would be expected in a simple world.

Miracle Argument

Planetary retrograde motion

Mars

Earth

Sun

Miracle Argument

- Simple data would be a miracle in a complex

world. - Simple data would be expected in a simple world.

epicycle

q

lapping

Complex theory

Simple theory

Miracle Argument

- Simple data would be a miracle in a complex

world. - Simple data would be expected in a simple world.

epicycle

lapping

q

Simple theory

Complex theory

However

- Simple data would not be a miracle if the

complex theorys parameter were set near q

epicycle

q

lapping

Complex theory

Simple theory

The Real Miracle

Ignorance about model p(S) ?

p(C) Ignorance about parameter settings

within theories p(C(q) C) ? p(C(q ) C).

Knowledge about parameter settings across

theories p(C(q)) ltlt p(S).

Is it knognorance or Ignoredge?

The Ellsberg Paradox

3 ball colors with these frequencies

The Ellsberg Paradox

p

q

r

Human betting preferences

p

q

gt

The Ellsberg Paradox

p

q

r

Human betting preferences

!

p

q

gt

r

lt

p

q

r

Diagnosis

p

q

r

ignorance

knowledge

Robust Bayesianism (Levi, Kadane, Seidenfeld)

knowledge

ignorance

1/3

?

?

p

q

r

. . .

Credence is range of probs.

1/3

1/3

1/3

. . .

1/3

2/3

0

Choose the act with highest worst-case expected

value.

Worst-case Expected Values

1/3

?

?

1/3

?

?

1/3

0

gt

gt

lt

1/3

0

2/3

Whither Ockham?

Since you dont really know that complex worlds

wont produce simple data, shouldnt your

ignorance include distributions concentrated on

such possibilities?

I prefer ignoredge.

In Any Event

The coherentist foundations of Bayesianism have

nothing to do with short-run truth-conduciveness.

Temptation

If only the probabilities p(C(q ) C) were

chances rather than opinions. Then the alleged

miracle would be a proper miracle.

Proof of God (R. Koons 1999)

- Natural chance is determined by the fundamental

theory of natural chance. - If Ockhams razor reliably infers the theory of

natural chance, the chance that a complex theory

of natural chance would have its parameters set

to produce simple data must be low. - But since natural chance is determined by the

free parameters of the fundamental theory of

natural chance, the parameter setting is not

governed by natural chance. - Hence, it must be governed by non-natural chance.
- Holy water is available at the exit.

Moral

- The basic point is right.
- Solution
- Keep naturalism
- Keep fundamental scientific knowledge
- Dump short-run reliability as explication of

truth-conduciveness.

Externalist Magic

- Simplicity informs via hidden causes or tracking

mechanisms.

G

With Friends Like Those

- Practice and data are the same.
- Knowledge vs. non-knowledge depends on hidden

causes. - By Ockhams razor, better to explain Ockhams

razor without the hidden causes.

?

The Last Gasp Convergence

Bayes (washing out of the prior) BIC

(Schwarz) Structural Risk Minimization (Vapnik,

Harman) TETRAD (Spirtes, Glymour, Scheines)

truth

Complexity

The Last Gasp Convergence

truth

Plink!

Blam!

Complexity

The Last Gasp Convergence

truth

Plink!

Blam!

Complexity

The Last Gasp Convergence

truth

Plink!

Blam!

Complexity

Logic is Backwards

- Ockham methods are sufficient for convergence.
- But every finite variant of a convergent method

converges (Salmon). - So Ockhams razor is not necessary for

convergence.

truth

Alternative ranking

Truth Conduciveness

- Reliability
- Too strong
- Circles or magic required.
- Convergence
- Too weak
- Doesnt single out simplicity

Complex

Simple

Simple

Complex

Truth Conduciveness

- Indication or tracking
- Too strong
- Circles or magic required.
- Convergence
- Too weak
- Doesnt single out simplicity
- Straightest convergence
- Just right?

Complex

Simple

Simple

Complex

Complex

Simple

Truth-conduciveness as Straightest Convergence

Complex

Simple

Ancient Roots

"Living in the midst of ignorance and considering

themselves intelligent and enlightened, the

senseless people go round and round, following

crooked courses, just like the blind led by the

blind." Katha Upanishad, I. ii. 5, c. 600 BCE.

Retraction

- New output does not entail previous output.

Retracted Content

t

t 1

Eliminate Needless Retractions

Truth

Necessary Retractions are Virtuous

Truth

Demons Role as Justifier

Truth

I can force every convergent method to retract

this often, so your retractions are justified by

my power.

Eliminate Needless Delays to Retractions

theory

Eliminate Needless Delays to Retractions

application

theory

application

application

application

corollary

application

application

application

corollary

application

corollary

Easy Comparisons

at least as bad at least as many retractions

at least as late

retractions

time

Worst-case Retraction Time Bounds

(1, 2, 8)

. . .

