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CAUSAL MODELING AND THE LOGIC OF SCIENCE. Judea Pearl Computer Science and Statistics UCLA www.cs.ucla.edu/~judea/. OVERVIEW Scope and Language in Scientific Theories. Statistical models ( observtions , PL ) Causal models 2.1 Stochastic causal model

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CAUSAL MODELING

AND THE

LOGIC OF SCIENCE

Judea Pearl

Computer Science and Statistics

UCLA

www.cs.ucla.edu/~judea/


OVERVIEW

Scope and Language in Scientific Theories

  • Statistical models

    • (observtions, PL)

  • Causal models

    • 2.1 Stochastic causal model

    • (interventions, PL + modality)

    • 2.2 Functional causal models

    • (counterfactuals, PL + subjunctives)

  • General equational models

    • (explicit interventions, PL)

    • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •

  • General Scientific theories

    • (objects-properties, FOL-SOL ...)


  • OUTLINE

    • Modeling: Statistical vs. Causal

    • Causal models and identifiability

    • Inference to three types of claims:

    • Effects of potential interventions,

    • Claims about attribution (responsibility)

    • Claims about direct and indirect effects

    • Falsifiability and Corroboration


    TRADITIONAL STATISTICAL

    INFERENCE PARADIGM

    P

    Joint

    Distribution

    Q(P)

    (Aspects of P)

    Data

    Inference

    e.g.,

    Infer whether customers who bought product A

    would also buy product B.

    Q = P(B|A)


    THE CAUSAL INFERENCE

    PARADIGM

    M

    Data-generating

    Model

    Q(M)

    (Aspects of M)

    Data

    Inference

    Some Q(M) cannot be inferred from P.

    e.g.,

    Infer whether customers who bought product A

    would still buy A if we double the price.


    Probability and statistics deal with static relations

    Statistics

    Probability

    inferences

    from passive

    observations

    joint

    distribution

    Data

    FROM STATISTICAL TO CAUSAL ANALYSIS:

    1. THE DIFFERENCES


    FROM STATISTICAL TO CAUSAL ANALYSIS:

    1. THE DIFFERENCES

    Probability and statistics deal with static relations

    Statistics

    Probability

    inferences

    from passive

    observations

    joint

    distribution

    Data

    • Causal analysis deals with changes (dynamics)

    • i.e. What remains invariant when P changes.

    • P does not tell us how it ought to change

    • e.g. Curing symptoms vs. curing diseases

    • e.g. Analogy: mechanical deformation


    Probability and statistics deal with static relations

    Statistics

    Probability

    inferences

    from passive

    observations

    joint

    distribution

    Data

    Causal analysis deals with changes (dynamics)

    • Effects of

      • interventions

    Data

    Causal

    Model

    • Causes of

      • effects

    Causal

    assumptions

    • Explanations

    Experiments

    FROM STATISTICAL TO CAUSAL ANALYSIS:

    1. THE DIFFERENCES


    CAUSAL

    Spurious correlation

    Randomization

    Confounding / Effect

    Instrument

    Holding constant

    Explanatory variables

    STATISTICAL

    Regression

    Association / Independence

    “Controlling for” / Conditioning

    Odd and risk ratios

    Collapsibility

    FROM STATISTICAL TO CAUSAL ANALYSIS:

    1. THE DIFFERENCES (CONT)


    CAUSAL

    Spurious correlation

    Randomization

    Confounding / Effect

    Instrument

    Holding constant

    Explanatory variables

    STATISTICAL

    Regression

    Association / Independence

    “Controlling for” / Conditioning

    Odd and risk ratios

    Collapsibility

    • No causes in – no causes out (Cartwright, 1989)

    }

    statistical assumptions + data

    causal assumptions

    causal conclusions

    FROM STATISTICAL TO CAUSAL ANALYSIS:

    1. THE DIFFERENCES (CONT)

    • Causal assumptions cannot be expressed in the mathematical language of standard statistics.


