Causal Inference and Graphical Models Peter Spirtes Carnegie Mellon University

Overview Manipulations Assuming no Hidden Common Causes From DAGs to Effects of Manipulation From Data to Sets of DAGs From Sets of Dags to Effects of Manipulation May be Hidden Common Causes From Data to Sets of DAGs From Sets of DAGs to Effects of Manipulations

If I were to force a group of people to smoke one pack a day, what what percentage would develop lung cancer? The Evidence

P(Lung cancer = yes) = 1/2

P(Lung Cancer = yes|Teeth white = yes) = 1/4 Conditioning on Teeth white = yes

Manipulating Teeth white = yes

P(Lung Cancer = yes ||White teeth = yes) = 1/2 Manipulating Teeth white = yes - After Waiting P(Lung Cancer = yes|White teeth = yes) = 1/4 

Smoking Decision Setting insurance rates for smokers - conditioning Suppose the Surgeon General is considering banning smoking? Will this decrease smoking? Will decreasing smoking decrease cancer? Will it have negative side-effects – e.g. more obesity? How is greater life expectancy valued against decrease in pleasure from smoking?

Manipulations and Distributions Since Smoking determines Teeth white, P(T,L,R,W) = P(S,L,R,W) But the manipulation of Teeth white leads to different results than the manipulation of Smoking Hence the distribution does not always uniquely determine the results of a manipulation

Causation We will infer average causal effects. We will not consider quantities such as probability of necessity, probability of sufficiency, or the counterfactual probability that I would get a headache conditional on taking an aspirin, given that I did not take an aspirin The causal relations are between properties of a unit at a time, not between events. Each unit is assumed to be causally isolated. The causal relations may be genuinely indeterministic, or only apparently indeterministic.

Causal DAGs Probabilistic Interpretation of DAGs A DAG represents a distribution P when each variable is independent of its non- descendants conditional on its parents in the DAG Causal Interpretation of DAGs There is a directed edge from A to B (relative to V) when A is a direct cause of B. An acyclic graph is not a representation of reversible or feedback processes

Conditioning Conditioning maps a probability distribution and an event into a new probability distribution: f(P(V),e)  P’(V), where P’(V=v) = P(V=v)/P(e)

Manipulating A manipulation maps a population joint probability distribution, a causal DAG, and a set of new probability distributions for a set of variables, into a new joint distribution Manipulating: for {X 1,…,X n }  V f: P(V), population distribution G, causal DAG {P’(X 1 |Non-Descendants(G,X 1 )),…, manipulated variables P’(X n |Non-Descendants(G,X n ))}  P’(V) manipulated distribution (assumption that manipulations are independent)

Manipulation Notation - Adapting Lauritzen The distribution of Lung Cancer given the manipulated distribution of Smoking P(Lung Cancer||P’(Smoking)) The distribution of Lung Cancer conditional on Radon given the manipulated distribution of Smoking P(Lung Cancer|Radon||P’(Smoking)) = P(Lung Cancer,Radon||P’(Smoking))/ P(Radon||P’(Smoking)) First manipulate, then condition

Ideal Manipulations No fat hand Effectiveness Whether or not any actual action is an ideal manipulation of a variable Z is not part of the theory - it is input to the theory. With respect to a system of variables containing murder rates, outlawing cocaine is not an ideal manipulation of cocaine usage It is not entirely effective - people still use cocaine It affects murder rates directly, not via its effect on cocaine usage, because of increased gang warfare

3 Representations of Manipulations Structural Equation Policy Variable Potential Outcomes

College Plans Sewell and Shah (1968) studied five variables from a sample of 10,318 Wisconsin high school seniors. SEX [male = 0, female = 1] SEX [male = 0, female = 1] IQ = Intelligence Quotient, [lowest = 0, highest = 3] IQ = Intelligence Quotient, [lowest = 0, highest = 3] CP = college plans[yes = 0, no = 1] CP = college plans[yes = 0, no = 1] PE = parental encouragement [low = 0, high = 1] PE = parental encouragement [low = 0, high = 1] SES = socioeconomic status[lowest = 0, highest = 3] SES = socioeconomic status[lowest = 0, highest = 3]

College Plans - A Hypothesis SES SEX PE CP SEX PE CP IQ IQ

Equational Representation x i = f(pa i (G),  i ) If the  i are causes of two or more variables, they must be included in the analysis There is a distribution over the  i The equations and the distribution over the  i determine a distribution over the x i When manipulating variable to a value, replace with x i = c

