Download Causal Inference and Graphical Models

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
Conditioning on Teeth white = yes
P(Lung Cancer = yes|Teeth white = yes) = 1/4
Manipulating Teeth white = yes
Manipulating Teeth white = yes - After Waiting
P(Lung Cancer = yes ||White teeth = yes) = 1/2

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 nondescendants 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 {X1,…,Xn}  V
f: P(V),
G,
{P’(X1|Non-Descendants(G,X1)),…,
P’(Xn|Non-Descendants(G,Xn))}




P’(V)
population distribution
causal DAG
manipulated variables
manipulated distribution
P '(X)   P '(Xi | Non - Descendants(G, Xi ))
i
(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
IQ = Intelligence Quotient,
CP = college plans
PE = parental encouragement
SES = socioeconomic status
[male = 0, female = 1]
[lowest = 0, highest = 3]
[yes = 0, no = 1]
[low = 0, high = 1]
[lowest = 0, highest = 3]
College Plans - A Hypothesis
SES
SEX
IQ
PE
CP
Equational Representation





xi = f(pai(G), ei)
If the ei are causes of two or more variables,
they must be included in the analysis
There is a distribution over the ei
The equations and the distribution over the ei
determine a distribution over the xi
When manipulating variable to a value,
replace with xi = 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))
SES
SEX




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)
SES
PE
IQ
Pre-manipulation
CP
SEX
PE
IQ
Post-manipulation
CP
From DAG to Effects of
Manipulation
Effect of Manipulation
Causal DAGs
Background Knowledge
Causal Axioms, Prior
Population Distribution
Sample
Sampling and
Distributional
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.
SES
SEX
IQ
PE
CP
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)
SES
SEX
IQ
PE
CP
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 <X,Y|Z>) in
the causal graph, then X is independent of Y conditional on Z in the
population distribution) denoted I(X,Y|Z)).
SES
SEX
IQ
PE
CP
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 x2
 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)
SES
SEX
IQ
Policy
PE
CP
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 premanipulation distribution
This is equivalent to one of the rules of the Do
Calculus and can also be applied to latent variable
models
SES
Policy
SEX
IQ
PE
CP
SES
Calculating Effects
P(cp || P '(PE)) 
 P(cp | pe || P '( pe))P( pe || P '(PE)) 
SEX
Policy
PE
CP
IQ
(chain rule)
PE
 P(cp | pe || P '( pe))P '( pe) 
(definition of P '(PE))
PE


P(cp
|
pe,
ses,iq
||
P
'(PE))

P(iq
|
pe,
ses
||
P
'(PE))

P(ses
|
pe
||
P
'(PE))
  
 P '( pe) 

PE IQ,SES
(chain rule)


P(cp
|
pe,
ses,iq
||
P
'(PE))

P(iq
|
ses
||
P
'(PE))

P(ses
||
P
'(PE))
  
 P '( pe)

PE IQ,SES
(d-separation in manipulated DAG)


   P(cp | pe, ses,iq)  P(iq | ses)  P(ses) P '( pe)
PE IQ,SES
(invariance)
From Sample to Sets of DAGs
Effect of Manipulation
Causal DAGs
Background Knowledge
Causal Axioms, Prior
Population Distribution
Sample
Sampling and
Distributional
Assumptions, Prior
From Sample to Population to
DAGs

Constraint - Based

Uses tests of
conditional
independence
 Goal: Find set of
DAGs whose dseparation 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
Constraint Score
Use non conditional
No
independence information
Yes
Quantitative comparison
of models
No
Yes
Single test result leads
astray
Yes
No
Easy to apply to latent
Yes
No

Bayesian Information Criterion
ˆ ,G)  (d / 2) log N
log P(D | 
G

D is the sample data
 G is a DAG
ˆ is the vector of maximum likelihood estimates of
 
G
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
SES
SES
SEX
PE
CP
True Model
CP
Alternative 1
SES
SES
IQ
PE
IQ
IQ
SEX
SEX
PE
Alternative 3
CP
SEX
PE
IQ
Alternative 2
CP
Alternative Causal Models
SES
SES
SEX
PE
IQ
True Model


CP
SEX
PE
CP
IQ
Alternative 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
SES
SES
SEX
PE
IQ
True Model

CP
SEX
PE
CP
IQ
Alternative 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.
Both Assumptions Can Be False
Independence holds
only for parameters
on lower dimensional
surface - Lebesgue
measure 0
Independence holds
for all values of
parameters
Alternative 2
True Model
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
SES
SES
SEX
PE
IQ
True Model

CP
SEX
PE
CP
IQ
Alternative 3
Constraint - Based: Alternative 2 entails the same set of
conditional independence relations - there is no principled way
to choose.
Alternative Causal Models
SES
SES
SEX
PE
IQ
True Model

