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CAUSES ANDCOUNTERFACTUALS
OR
THE SUBTLE WISDOMOF BRAINLESS ROBOTS
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ANTIQUITY TO ROBOTICS
“I would rather discover one causal relation than beKing of Persia”
Democritus (430-380 BC)
Development of Western science is based on two great achievements: the invention of the formal logical system (in Euclidean geometry) by the Greek philosophers, and the discovery of the possibility to find out causal relationships by systematic experiment (during the Renaissance).
A. Einstein, April 23, 1953
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David Hume (1711–1776)
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HUME’S LEGACYHUME’S LEGACY
1. Analytical vs. empirical claims
2. Causal claims are empirical
3. All empirical claims originate from experience.
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THE TWO RIDDLESTHE TWO RIDDLESOF CAUSATIONOF CAUSATION
What empirical evidence legitimizes a cause-effect connection?
What inferences can be drawn from causal information? and how?
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““Easy, man! that hurts!”Easy, man! that hurts!”
The Art ofCausal Mentoring
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1. How should a robot acquire causal information from the environment?
2. How should a robot process causal information received from its creator-programmer?
OLD RIDDLES IN NEW DRESSOLD RIDDLES IN NEW DRESS
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Input:1. “If the grass is wet, then it rained”2. “if we break this bottle, the grass
will get wet”
Output:“If we break this bottle, then it rained”
CAUSATION AS A CAUSATION AS A PROGRAMMER'S NIGHTMAREPROGRAMMER'S NIGHTMARE
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CAUSATION AS ACAUSATION AS APROGRAMMER'S NIGHTMARE PROGRAMMER'S NIGHTMARE
(Cont.) ( Lin, 1995)(Cont.) ( Lin, 1995)
Input:1. A suitcase will open iff both
locks are open.2. The right lock is open
Query:What if we open the left lock?
Output:The right lock might get closed.
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Y = 2X
BRAINLESS FIRST DISCOVERY:PHYSICS DESERVES A NEW ALGEBRA
Had X been 3, Y would be 6.If we raise X to 3, Y would be 6.Must “wipe out” X = 1.
X = 1 Y = 2
The solutionProcess information
Y := 2X
Correct notation:
X = 1
e.g., Length (Y) equals a constant (2) times the weight (X)
Scientific Equations (e.g., Hooke’s Law) are non-algebraic
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MATHEMATICAL EXTRAPOLATION:THE WORLD AS A COLLECTION
OF SPRINGS
Definition: A structural causal model is a 4-tupleV,U, F, P(u), where• V = {V1,...,Vn} are endogeneas variables• U = {U1,...,Um} are background variables• F = {f1,..., fn} are functions determining V,
vi = fi(v, u)• P(u) is a distribution over UP(u) and F induce a distribution P(v) over observable variables
Yuxy e.g.,
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Z
YX
INPUT OUTPUT
FAMILIAR CAUSAL MODELORACLE FOR COUNTERFACTUALS
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)()( uYuY xMx
The Fundamental Equation of Counterfactuals:
BRAINLESS SECOND DISCOVERY:COUNTERFACTUALS ARE EMBARRASINGLY SIMPLE
Definition: The sentence: “Y would be y (in situation u), had X been x,”
denoted Yx(u) = y, means:The solution for Y in a mutilated model Mx, (i.e., the equations
for X replaced by X = x) with input U=u, is equal to y.
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),|(),|'(
)()()|(
')(':'
)(:
yxuPyxyYPN
uPyYPyP
yuxYux
yuxYux
In particular:
)(xdo
BRAINLESS SECOND DISCOVERY:COUNTERFACTUALS ARE EMBARRASINGLY SIMPLE
Definition: The sentence: “Y would be y (in situation u), had X been x,”
denoted Yx(u) = y, means:The solution for Y in a mutilated model Mx, (i.e., the equations
for X replaced by X = x) with input U=u, is equal to y.•
)(),()(,)(:
uPzZyYPzuZyuYu
wxwx
Joint probabilities of counterfactuals:
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Data
Inference
Q(M)(Aspects of M)
Data Generating
Model
M – Invariant strategy (mechanism, recipe, law, protocol) by which Nature assigns values to variables in the analysis.
JointDistribution
THE STRUCTURAL MODELPARADIGM
M
“Think Nature, not experiment!”•
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THE PUZZLE OF COUNTERFACTUAL CONSENSUS
• Indicative: “If Oswald didn’t kill Kennedy, someone else did,”
• Subjunctive: “If Oswald hadn’t killed Kennedy, someone else would have.”
(Adams 1975)
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THE PUZZLING UBIQUITYOF COUNTERFACTUALS
Hume’s Definition of “cause”: We may define a cause to be an object followed by another, and where all the objects, similar to the first, are followed by objects similar to the second, Or, in other words, where, if the first object had not been, the second never had existed (Hume 1748/1958, sec. VII).
Lewis’s Definition of “cause”: “A has caused B” if “B would not have occurred if it were not
for A (Lewis 1986).
• Why not define counterfactuals in terms of causes?(Pearl 2000)
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w
STRUCTURAL AND SIMILARITY-BASED COUNTERFACTUALS
Lewis’s account (1973): The counterfactual “B if it were A” is true in a world w just in case B is true in all the closest A-worlds to w.
B
A
.)( yuY xM
Structural account (1995): “Y would be y if X were x” is true in situation u just in case
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OS
true
false
P (SE) = 1
true
true
true
P (SE) = P (SE)
S1: “IF OSWALD DIDN’T KILL KENNEDY, SOMEONE ELSE DID”
MOS
true
Oswald killed Kennedy
K
MSE
SE OS
true
Prior knowledge
OS
MSE MOS
K
SE
Realizing Oswald did not
kill Kennedy
K
MSE MOS
SE OS
true
false
P (SE)
MOS = true
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K
SE OS
MSE
true
true
After learning Oswald killed
Kennedy
MOSMOS = true
P (SE) = P (SE)
S2: “IF OSWALD HADN’T KILLED KENNEDY, SOMEONE ELSE WOULD HAVE?”
Prior knowledge
K
SE OS
MSE MOS
Oswald refraining from
killing
P (SE)
K
MOS
SE OS
trueMSE
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K
SE OS
true
true
MSE M
OS = true
S2: “IF OSWALD HADN’T KILLED KENNEDY, SOMEONE ELSE WOULD HAVE?”
Prior knowledge
K
SE OS
MSE MOS
Oswald refraining from
killing
After learning Oswald killed
Kennedy
P (SE)
K
MOS
SE OS
trueMSE
P (SE) = P (SE)
false
P (SE) = P (SE)
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BRAINLESS THIRD DISCOVERY:HIGH SCHOOL COUNTERFACTUALS
CAN BE USEFUL
• Solidify and unify (all?) other approaches to causation(e.g., PO, SEM, DA, Prob., SC)
• Demystify enigmatic notions and weed out myths and misconceptions(e.g., ignorability, exogeneity, exchangeability, confounding, mediation, attribution, regret)
• Algorithmitize causal inference tasks (e.g., covariate-selection, identification, c-equivalence, effect-restoration, experimental integration, sufficiency)
• Resolve lingering puzzles
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CONCLUSIONS
• If Oswald had not used counterfactuals, brainless would have.
• Much of modern thinking is owed to brainless robots.
• I compute, therefore I think.