Policy Variable Representation P(PE,SES,SEX,IQ,CP) Suppose P’(PE=1)=1 P(SES,SEX,IQ,CP,PE=1||P’(PE)) P(CP|PE||P’(PE)) P(PE,SES,SEX,IQ,CP|policy = off) P(PE=1|policy = on) = 1 P(SES,SEX,IQ,CP,PE=1|policy=on) P(CP|PE|policy = on) PE CP SEX SES IQ SES Pre-manipulation PE CP SEX SES IQ Post-manipulation

From DAG to Effects of Manipulation Effect of Manipulation Causal DAGs Background Knowledge Causal Axioms, Prior Population Distribution Sampling and Distributional Sample Assumptions, Prior

Causal Sufficiency A set of variables is causally sufficient if every cause of two variables in the set is also in the set. {PE,CP,SES} is causally sufficient {IQ,CP,SES} is not causally sufficient. PE CP SEX SES IQ SES

Causal Markov Assumption For a causally sufficient set of variables, the joint distribution is the product of each variable conditional on its parents in the causal DAG. P(SES,SEX,PE,CP,IQ) = P(SES)P(SEX)P(IQ|SES)P(PE|SES,SEX,IQ)P(CP|PE) PE CP SEX SES IQ SES

Equivalent Forms of Causal Markov Assumption In the population distribution, each variable is independent of its non-descendants in the causal DAG (non-effects) conditional on its parents (immediate causes). If X is d-separated from Y conditional on Z (written as ) in the causal graph, then X is independent of Y conditional on Z in the population distribution) denoted I(X,Y|Z)). PE CP SEX SES IQ SES

Causal Markov Assumption Causal Markov implies that if X is d-separated from Y conditional on Z in the causal DAG, then X is independent of Y conditional on Z. Causal Markov is equivalent to assuming that the causal DAG represents the population distribution. What would a failure of Causal Markov look like? If X and Y are dependent, but X does not cause Y, Y does not cause X, and no variable Z causes both X and Y.

Causal Markov Assumption Assumes that no unit in the population affects other units in the population If the “natural” units do affect each other, the units should be re-defined to be aggregations of units that don’t affect each other For example, individual people might be aggregated into families Assumes variables are not logically related, e.g. x and x 2 Assumes no feedback

Manipulation Theorem - No Hidden Variables P(PE,SES,SEX,CP,IQ||P’(PE)) = P(PE)P(SEX)P(CP|PE,SES,IQ)P(IQ|SES)P(PE|policy =on) = P(PE)P(SEX)P(CP|PE,SES,IQ)P(IQ|SES)P’(PE) PE CP SEX SES IQ SES Policy

Invariance Note that P(CP|PE,SES,IQ,policy = on) = P(CP|PE,SES,IQ,policy = off) because the policy variable is d-separated from CP conditional on PE,SES,IQ We say that P(CP|PE,SES,IQ) is invariant An invariant quantity can be estimated from the pre- manipulation distribution This is equivalent to one of the rules of the Do Calculus and can also be applied to latent variable models IQ PE CP SEX SES Policy

Calculating Effects IQ PE CP SEX SES Policy

From Sample to Sets of DAGs Effect of Manipulation Causal DAGs Background Knowledge Causal Axioms, Prior Population Distribution Sampling and Distributional Sample Assumptions, Prior

From Sample to Population to DAGs Constraint - Based Uses tests of conditional independence Goal: Find set of DAGs whose d- separation relations match most closely the results of conditional independenc tests Score - Based Uses scores such as Bayesian Information Criterion or Bayesian posterior Goal: Maximize score

Two Kinds Of Search ConstraintScore Use non conditional independence information NoYes Quantitative comparison of models NoYes Single test result leads astray YesNo Easy to apply to latent YesNo

Bayesian Information Criterion D is the sample data G is a DAG is the vector of maximum likelihood estimates of the parameters for DAG G N is the sample size d is the dimensionality of the model, which in DAGs without latent variables is simply the number of free parameters in the model

3 Kinds of Alternative Causal Models PE CP SEX SES IQ SES PE CP SEX SES IQ SES PE CP SEX SES IQ SES PE CP SEX SES IQ SES True ModelAlternative 1 Alternative 3Alternative 2

Alternative Causal Models PE CP SEX SES IQ SES PE CP SEX SES IQ SES True ModelAlternative 1 Constraint - Based: Alternative 1 violates Causal Markov Assumption by entailing that SES and IQ are independent Score - Based: Use a score that prefers a model that contains the true distribution over one that does not.

