Three Applications of Means-EndsEpistemology

Oliver SchulteDepartment of Philosophy andSchool of Computing ScienceSimon Fraser UniversityVancouver, Canadaoschulte@cs.sfu.ca `

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Outline

The Means-Ends Approach to InductiveInference Induction in Particle Physics Will a particle reaction be observed? Find conservation laws explaining which

reactions are observed. A Goodmanian Riddle of Induction Explaining Observed Correlations withCausal Models

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Philosophical Roots of Means-EndsEpistemology (Learning Theory)

Peirce

Reichenbach

Putnam

Glymour, Kelly

Peirce:

• In inquiry, some things are settled,some are in doubt.

•Inquiry settles on true opinion in thelimit.

Reichenbach, “Vindication of Induction”:guaranteed convergence to the rightopinion.

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Convergence to the Correct Theory inthe Limit of Inquiry

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Inductive Axiology

• Consider epistemic goals in addition tolong-run convergence (Putnam 1965)

• Analyze relationship between different goals,e.g.:

• Fast Convergence

• Stable Convergence - avoid theory changes(Plato, Putnam 1965, “epistemic conservatism”Sklar 1975, Kuhn)

• Mistake Bounds - avoid false predictions

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Fast Convergence to the Truth

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Reliable Stable Convergence: AnExample

Is a certain reaction possible?, e.g.r = n + n → p + p + e- + e-

Rules

• Inquirer makesconjecture (“yes”,“no”, ?)

• Demon showsexperimental outcomes(“observed” or not).

• Inquirer pays forabandoning “yes” or“no”.

r observed

nor

r observed

nor

r observed

nor

….

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Mean-Ends Analysis for “Is thisreaction possible?”

Proposition All inference rules that converge to the right answer with atmost one mind change conjecture ?or “reaction is forbidden” until thereaction is observed.

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Induction in Particle Physics

ParticleReview2002:no2v → 2p+ 2e-

ParticleReview2003:no2v → 2p+ 2e-

ParticleReview2004:no2v → 2p+ 2e-

ParticleReview2005:2v →2p + 2e-

observed

I see! 2v → 2p + 2e- is impossible.

I must find a conservation lawthat explains this.

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Additive Conservation Laws

Assuming the empirical adequacy ofConservation Principles entailsunobserved reactions

Σ- → π- + n π- → µ- + νµ µ

- → e- + νµ + νe n → e- + νe + pp + p → p + p + π

observed reactions not yet observed reactions

n → e- + νe

p + p → p + p + π + π

entailed

Hypothetical Scenario

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The Strict Inference Method

Strict Method: suppose that reaction r has notbeen observed so far. If there is no possible conservation principle that

rules out r, conjecture that r is possible. If some possible conservation principle rules out r,

conjecture that r is forbidden, and introduce aconservation principle to explain why.

• Nobel Laureate Cooper (1970): “In the analysis of events among thesenew particles, where the forces are unknown and the dynamical analysis,if they were known, is almost impossibly difficult, one has tried byobserving what does not happen to find selection rules, quantumnumbers, and thus the symmetries of the interactions that are relevant.”

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Means-Ends Justification forMaximally Strict Inferences

Theorem. Suppose we have n knownparticles. The strict inference methodis the only inference rule that

1. is guaranteed to eventually arrive atan empirically adequate set ofconservation principles, and

2. changes its predictions at most ntimes.

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The Naturalist’s Question:Comparison with Practice

Finding: The standard laws Electric Charge,Baryon#, Muon#, Electron#, Tau# form amaximally strict set for the current reactiondata.

Physicists have acted as if they are followingthe methodology described so far.Program

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Green and GrueGoodman (1983). “Grue applies to all things examined before tjust in case they are green but to other things just in case theyare blue.”

Rules

• Inquirer projectsgeneralization (e.g. “allgreen”)

• Demon chooses colorof next emerald.

• Inquirer pays formistaken predictions.

Allgreen….

…. All blue

…. All grue(1)

…. All grue(2)

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Means-Ends Justification for TheNatural Projection Rule

Theorem. Given the possible observationsequences shown before, the naturalprojection rule (project “all green”) is theonly projection rule that

1. is guaranteed to eventually arrive at anempirically adequate generalization aboutemerald colors, and

2. makes at most 1 false prediction.

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Learning Causal Models fromCorrelations

Hume: causes are inferred from “constantconjunctions”.Reichenbachian principle: every correlation has acausal explanation, e.g. by common cause.Need to consider conditional correlations.

Father’s Eye Color Mother’s Eye Color

Child’s Eye Color

ME correlates with FEgiven CE = green,o.w. independent.

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Causal Graphs and ConditionalCorrelations

A 3-variable graph(A,B,C) entails thefollowing correlations. A correlates with B if

there is a link A—B(either direction), or apath A→C→B or A←C←Bor A←C→B.

A correlates with B givenC if there is a link A—Bor a path A→C←B.

Father’s Eye Color Mother’s Eye Color

Child’s Eye Color

• FE corr CE

• ME corr CE

• FE corr ME given CE

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Causal Learning from CorrelationsChildren may show spots on their stomachs. A doctor wonderswhether there is a causal connection with measles or an allergicreaction.

Rules

• Inquirer proposescausal graph or ?.

• Demon chooses nextcorrelation(s) or“nothing new”.

• Inquirer pays forabandoning a causalmodel.

Measles Allergy

Spots

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The Mind-Change Optimal CausalLearner

Consider the following causal learner: Given alist of observed correlations, if there is a graphG consistent with the observations such thatall other consistent graphs have more edgesthan G, output G. Otherwise output ?.Explain the observed correlations with thefewest direct causal links.Theorem Let n variables (nodes) be given.The causal learner above succeeds withmind changes.

!

n

2

"

# $ %

& '

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Notes on Causal Learning Result

The edge-minimizer is the fastest causallearner that minimizes mind changes nomatter what correlations are observed.A variant: conjecture the set of graphsthat are consistent with the observedcorrelations and have a minimum numberof edges.There is no fast program for computingminimum-edge graphs (Schulte,Greiner,Luo 2007).

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Extensions and Directions

Characterize deep structure of inductiveproblem (point-set topology).Relate to other aims (speed, simplicity).Relate to “categorical imperatives”, e.g.Bayesian conditioning, minimal belief changepostulates.Address computational questions →algorithmic learning theory.

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Summary

Means-Ends Analysis: Convergence to the truth, stableand fast convergence, correct predictions.For specific problems, a set of inductive goals →methods that achieve these goals.3 Illustrations/Applications: Conservation laws in particle physics: conjecture maximally

strict law sets. Goodmanian Riddle: conjecture “all emeralds are green”. Learning Causal Models: Conjecture the model that explains

the observed correlations with a minimum number of directcausal links (edges).

These are instantiations of a single generic mindchange optimal inference rule.

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References

1. “Inferring Conservation Principles in Particle Physics: A CaseStudy in the Problem of Induction”.Schulte, O. (2000). British Journal for the Philosophy ofScience.

2. "Mind-Change Efficient Learning", W. Luo and O. Schulte(2006). Information and Computation,204:989--1011.

3. "Mind Change Optimal Learning of Bayes Net Structure".O.Schulte, W. Luo and R. Greiner (2007), in Proceedings ofthe 20th Annual Conference on Learning Theory, pp. 187-202.

4. “Causation, Prediction and Search”, Sprites, Glymour andScheines (2000).

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THE END