    CAUSAL

    Spurious correlation

    Randomization

    Confounding / Effect

    Instrument

    Holding constant

    Explanatory variables

    STATISTICAL

    Regression

    Association / Independence

    “Controlling for” / Conditioning

    Odd and risk ratios

    Collapsibility

    • No causes in – no causes out (Cartwright, 1989)

    }

    statistical assumptions + data

    causal assumptions

    causal conclusions

    FROM STATISTICAL TO CAUSAL ANALYSIS:

    1. THE DIFFERENCES (CONT)

    • Causal assumptions cannot be expressed in the mathematical language of standard statistics.

    • Non-standard mathematics:

      • Structural equation models (SEM)

      • Counterfactuals (Neyman-Rubin)

      • Causal Diagrams (Wright, 1920)


    WHAT'SIN A CAUSAL MODEL?

    Oracle that assigns truth value to causal

    sentences:

    Action sentences:B if wedoA.

    Counterfactuals:B would be different if

    Awere true.

    Explanation:B occurredbecauseof A.

    Optional:with whatprobability?


    FAMILIAR CAUSAL MODEL

    ORACLE FOR MANIPILATION

    X

    Y

    Z

    INPUT

    OUTPUT


    Definition: A causal model is a 3-tuple

    M = V,U,F

    with a mutilation operator do(x): MMx where:

    (i) V = {V1…,Vn} endogenous variables,

    (ii) U = {U1,…,Um} background variables

    (iii) F = set of n functions, fi : V \ ViU Vi

    vi = fi(pai,ui)PAi V \ ViUi U

    CAUSAL MODELS AND

    CAUSAL DIAGRAMS


    Definition: A causal model is a 3-tuple

    M = V,U,F

    with a mutilation operator do(x): MMx where:

    (i) V = {V1…,Vn} endogenous variables,

    (ii) U = {U1,…,Um} background variables

    (iii) F = set of n functions, fi : V \ ViU Vi

    vi = fi(pai,ui)PAi V \ ViUi U

    I

    W

    Q

    P

    CAUSAL MODELS AND

    CAUSAL DIAGRAMS

    U1

    U2

    PAQ


    Definition: A causal model is a 3-tuple

    M = V,U,F

    with a mutilation operator do(x): MMx where:

    (i) V = {V1…,Vn} endogenous variables,

    (ii) U = {U1,…,Um} background variables

    (iii) F = set of n functions, fi : V \ ViU Vi

    vi = fi(pai,ui)PAi V \ ViUi U

    CAUSAL MODELS AND

    MUTILATION

    (iv) Mx= U,V,Fx, X  V, x  X

    where Fx = {fi: Vi X }  {X = x}

    (Replace all functions ficorresponding to X with the constant functions X=x)


    Definition: A causal model is a 3-tuple

    M = V,U,F

    with a mutilation operator do(x): MMx where:

    (i) V = {V1…,Vn} endogenous variables,

    (ii) U = {U1,…,Um} background variables

    (iii) F = set of n functions, fi : V \ ViU Vi

    vi = fi(pai,ui)PAi V \ ViUi U

    I

    W

    Q

    CAUSAL MODELS AND

    MUTILATION

    (iv)

    U1

    U2

    P


    Definition: A causal model is a 3-tuple

    M = V,U,F

    with a mutilation operator do(x): MMx where:

    (i) V = {V1…,Vn} endogenous variables,

    (ii) U = {U1,…,Um} background variables

    (iii) F = set of n functions, fi : V \ ViU Vi

    vi = fi(pai,ui)PAi V \ ViUi U

    I

    W

    Q

    CAUSAL MODELS AND

    MUTILATION

    (iv)

    Mp

    U1

    U2

    P

    P = p0


    Definition: A causal model is a 3-tuple

    M = V,U,F

    with a mutilation operator do(x): MMx where:

    (i) V = {V1…,Vn} endogenous variables,

    (ii) U = {U1,…,Um} background variables

    (iii) F = set of n functions, fi : V \ ViU Vi

    vi = fi(pai,ui)PAi V \ ViUi U

    PROBABILISTIC

    CAUSAL MODELS

    (iv) Mx= U,V,Fx, X  V, x  X

    where Fx = {fi: Vi X }  {X = x}

    (Replace all functions ficorresponding to X with the constant functions X=x)

    Definition (Probabilistic Causal Model):

    M, P(u)

    P(u) is a probability assignment to the variables in U.


