causes and counterfactuals or the subtle wisdom of brainless robots

CAUSES ANDCOUNTERFACTUALS

OR

THE SUBTLE WISDOMOF BRAINLESS ROBOTS

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

David Hume (1711–1776)

HUME’S LEGACYHUME’S LEGACY

1. Analytical vs. empirical claims

2. Causal claims are empirical

3. All empirical claims originate from experience.

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?

““Easy, man! that hurts!”Easy, man! that hurts!”

The Art ofCausal Mentoring

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

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

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.

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

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.,

Z

YX

INPUT OUTPUT

FAMILIAR CAUSAL MODELORACLE FOR COUNTERFACTUALS

)()( 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.

),|(),|'(

)()()|(

')(':'

)(:

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:

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!”•

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)

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)

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

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

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

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)

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

CONCLUSIONS

• If Oswald had not used counterfactuals, brainless would have.

• Much of modern thinking is owed to brainless robots.

• I compute, therefore I think.