Alternative Causal Models PE CP SEX SES IQ SES True ModelAlternative 2 Constraint - Based: Assume that if Sex and CP are independent (conditional on some subset of variables such as PE, SES, and IQ) then Sex and CP are adjacent - Causal Adjacency Faithfulness Assumption. Score - Based: Use a score such that if two models contain the true distribution, choose the one with fewer parameters. The True Model has fewer parameters. PE CP SEX SES IQ SES

Both Assumptions Can Be False Alternative 2 True Model Independence holds for all values of parameters Independence holds only for parameters on lower dimensional surface - Lebesgue measure 0

When Not to Assume Faithfulness Deterministic relationships between variables entail “extra” conditional independence relations, in addition to those entailed by the global directed Markov condition. If A  B  C, and B = A, and C = B, then not only I(A,C|B), which is entailed by the global directed Markov condition, but also I(B,C|A), which is not. The deterministic relations are theoretically detectible, and when present, faithfulness should not be assumed. Do not assume in feedback systems in equilibrium.

Alternative Causal Models PE CP SEX SES IQ SES True ModelAlternative 3 PE CP SEX SES IQ SES Constraint - Based: Alternative 2 entails the same set of conditional independence relations - there is no principled way to choose.

Alternative Causal Models PE CP SEX SES IQ SES True ModelAlternative 2 PE CP SEX SES IQ SES Score - Based: Whether or not one can choose depends upon the parametric family. For unrestricted discrete, or linear Gaussian, there is no way to choose - the BIC scores will be the same. For linear non-Gaussian, the True Model will be preferred (because while the two models entail the same second order moments, they entail different fourth order moments.)

Patterns A pattern (or p-dag) represents a set of DAGs that all have the same d-separation relations, i.e. a d- separation equivalence class of DAGs. The adjacencies in a pattern are the same as the adjacencies in each DAG in the d-separation equivalence class. An edge is oriented as A  B in the pattern if it is oriented as A  B in every DAG in the equivalence class. An edge is oriented as A  B in the pattern if the edge is oriented as A  B in some DAGs in the equivalence class, and as A  B in other DAGs in the equivalence class.

Patterns to Graphs All of the DAGs in a d-separation equivalence class can be derived from the pattern that represents the d-separation equivalence class by orienting the unoriented edges in the pattern. Every orientation of the unoriented edges is acceptable as long as it creates no new unshielded colliders. That is A  B  C can be oriented as A  B  C, A  B  C, or A  B  C, but not as A  B  C.

Patterns PE CP SEX SES IQ SES PE CP SEX SES IQ SES PE CP SEX SES IQ SES D-separation Equivalence Class Pattern

Search Methods Constraint Based: PC (correct in limit) Variants of PC (correct in limit, better on small sample sizes) Score - Based: Greedy hill climbing Simulated annealing Genetic algorithms Greedy Equivalence Search (correct in limit)

From Sets of DAGs to Effects of Manipulation Effect of Manipulation Causal DAGs Background Knowledge Causal Axioms, Prior Population Distribution Sampling and Distributional Sample Assumptions, Prior

Causal Inference in Patterns Is P(IQ) invariant when SES is manipulated to a constant? Can’t tell. If SES  IQ, then policy is d-connected to IQ given empty set - no invariance. If SES  IQ, then policy is not d-connected to IQ given empty set - invariance. PE CP SEX SES IQ SES policy ?

Causal Inference in Patterns Different DAGs represented by pattern give different answers as to the effect of manipulating SES on IQ - not identifiable. In these cases, should ouput “can’t tell”. Note the difference from using Bayesian networks for classification - we can use either DAG equally well for correct classification, but we have to know which one is true for correct inference about the effect of a manipulation. PE CP SEX SES IQ SES policy ?

Causal Inference in Patterns Is P(CP|PE,SES,IQ) invariant when PE is manipulated to a constant? Can tell. policy variable is d-separated from CP given PE, SES, IQ regardless of which way the edge points - invariance in every DAG represented by the pattern. PE CP SEX SES IQ SES policy ?

College Plans SES SEX PE CP SEX PE CP IQ IQ not invariant, but is identifiable invariant

Good News Effect of Manipulation Causal DAGs Background Knowledge Causal Axioms, Prior Population Distribution Sampling and Distributional Sample Assumptions, Prior In the large sample limit, there are algorithms (PC, Greedy Equivalence Search) that are arbitrarily close to correct (or output “can’t tell”) with probability 1 (pointwise consistency).

Bad News Effect of Manipulation Causal DAGs Background Knowledge Causal Axioms, Prior Population Distribution Sampling and Distributional Sample Assumptions, Prior At every finite sample size, every method will be far from truth with high probability for some values of the truth (no uniform consistency.) (Typically not true of classification problems.)