    CAUSAL MODELS AND COUNTERFACTUALS

    Definition: Potential Response

    The sentence: “Y would be y (in unit u), had X been x,”

    denoted Yx(u) = y, is the solution for Y in a mutilated model

    Mx, with the equations for X replaced by X = x.

    (“unit-based potential outcome”)


    CAUSAL MODELS AND COUNTERFACTUALS

    Joint probabilities of counterfactuals:

    Definition: Potential Response

    The sentence: “Y would be y (in unit u), had X been x,”

    denoted Yx(u) = y, is the solution for Y in a mutilated model

    Mx, with the equations for X replaced by X = x.

    (“unit-based potential outcome”)


    CAUSAL MODELS AND COUNTERFACTUALS

    In particular:

    Definition: Potential Response

    The sentence: “Y would be y (in unit u), had X been x,”

    denoted Yx(u) = y, is the solution for Y in a mutilated model

    Mx, with the equations for X replaced by X = x.

    (“unit-based potential outcome”)

    Joint probabilities of counterfactuals:


    3-STEPS TO COMPUTING

    COUNTERFACTUALS

    U

    U

    TRUE

    TRUE

    C

    C

    FALSE

    FALSE

    A

    B

    A

    B

    D

    D

    TRUE

    TRUE

    S5. If the prisoner is dead, he would still be dead

    if A were not to have shot. DDA

    Abduction

    Action

    Prediction

    U

    TRUE

    C

    A

    B

    D


    COMPUTING PROBABILITIES

    OF COUNTERFACTUALS

    U

    U

    P(u|D)

    P(u)

    P(u|D)

    P(u|D)

    C

    C

    FALSE

    FALSE

    A

    B

    A

    B

    D

    D

    TRUE

    P(DA|D)

    P(S5). The prisoner is dead. How likely is it that he would be dead

    if A were not to have shot. P(DA|D) = ?

    Abduction

    Action

    Prediction

    U

    C

    A

    B

    D


    CAUSAL INFERENCE

    MADE EASY (1985-2000)

    • Inference with Nonparametric Structural Equations

      • made possible through Graphical Analysis.

    • Mathematical underpinning of counterfactuals

      • through nonparametric structural equations

    • Graphical-Counterfactuals symbiosis


    IDENTIFIABILITY

    Definition:

    Let Q(M) be any quantity defined on a causal

    model M, andlet A be a set of assumption.

    Q is identifiable relative to A iff

    P(M1) = P(M2) ÞQ(M1) = Q(M2)

    for all M1, M2, that satisfy A.


    IDENTIFIABILITY

    Definition:

    Let Q(M) be any quantity defined on a causal

    model M, andlet A be a set of assumption.

    Q is identifiable relative to A iff

    P(M1) = P(M2) ÞQ(M1) = Q(M2)

    for all M1, M2, that satisfy A.

    In other words, Q can be determined uniquely

    from the probability distribution P(v) of the

    endogenous variables, V, and assumptions A.


    IDENTIFIABILITY

    Definition:

    Let Q(M) be any quantity defined on a causal

    model M, andlet A be a set of assumption.

    Q is identifiable relative to A iff

    P(M1) = P(M2)ÞQ(M1) = Q(M2)

    for all M1, M2, that satisfy A.

    In this talk:

    A: Assumptions encoded in the diagram

    Q1: P(y|do(x)) Causal Effect (= P(Yx=y))

    Q2: P(Yx =y | x, y) Probability of necessity

    Q3: Direct Effect


    THE FUNDAMENTAL THEOREM

    OF CAUSAL INFERENCE

    Causal Markov Theorem:

    Any distribution generated by Markovian structural model M

    (recursive, with independent disturbances) can be factorized as

    Where pai are the (values of) the parents of Viin the causal

    diagram associated with M.


