Truth-conduciveness Without Reliability: A Skeptical Derivation of Ockham

1
Truth-conduciveness Without Reliability A
Skeptical Derivation of Ockhams Razor

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

2
Naivete
Lo! An apple.
3
Skeptical Hypothesis
Lo! An apple.
Maybe you are a brain in a vat. Everything would
look the same.
4
Skeptical Hypothesis
poof
Maybe you are a brain in a vat. Everything would
look the same.
5
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?
6
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?
7
Unsatisfying

• Possibilities delimited a priori circular
  account.
• Possibilities delimited a posteriori how do we
  seek knowledge?

So there!
8
Zen Approach

• Dont rush to defeat the demon.

Grrrr!
9
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.

10
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).

11
Scientific Theory Choice
Which theory is true?
12
Ockham Says
Choose the Simplest!
13
Skeptical Hypothesis
Maybe a complex theory is true but the data are
simple
14
Puzzle

• An indicator must be sensitive to what it
  indicates.

simple
15
Puzzle

• An indicator must be sensitive to what it
  indicates.

complex
16
Puzzle

• But Ockhams razor always points at simplicity.

simple
17
Puzzle

• But Ockhams razor always points at simplicity.

complex
18
Meno

• If we know that the truth is simple, we dont
  need Ockhams razor.

simple
19
Meno

• If we dont know that the truth is simple, what
  good is Ockams razor?

complex
20
Some Standard Responses
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Simple Theories are Virtuous

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

22
But the Truth Might Not be Virtuous

• To conclude that a theory is true because it is
  virtuous is wishful thinking (van Fraassen).

23
Overfitting (Akaike, Sober, Forster)

• Empirical estimates based on complex models have
  greater mean squared distance from the truth

Truth
24
Overfitting (Akaike, Sober, Forster)

• Empirical estimates based on complex models have
  greater mean squared distance from the truth.

Pop! Pop! Pop! Pop!
25
Overfitting (Akaike, Sober, Forster)

• Empirical estimates based on complex models have
  greater mean squared distance from the truth.

Truth
clamp
26
Overfitting (Akaike, Sober, Forster)

• Empirical estimates based on complex models have
  greater mean squared distance from the truth.

Pop! Pop! Pop! Pop!
Truth
clamp
27
Does Not Aim at True Theory

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

Four eyes!
clamp
28
Miracle Argument (Putnam, Rosenkrantz)

• Simple data would be a miracle in a complex
  world.
• Simple data would be expected in a simple world.

29
Miracle Argument
Planetary retrograde motion
Mars
Earth
Sun
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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
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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
32
However

• Simple data would not be a miracle if the
  complex theorys parameter were set near q

epicycle
q
lapping
Complex theory
Simple theory
33
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?
34
The Ellsberg Paradox
3 ball colors with these frequencies
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The Ellsberg Paradox
p
q
r
Human betting preferences
p
q
gt
36
The Ellsberg Paradox
p
q
r
Human betting preferences
!
p
q
gt
r
lt
p
q
r
37
Diagnosis
p
q
r
ignorance
knowledge
38
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.
39
Worst-case Expected Values
1/3
?
?
1/3
?
?
1/3
0
gt
gt
lt
1/3
0
2/3
40
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.
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In Any Event
The coherentist foundations of Bayesianism have
nothing to do with short-run truth-conduciveness.
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Temptation
If only the probabilities p(C(q ) C) were
chances rather than opinions. Then the alleged
miracle would be a proper miracle.
43
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.

44
Moral

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

45
Externalist Magic

• Simplicity informs via hidden causes or tracking
  mechanisms.

G
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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.

?
47
The Last Gasp Convergence
Bayes (washing out of the prior) BIC
(Schwarz) Structural Risk Minimization (Vapnik,
Harman) TETRAD (Spirtes, Glymour, Scheines)
truth
Complexity
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The Last Gasp Convergence
truth
Plink!
Blam!
Complexity
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The Last Gasp Convergence
truth
Plink!
Blam!
Complexity
50
The Last Gasp Convergence
truth
Plink!
Blam!
Complexity
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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
52
Truth Conduciveness

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

Complex
Simple
Simple
Complex
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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
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Truth-conduciveness as Straightest Convergence
Complex
Simple
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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.
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Retraction

• New output does not entail previous output.

Retracted Content
t
t 1
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Eliminate Needless Retractions
Truth
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Necessary Retractions are Virtuous
Truth
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Demons Role as Justifier
Truth
I can force every convergent method to retract
this often, so your retractions are justified by
my power.
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Eliminate Needless Delays to Retractions
theory
61
Eliminate Needless Delays to Retractions
application
theory
application
application
application
corollary
application
application
application
corollary
application
corollary
62
Easy Comparisons
at least as bad at least as many retractions
at least as late
retractions
time
63
Worst-case Retraction Time Bounds
(1, 2, 8)
. . .
. . .
. . .
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Empirical Complexity
Hopeless ideas Syntactic length Computational
incompressibility
By what miracle do notational conventions
indicate truth?
65
Empirical Complexity
Close but no cigar Free parameters Broken
symmetries
Meno, I want simplicity itself, not parts of
simplicity.
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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
67
Polynomial Order

• Data open intervals around Y at rational values
  of X.

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Polynomial Order

• Demon shows flat line until convergent method
  takes bait.

Zero degree curve
69
Polynomial Order

• Demon shows flat line until convergent method
  takes bait.

Zero degree curve
70
Polynomial Order

• Then switches to tilted line until convergent
  method takes the bait.

First degree curve
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Polynomial Order

• Then switches to parabola until convergent method
  takes the bait

Second degree curve
72
Complexity can be Complex
Complexity given e
T2
3
T7
T4
2
T8
T5
1
0
T3
73
Complexity Relative to Data
Complexity given e e
T2
3
T7
T4
2
T8
T5
1
0
T3
74
Complexity Relative to Data
Complexity given e e
3
2
T2
1
0
T5
T4
T7
75
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
76
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
77
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
78
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
79
Demons for Ockham
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Ockhams Razor

• Dont select a theory unless it is uniquely
  simplest in light of experience.

3
2
?
T2
1
0
T5
T4
T7
81
Ockhams Razor

• Dont select a theory unless it is uniquely
  simplest in light of experience.

3
2
T7
T2
1
0
T7
82
Stalwartness

• Dont retract your answer while it remains
  uniquely simplest

3
2
T7
T7,
T2
1
0
T7
83
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.

84
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.

85
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).

86
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)
87
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
88
Forcible Sequence of Causal Theories
Y1
X2
X3
W
X1
Y2
89
Forcible Sequence of Causal Theories
Y1
Y3
X2
X3
W
X1
Y2
Y4
90
Forcible Sequence of Causal Theories
Y1
Y3
X2
X3
W
X1
Y2
Y4
Y5
91
Forcible Sequence of Causal Theories
Y1
Y3
X2
X3
W
X1
Y2
Y4
Y5
Y4
92
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.

93
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.