Why Bad News? Effect of Manipulation Causal DAGs Background Knowledge Causal Axioms, Prior Population Distribution Sampling and Distributional Sample Assumptions, Prior The problem - small differences in population distribution can lead to big changes in inference to causal DAGs.

Strengthening Faithfulness Assumption Strong versus weak Weak adjacency faithfulness assumes a zero conditional dependence between X and Y entails a zero-strength edge between X and Y Strong adjacency faithfulness assumes in addition that a weak conditional dependence between X and Y entails a weak-strength edge between X and Y Under this assumption, there are uniform consistent estimators of the effects of manipulations.

Obstacles to Causal Inference from Non-experimental Data unmeasured confounders measurement error, or discretization of data mixtures of different causal structures in the sample feedback reversibility the existence of a number of models that fit the data equally well an enormous search space low power of tests of independence conditional on large sets of variables selection bias missing values sampling error complicated and dense causal relations among sets of variables, complcated probability distributions

From Data to Sets of DAGs - Possible Hidden Variables Effect of Manipulation Causal DAGs Background Knowledge Causal Axioms, Prior Population Distribution Sampling and Distributional Sample Assumptions, Prior

Why Latent Variable Models? For classification problems, introducing latent variables can help get closer to the right answer at smaller sample sizes - but they are needed to get the right answer in the limit. For causal inference problems, introducing latent variables are needed to get the right answer in the limit.

Score-Based Search Over Latent Models Structural EM interleaves estimation of parameters with structural search Can also search over latent variable models by calculating posteriors But there are substantial computational and statistical problems with latent variable models

DAG Models with Latent Variables Facilitates construction of causal models Provides a finite search space ‘Nice’ statistical properties: Always identified Correspond to a set of distributions characterized by independence relations Have a well-defined dimension Asymptotic existence of ML estimates

Solution Embed each latent variable model in a ‘larger’ model without latent variables that is easier to characterize. Disadvantage - uses only conditional independence information in the distribution. Latent variable model Model imposing only independence constraints on observed variables Sets of distributions

Alternative Hypothesis and Some D-separations SES SEX PE CP SEX PE CP L 1 L 2 L 1 L 2 IQ IQ <SEX,CP|{PE,SES}) These entail conditional independence relations in population.

D-separations Among Observed SES SEX PE CP SEX PE CP L 1 L 2 L 1 L 2 IQ IQ <SEX,CP|{PE,SES})

D-separations Among Observed It can be shown that no DAG with just the measured variables has exactly the set of d-separation relations among the observed variables. In this sense, DAGs are not closed under marginalization. SES SEX PE CP SEX PE CP L 1 L 2 L 1 L 2 IQ IQ

Mixed Ancestral Graphs Under a natural extension of the concept of d- separation to graphs with , MAG(G) is a graphical object that contains only the observed variables, and has exactly the d-separations among the observed variables. SES SEX PE CP SEX PE CP IQ IQ SES SEX PE CP SEX PE CP IQ IQ L1L1L1L1 L2L2L2L2 Latent Variable DAG Corresponding MAG

There is an edge between A and B if and only if for every, there is a latent variable in C. If A and B are adjacent, then A  B if and only if A is an ancestor of B. If A and B are adjacent, then A  B if and only if A is not an ancestor of B and B is not an ancestor of A. Mixed Ancestral Graph Construction

Suppose SES Unmeasured SEX PE CP SEX PE CP IQ IQ SES SEX PE CP SEX PE CP IQ IQ L1L1L1L1 L2L2L2L2 SEX PE CP SEX PE CP IQ IQ L1L1L1L1 L2L2L2L2 DAG Corresponding MAG Another DAG with the same MAG

Mixed Ancestral Models Can score and evaluate in the usual ways Not every parameter is directly interpreted as a structural (causal) coefficient Not every part of marginal manipulated model can be predicted from mixed ancestral graph Because multiple DAGs can have the same MAG, they might not all agree on the effect of a manipulation. It is possible to tell from the MAG when all of the DAGs with that MAG all agree on the effect of a manipulation.

Mixed Ancestral Graph Mixed ancestral models are closed under marginalization. In the linear normal case, the parameterization of a MAG is just a special case of the parameterization of a linear structural equation model. There is a maximum liklihood estimator of the parameters (Drton). The BIC score is easy to calculate. In the discrete case, it is not known how to parameterize a MAG - some progress has been made.