    Corollary: (Truncated factorization, Manipulation Theorem)

    The distribution generated by an intervention do(X=x)

    (in a Markovian model M) is given by the truncated factorization

    THE FUNDAMENTAL THEOREM

    OF CAUSAL INFERENCE

    Causal Markov Theorem:

    Any distribution generated by Markovian structural model M

    (recursive, with independent disturbances) can be factorized as

    Where pai are the (values of) the parents of Viin the causal

    diagram associated with M.


    Given P(x,y,z),should we ban smoking?

    U (unobserved)

    U (unobserved)

    X = x

    Y

    Z

    X

    Y

    Z

    Smoking

    Tar in

    Lungs

    Cancer

    Smoking

    Tar in

    Lungs

    Cancer

    RAMIFICATIONS OF THE FUNDAMENTAL THEOREM


    Given P(x,y,z),should we ban smoking?

    U (unobserved)

    U (unobserved)

    X = x

    Y

    Z

    X

    Y

    Z

    Smoking

    Tar in

    Lungs

    Cancer

    Smoking

    Tar in

    Lungs

    Cancer

    RAMIFICATIONS OF THE FUNDAMENTAL THEOREM

    Pre-intervention

    Post-intervention


    Given P(x,y,z),should we ban smoking?

    U (unobserved)

    U (unobserved)

    X = x

    Y

    Z

    X

    Y

    Z

    Smoking

    Tar in

    Lungs

    Cancer

    Smoking

    Tar in

    Lungs

    Cancer

    RAMIFICATIONS OF THE FUNDAMENTAL THEOREM

    Pre-intervention

    Post-intervention

    To compute P(y,z|do(x)), wemust eliminate u. (graphical problem).


    G

    Gx

    THE BACK-DOOR CRITERION

    Graphical test of identification

    P(y | do(x)) is identifiable in G if there is a set Z of

    variables such that Zd-separates X from Y in Gx.

    Z1

    Z1

    Z2

    Z2

    Z

    Z3

    Z3

    Z4

    Z5

    Z5

    Z4

    X

    X

    Z6

    Y

    Y

    Z6


    G

    Gx

    Moreover, P(y | do(x)) = åP(y | x,z) P(z)

    (“adjusting” for Z)

    z

    THE BACK-DOOR CRITERION

    Graphical test of identification

    P(y | do(x)) is identifiable in G if there is a set Z of

    variables such that Zd-separates X from Y in Gx.

    Z1

    Z1

    Z2

    Z2

    Z

    Z3

    Z3

    Z4

    Z5

    Z5

    Z4

    X

    X

    Z6

    Y

    Y

    Z6


    RULES OF CAUSAL CALCULUS

    • Rule 1:Ignoring observations

      • P(y |do{x},z, w) = P(y | do{x},w)

    • Rule 2:Action/observation exchange

      • P(y |do{x}, do{z}, w) = P(y|do{x},z,w)

    • Rule 3: Ignoring actions

      • P(y |do{x},do{z},w) = P(y|do{x},w)


    DERIVATION IN CAUSAL CALCULUS

    Genotype (Unobserved)

    Smoking

    Tar

    Cancer

    Probability Axioms

    P (c |do{s})=tP (c |do{s},t) P (t |do{s})

    Rule 2

    = tP (c |do{s},do{t})P (t |do{s})

    Rule 2

    = tP (c |do{s},do{t})P (t | s)

    Rule 3

    = tP (c |do{t})P (t | s)

    Probability Axioms

    = stP (c |do{t},s) P (s|do{t})P(t |s)

    Rule 2

    = stP (c | t, s) P (s|do{t})P(t |s)

    Rule 3

    = stP (c | t, s) P (s) P(t |s)


    OUTLINE

    • Modeling: Statistical vs. Causal

    • Causal models and identifiability

    • Inference to three types of claims:

    • Effects of potential interventions,

    • Claims about attribution (responsibility)


    DETERMINING THE CAUSES OF EFFECTS

    (The Attribution Problem)

    • Your Honor! My client (Mr. A) died BECAUSE

      • he used that drug.