. . .

. . .

Empirical Complexity

Hopeless ideas Syntactic length Computational

incompressibility

By what miracle do notational conventions

indicate truth?

Empirical Complexity

Close but no cigar Free parameters Broken

symmetries

Meno, I want simplicity itself, not parts of

simplicity.

Empirical Complexity

Empirical complexity of T in G the length of

the maximum path (T1, , Tn, T) of answers in G

the demon can force from an arbitrary convergent

method.

Keep up!

T

T3

T2

T1

Polynomial Order

- Data open intervals around Y at rational values

of X.

Polynomial Order

- Demon shows flat line until convergent method

takes bait.

Zero degree curve

Polynomial Order

- Demon shows flat line until convergent method

takes bait.

Zero degree curve

Polynomial Order

- Then switches to tilted line until convergent

method takes the bait.

First degree curve

Polynomial Order

- Then switches to parabola until convergent method

takes the bait

Second degree curve

Complexity can be Complex

Complexity given e

T2

3

T7

T4

2

T8

T5

1

0

T3

Complexity Relative to Data

Complexity given e e

T2

3

T7

T4

2

T8

T5

1

0

T3

Complexity Relative to Data

Complexity given e e

3

2

T2

1

0

T5

T4

T7

Timed Retraction Bounds

- r(M, e, n) the least timed retraction bound for

worlds satisfying theories of complexity n and

producing finite input history e.

M

. . .

. . .

Empirical Complexity

0

1

2

3

M is Efficient at e

- For each convergent M that agrees with M along

finite input history e, - for each complexity n
- r(M, e, n) ? r(M, e, n)

M

M

. . .

. . .

Empirical Complexity

0

1

2

3

M is Strongly Beaten at e

- There exists convergent M that agrees with M up

to the end of e, such that - for each complexity n
- r(M, e, n) gt r(M, e, n).

M

M

. . .

. . .

Empirical Complexity

0

1

2

3

M is Weakly Beaten at e

- There exists convergent M that agrees with M up

to the end of e, such that - For each n, r(M, e, n) ? r(M, e, n)
- Exists n, r(M, e, n) gt r(M, e, n).

M

M

. . .

. . .

Empirical Complexity

0

1

2

3

Demons for Ockham

Ockhams Razor

- Dont select a theory unless it is uniquely

simplest in light of experience.

3

2

?

T2

1

0

T5

T4

T7

Ockhams Razor

- Dont select a theory unless it is uniquely

simplest in light of experience.

3

2

T7

T2

1

0

T7

Stalwartness

- Dont retract your answer while it remains

uniquely simplest

3

2

T7

T7,

T2

1

0

T7

Argument Sketch

- No matter what convergent M has done in the past,

nature can force M to produce each answer down an

arbitrary effect path, arbitrarily often. - Nature can also force violators of Ockhams razor

or stalwartness either into an extra retraction

or a late retraction in each complexity class.

Ockham Efficiency Theorem

- Let M converge to the true theory in problem P.

The following are equivalent - M is always Ockham and stalwart in P
- M is always efficient in P
- M is never weakly beaten in P.

Policy Retractions

- Many explanations have been offered to make sense

of the here-today-gone-tomorrow nature of medical

wisdom what we are advised with confidence one

year is reversed the next but the simplest one

is that it is the natural rhythm of science. - (Do We Really Know What Makes us Healthy, NY

Times Magazine, Sept. 16, 2007).

Causal Inference

- Causal graph theory more correlations ? more

causes. - Idealized data list of conditional dependencies

discovered so far. - Anomaly the addition of a conditional

dependency to the list.

partial correlations

S

G(S)

Causal Axioms (Pearl, Glymour)

- Screening off X is statistically independent of

its non-descendents given its parents. - No invisible causes The only true independence

relations are those entailed by condition 1.

N1

N1

P1

P2

P2

P1

N2

X

D

Forcible Sequence of Causal Theories

Y1

X2

X3

W

X1

Y2

Forcible Sequence of Causal Theories

Y1

Y3

X2

X3

W

X1

Y2

Y4

Forcible Sequence of Causal Theories

Y1

Y3

X2

X3

W

X1

Y2

Y4

Y5

Forcible Sequence of Causal Theories

Y1

Y3

X2

X3

W

X1

Y2

Y4

Y5

Y4

Moral

- In counterfactual prediction, form of model

matters and retractions are unavoidable. - Ockham efficiency agrees very closely with best

contemporary practice. - Maybe thats all there is to it.

Conclusions

- Ockhams razor is necessary for staying on the

straightest path to the truth - Does not reliably point at or indicate the truth.
- Demonstrably works without circles, evasions, or

magic. - Such a theory is motivated in counterfactual

inference and estimation.