Some Markov Equivalent Mixed Ancestral Graphs SEX PE CP IQ IQ These different MAGs all have the same d-separation relations. SEX PE CP IQ IQ SEX PE CP IQ IQ SEX PE CP IQ IQ

Partial Ancestral Graphs SEX PE CP IQ IQ Partial Ancestral Graph SEX PE CP IQ IQ SEX PE CP IQ IQ SEX PE CP IQ IQ SEX PE CP SEX PE CP IQ IQ o o o o o o

Partial Ancestral Graph represents MAG M A is adjacent to B iff A and B are adjacent in M. A  B iff A is an ancestor of B in every MAG d-separation equivalent to M. A  B iff A and B are not ancestors of each other in every MAG d-separation equivalent to M. A o  B iff B is not an ancestor of A in every MAG d-separation equivalent to M, and A is an ancestor of B in some MAGs d- separation equivalent to M, but not in others. A o  o B iff A is an ancestor of B in some MAGs d-separation equivalent to M, but not in others, and B is an ancestor of A in some MAGs d-separation equivalent to M, but not in others.

Partial Ancestral Graph represents ancestor features common to MAGs that are d-separation equivalent d-separation relations in the d-separation equivalence class of MAGs. Can be parameterized by turning it into a mixed ancestral graph Can be scored and evaluated like MAG

FCI Algorithm In the large sample limit, with probability 1, the output is a PAG that represents the true graph over O If the algorithm needs to test high order conditional independence relations then Time consuming - worst case number of conditional independence tests (complete PAG) Unreliable (low power of tests) Modified versions can halt at any given order of conditional independence test, at the cost of more “Can’t tell” answers. Not useful information when each pair of variables have common hidden cause. There is a provably correct score-based search, but it outputs “can’t tell” in most cases

Output for College Plans SES SEX PE CP SEX PE CP IQ IQ o o SES SEX PE CP SEX PE CP IQ IQ o o o o Output of FCI Algorithm PAG Corresponding to Output of PC Algorithm These are different because no DAG can represent the d- separations in the output of the FCI algorithm.

From Sets of DAGs to Effects of Manipultions - May Be Hidden Common Causes Effect of Manipulation Causal DAGs Background Knowledge Causal Axioms, Prior Population Distribution Sampling and Distributional Sample Assumptions, Prior

Manipulation Model for PAGs A PAG can be used to calculate the results of manipulations for which every DAG represented by the PAG gives the same answer. It is possible to tell from the PAG that the policy variable for PE is d-separated from CP given PE. Hence P(CP|PE) is invariant. SES SEX PE CP SEX PE CP IQ IQ o o

Comparison with non-latent case FCI P(cp|pe||P’(PE)) = P(cp|pe). P(CP=0|PE=0||P’(PE)) =.063 P(CP=1|PE=0||P’(PE)) =.937 P(CP=0|PE=1||P’(PE)) =.572 P(CP=1PE=1||P’(PE)) =.428 PC P(CP=0|PE=0||P’(PE)) =.095 P(CP=1|PE=0||P’(PE)) =.905 P(CP=0|PE=1||P’(PE)) =.484 P(CP=1PE=1||P’(PE)) =.516

Good News Effect of Manipulation Causal DAGs Background Knowledge Causal Axioms, Prior Population Distribution Sampling and Distributional Sample Assumptions, Prior In the large sample limit, there is an algorithm (FCI) whose output is arbitrarily close to correct (or output “can’t tell”) with probability 1 (pointwise consistency).

Bad News Effect of Manipulation Causal DAGs Background Knowledge Causal Axioms, Prior Population Distribution Sampling and Distributional Sample Assumptions, Prior At every finite sample size, every method will be arbitrarily far from truth with high probability for some values of the truth (no uniform consistency.)

Other Constraints The disadvantage of using MAGs or FCI is they only use conditional independence information In the case of latent variable models, there are constraints implied on the observed margin that are not conditional independence relations, regardless of the family of distributions These can be used to choose between two different latent variable models that have the same d-separation relations over the observed variables In addition, there are constraints implied on the observed margin that are particular to a family of distributions

Examples of Open Questions Complete non-parametric manipulation calculations for partially known DAGs with latent variables Define strong faithfulness for the latent case. Calculating constraints (non-parametric or parametric) from latent variable DAGs Using constraints (non-parametric or parametric) to guide search for latent variable DAGs Latent variable score-based search over PAGs Parameterizations of MAGs for other families of distsributions Completeness of do-calculus for PAGs Time series inference

Introductory Books on Graphical Causal Inference Causation, Prediction, and Search, by P. Spirtes, C. Glymour, R. Scheines, MIT Press, Causality: Models, Reasoning, and Inference by J. Pearl, Cambridge University Press, Computation, Causation, and Discovery (Paperback), ed. by C. Glymour and G. Cooper, AAAI Press, 1999.