    DETERMINING THE CAUSES OF EFFECTS

    (The Attribution Problem)

    • Your Honor! My client (Mr. A) died BECAUSE

      • he used that drug.

    • Court to decide if it is MORE PROBABLE THAN

      • NOT that A would be alive BUT FOR the drug!

      • P(? | A is dead, took the drug) > 0.50


    THE PROBLEM

    • Theoretical Problems:

    • What is the meaning of PN(x,y):

    • “Probability that event y would not have occurred if it were not for event x, given that x and y did in fact occur.”


    THE PROBLEM

    • Theoretical Problems:

    • What is the meaning of PN(x,y):

    • “Probability that event y would not have occurred if it were not for event x, given that x and y did in fact occur.”

    • Answer:


    THE PROBLEM

    • Theoretical Problems:

    • What is the meaning of PN(x,y):

    • “Probability that event y would not have occurred if it were not for event x, given that x and y did in fact occur.”

    • Under what condition can PN(x,y) be learned from statistical data, i.e., observational, experimental and combined.


    WHAT IS INFERABLE FROM EXPERIMENTS?

    Simple Experiment:

    Q = P(Yx= y | z)

    Z nondescendants of X.

    Compound Experiment:

    Q = P(YX(z) = y | z)

    Multi-Stage Experiment:

    etc…


    CAN FREQUENCY DATA DECIDE LEGAL RESPONSIBILITY?

    ExperimentalNonexperimental

    do(x) do(x) xx

    Deaths (y) 16 14 2 28

    Survivals (y) 984 986 998 972

    1,000 1,000 1,000 1,000

    • Nonexperimental data: drug usage predicts longer life

    • Experimental data: drug has negligible effect on survival

    • Plaintiff: Mr. A is special.

    • He actually died

    • He used the drug by choice

    • Court to decide (given both data):

      • Is it more probable than not that A would be alive

      • but for the drug?


    TYPICAL THEOREMS

    (Tian and Pearl, 2000)

    • Identifiability under monotonicity (Combined data)

    • corrected Excess-Risk-Ratio

    • Bounds given combined nonexperimental and experimental data


    SOLUTION TO THE ATTRIBUTION PROBLEM (Cont)

    • From population data to individual case

    • Combined data tell more that each study alone


    OUTLINE

    • Modeling: Statistical vs. Causal

    • Causal models and identifiability

    • Inference to three types of claims:

    • Effects of potential interventions,

    • Claims about attribution (responsibility)

    • Claims about direct and indirect effects


    QUESTIONS ADDRESSED

    • What is the semantics of direct and indirect effects?

    • Can we estimate them from data? Experimental data?


    TOTAL, DIRECT, AND INDIRECT EFFECTS HAVE SIMPLE SEMANTICS

    IN LINEAR MODELS

    b

    X

    Z

    z = bx + 1

    y = ax + cz + 2

    a

    c

    Y

    a+bc

    a

    bc


    SEMANTICS BECOMES NONTRIVIAL

    IN NONLINEAR MODELS

    (even when the model is completely specified)

    X

    Z

    z = f (x, 1)

    y = g (x, z, 2)

    Y

    Dependent on z?

    Void of operational meaning?


    THE OPERATIONAL MEANING OF

    DIRECT EFFECTS

    X

    Z

    z = f (x, 1)

    y = g (x, z, 2)

    Y

    “Natural” Direct Effect of X on Y:

    The expected change in Y per unit change of X, when we keep Z constant at whatever value it attains before the change.

    In linear models, NDE = Controlled Direct Effect


    GENDER

    QUALIFICATION

    HIRING

    POLICY IMPLICATIONS

    (Who cares?)

    indirect

    What is the direct effect of X on Y?

    The effect of Gender on Hiring if sex discrimination

    is eliminated.