CP
SEX
PE
CP
IQ
Alternative 2
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 dseparation 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
SES
SES
SEX
PE
SEX
CP
PE
IQ
IQ
D-separation Equivalence Class
SES
SEX
PE
IQ
Pattern
CP
CP
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
limit)
(correct in
From Sets of DAGs to Effects of
Manipulation
Effect of Manipulation
Causal DAGs
Background Knowledge
Causal Axioms, Prior
Population Distribution
Sample
Sampling and
Distributional
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.
SES
?
policy
SEX
IQ
PE
CP
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.
SES
?
policy
SEX
IQ
PE
CP
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.
SES
?
SEX
policy
IQ
PE
CP
College Plans
not invariant, but is
identifiable
SES
SEX
PE
CP
IQ
P(cp | pe || P '(PE)) 

P(cp | pe, ses,iq || P '(PE))  P(iq | ses, pe || P '(PE))  P(ses | pe || P '(PE)) 
IQ,SES

P(cp | pe, ses,iq || P '(PE))  P(iq | ses || P '(PE))  P(ses || P '(PE)) 
IQ,SES

P(cp | pe, ses,iq)  P(iq | ses)  P(ses)
IQ,SES
invariant
Good News
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).
Effect of Manipulation
Causal DAGs
Background Knowledge
Causal Axioms, Prior
Population Distribution
Sample
Sampling and
Distributional
Assumptions, Prior
Bad News
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.)
Effect of Manipulation
Causal DAGs
Background Knowledge
Causal Axioms, Prior
Population Distribution
Sample
Sampling and
Distributional
Assumptions, Prior
Why Bad News?
The problem - small differences in population
distribution can lead to big changes in inference to
causal DAGs.
Effect of Manipulation
Causal DAGs
Background Knowledge
Causal Axioms, Prior
Population Distribution
Sample
Sampling and
Distributional
Assumptions, Prior
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
Sample
Sampling and
Distributional
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.
Model imposing
only independence
constraints
on observed variables
Latent variable
model
Sets of distributions
Alternative Hypothesis and Some
D-separations
SES
SEX
PE
L1
CP
L2
IQ
<CP,{IQ,L1,SEX}|{L2,PE,SES}>
<L2,{SES,L1,SEX, PE}|>
<SEX,{L1,SES,L2,IQ}|>
<L1,{SES,L2,SEX}|>
<SEX,CP|{PE,SES})
These entail conditional
independence relations in
<IQ,{SEX,PE,CP}|{L1,L2,SES}> population.
<PE,{IQ,L2}|{L1,SEX,SES}>
<SES,{SEX,IQ,L1,L2}|>
D-separations Among Observed
SES
SEX
PE
L1
CP
L2
IQ
<CP,{IQ,L1,SEX}|{L2,PE,SES}>
<PE,{IQ,L2}|{L1,SEX,SES}>
<IQ,{SEX,PE,CP}|{L1,L2,SES}>
<SES,{SEX,IQ,L1,L2}|>
<L2,{SES,L1,SEX, PE}|>
<SEX,{L1,SES,L2,IQ}|>
<L1,{SES,L2,SEX}|>
<SEX,CP|{PE,SES})
D-separations Among Observed
SES
SEX
PE
L1
CP
L2
IQ
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.
Mixed Ancestral Graphs

Under a natural extension of the concept of dseparation 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
SES
SEX
PE
L1
CP
SEX
PE
L2
IQ
Latent Variable DAG
IQ
Corresponding MAG
CP
Mixed Ancestral Graph
Construction

There is an edge between A and B if and only
if for every <{A},{B}|C>, 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.
Suppose SES Unmeasured
SES
SEX
PE
L1
CP
PE
CP
L2
IQ
IQ
DAG
Corresponding MAG
SEX
PE
L1
IQ
SEX
CP
L2
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
SEX
IQ
SEX
IQ
PE
CP
IQ
PE
CP
SEX
PE
CP
IQ
These different MAGs all have the same d-separation
relations.
Partial Ancestral Graphs
SEX
PE
CP
SEX
IQ
SEX
IQ
PE
CP
o
IQ
PE
CP
SEX
IQ
SEX o
PE
CP
PE o
o CP
o
IQ o
Partial Ancestral Graph
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 dseparation 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

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)
  n n 2 
O  2 
  2


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
o SES o
o SES
SEX
PE
oIQ
Output of FCI Algorithm
CP
SEX
PE
CP
oIQo
PAG Corresponding to
Output of PC Algorithm
These are different because no DAG can represent the dseparations 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
Sample
Sampling and
Distributional
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.
o
SES
SEX
oIQ
PE
CP
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 | pe || P'(PE)) 
 P(cp | pe,ses,iq) P(iq | ses) P(ses)
IQ, SES





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
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).
Effect of Manipulation
Causal DAGs
Background Knowledge
Causal Axioms, Prior
Population Distribution
Sample
Sampling and
Distributional
Assumptions, Prior
Bad News
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.)
Effect of Manipulation
Causal DAGs
Background Knowledge
Causal Axioms, Prior
Population Distribution
Sample
Sampling and
Distributional
Assumptions, Prior
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, 2000.
 Causality: Models, Reasoning, and
Inference by J. Pearl, Cambridge
University Press, 2000.
 Computation, Causation, and Discovery
(Paperback) , ed. by C. Glymour and G.
Cooper, AAAI Press, 1999.