    X

    Z

    IGNORE

    f

    Y


    THE OPERATIONAL MEANING OF

    INDIRECT EFFECTS

    X

    Z

    z = f (x, 1)

    y = g (x, z, 2)

    Y

    “Natural” Indirect Effect of X on Y:

    The expected change in Y when we keep X constant, say at x0, and let Z change to whatever value it would have under a unit change in X.

    In linear models, NIE = TE - DE


    LEGAL DEFINITIONS TAKE THE NATURAL CONCEPTION

    (FORMALIZING DISCRIMINATION)

    ``The central question in any employment-discrimination case is whether the employer would have taken the same action had the employee been of different race (age, sex, religion, national origin etc.) and everything else had been the same’’

    [Carson versus Bethlehem Steel Corp. (70 FEP Cases 921, 7th Cir. (1996))]

    x = male, x = female

    y = hire, y = not hire

    z = applicant’s qualifications

    NO DIRECT EFFECT

    YxZx= Yx, YxZx = Yx


    SEMANTICS AND IDENTIFICATION OF NESTED COUNTERFACTUALS

    Consider the quantity

    Given M, P(u), Q is well defined

    Given u, Zx*(u) is the solution for Z in Mx*,call it z

    is the solution for Y in Mxz

    Can Q be estimated from data?


    ANSWERS TO QUESTIONS

    • Graphical conditions for estimability from

      • experimental / nonexperimental data.

    • Graphical conditions hold in Markovian models


    ANSWERS TO QUESTIONS

    • Graphical conditions for estimability from

      • experimental / nonexperimental data.

    • Graphical conditions hold in Markovian models

    • Useful in answering new type of policy questions

      • involving mechanism blocking instead of variable fixing.


    THE OVERRIDING THEME

    • Define Q(M) as a counterfactual expression

    • Determine conditions for the reduction

    • If reduction is feasible, Q is inferable.

  • Demonstrated on three types of queries:

  • Q1: P(y|do(x)) Causal Effect (= P(Yx=y))

    Q2: P(Yx = y | x, y) Probability of necessity

    Q3: Direct Effect


    w

    x

    y

    z

    FALSIFIABILITY and CORROBORATION

    P*

    P*(M)

    Falsifiability: P*(M) P*

    D (Data)

    Constraints implied by M

    Data Dcorroborates model M if M is (i) falsifiable

    and (ii) compatible with D.

    Types of constraints:1. conditional independencies2. inequalities (for restricted domains)3. functional

    e.g.,


    OTHER TESTABLE CLAIMS

    Changes under interventions

    For all causal models:

    For all semi-Markovian models:

    For Markovian models (and ):

    For a given Markovian model:


    FROM CORROBORATING MODELS

    TO CORROBORATING CLAIMS

    A corroborated model can imply identifiable yet

    uncorroborated claims.

    e.g.,

    x

    x

    y

    y

    z

    z

    x

    y

    z

    a

    a

    b

    Some claims can be more corroborated than others.

    Definition:

    An identifiable claim C is corroborated by data if some minimal set of assumptions in M sufficient for identifying C is corroborated by the data.

    Graphical criterion: minimal submodel = maximal supergraph


    FROM CORROBORATING MODELS

    TO CORROBORATING CLAIMS

    A corroborated model can imply identifiable yet

    uncorroborated claims.

    e.g.,

    x

    x

    y

    y

    z

    z

    x

    y

    z

    a

    a

    b

    Some claims can be more corroborated than others.

    Definition:

    An identifiable claim C is corroborated by data if some minimal set of assumptions in M sufficient for identifying C is corroborated by the data.

    Graphical criterion: minimal submodel = maximal supergraph


    OVERVIEW

    Scope and Language in Scientific Theories

    • Statistical models

      • (observtions, PL)

  • Causal models

    • 2.1 Stochastic causal model

    • (interventions, PL + modality)

    • 2.2 Functional causal models

    • (counterfactuals, PL + subjunctives)

  • General equational models

    • (explicit interventions, PL)

    • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • •

  • General Scientific theories

    • (objects-properties, FOL-SOL ...)


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