Markov Properties of Directed
Acyclic Graphs
Peter Spirtes
Outline
Bayesian Networks
Simplicity and Bayesian Networks
Causal Interpretation of Bayesian Networks
Greedy Equivalence Search
2
Outline
Bayesian Networks
Simplicity and Bayesian Networks
Causal Interpretation of Bayesian Networks
Greedy Equivalence Search
3
Directed Acyclic Graph (DAG)
ng
bl
bd
ng – no gas
bl – battery light on
bd – battery dead
ns – no start
ns
The vertices are random variables.
All edges are directed.
There are no directed cycles.
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Definition of Conditional
Independence
Let P(X,Y,Z) be a joint distribution over X,
Y, Z.
X is independent of Y conditional on Z in P,
written as IP(X,Y|Z), if and only if P(X|Y,Z)
= P(X|Z) whenever P(y,z) > 0.
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Probabilistic Interpretation Local Markov Condition
ng
bl
bd
ns
Distribution P satisfies the
Local Markov Condition
for DAG G iff each vertex
is independent in P of all
vertices that are neither
parents nor descendants,
conditional on its parents.
IP(ng,{bd,bl}| ) IP(bl,{ng,ns}|bd)
IP(bd,ng| )
IP(ns,bl|{bd,ng}
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Probabilistic Interpretation: I-Map
ng
bl
bd
If distribution P satisfies
the Local Markov
Condition for G, G is an Imap (Independence-map)
of P.
ns
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Graphical Entailment
ng
bl
bd
ns
If G being an I-map of P entails
a conditional independence
relation I holds in P, G entails I.
Examples: In every distribution
P that G is an I-map of
IP(bd,ng| )
IP(bl,ng|{bd,ns})
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If I is Not Entailed by G
ng
bl
bd
ns
If conditional independence
relation I is not entailed by G,
then I may hold in some (but
not every) distribution P that G
is an I-map of.
Example: G is an I-map of
some P such that ~IP(ns,bl| )
and some P’ such that
IP’(ns,bl| )
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Factorization
ng
bl
bd
If G is an I-map of P, then
P(ng,bd,bl,ns) =
P(ng)P(bd)P(bl|bd)P(ns|bd,ng)
ns
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Example of Parametric Family
ng
bl
bd
ns
For binary variables,
the dimension of the
set of distributions that
G is an I-map of, is 8.
1.
2.
3.
4.
5.
6.
7.
8.
P(ng = 0)
P(bd = 0)
P(bl = 0|bd = 0)
P(bl = 0|bd = 1)
P(ns = 0|ng = 0, bd = 0)
P(ns = 0|ng = 0, bd = 1)
P(ns = 0|ng = 1, bd = 0)
P(ns = 0|ng = 1, bd = 1)
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Outline
Bayesian Networks
Simplicity and Bayesian Networks
Causal Interpretation of Bayesian Networks
Greedy Equivalence Search
12
Markov Equivalence
Two DAGs G1 and G2 are Markov
equivalent when they contain the same
variables, and for all disjoint X, Y, Z,
X is entailed to be independent from Y
conditional on Z in G1 if and only if X
is entailed to be independent from Y
conditional on Z in G2
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Markov Equivalence
Two DAGS over the same set of variables
are Markov equivalent iff they have:
the same adjacencies
the same unshielded colliders (X → Y ← Z,
and no X → Z or Z→ X edge)
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Markov Equivalence
ng
bl
ng
bl
bd
ns
DAG G
bd
ns
DAG G’
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Patterns
pattern represents a Markov
equivalence class of DAGs.
A
bl ng
bl ng
ng
DAG G
bd
bd
bd
ns
bl
ns
DAG G’
ns
Pattern(G)
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Patterns
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.
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Patterns
All of the conditional independence
relations entailed by a DAG G
represented by a pattern G’ can be read
off of G’.
So we can speak of a pattern being an Imap of P by extension.
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Faithfulness
P is faithful to G if
Every conditional independence entailed by G,
is true in P (Markov)
Every conditional independence true in P is
entailed by G
For every P, there is at most one, but
possibly no pattern that P is faithful to.
Unfaithfulness
X
Y
G
Z
X
Y
G’
Z
+ IP(X,Z| 
IP(X,Z| 
Both G and G’ are patterns that are I-maps
of a P in which the only independence is
I(X,Z| , but P is faithful only to G’, not G.
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Unfaithfulness
X
Y
Z
IP(X,Z|Y) + IP(X,Z| 
X
Y
Z
IP(X,Z|Y) + IP(X,Z| 
If both IP(X,Z|Y) + IP(X,Z|  then P is not
faithful to any DAG.
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Violations of Faithfulness
If conditional independence can be
expressed as rational function of the
parameters (e.g. any exponential family
including Gaussian and multinomial) then
violations of faithfulness are Lebesgue
measure 0.
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Minimality
G is Markov and Minimal for P if and only if
Every conditional independence entailed by G is
true in P (Markov)
No graph that entails a subset of the conditional
independencies entailed by G is Markov to P
(minimality).
For every P, there is at least one, and possibly
more than one pattern, Markov and minimal
for P.
Two Markov Minimal Graphs
for P
X
Y
Z
IP(X,Z|Y) + IP(X,Z| 
X
Y
Z
IP(X,Z|Y) + IP(X,Z| 
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Faithfulness and Minimality
If P is faithful to G, then G is a unique
Markov Minimal pattern for P.
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Dimensionality →
X
X
X
X
Y
Entailment Inclusion →
Lattice of I-maps
Y
Y
Y
Z
X
Z
Y
Z
Z
X
Y
Z
Z
X
X
Y
Z
X
Y
Z
X
Y
Z
X
Y
Y
Z
Z
Outline
Bayesian Networks
Simplicity and Bayesian Networks
Causal Interpretation of Bayesian
Networks
Greedy Equivalence Causal Search
27
Causal Interpretation of DAG
There is an edge X → Y in G if and only if
there is some pair of experimental
manipulations of all of the variables in G
other than Y, that
differ only in what manipulation is performed
on X; and
make the probability of Y different.
Causal Sufficiency
A set S of variables is causally sufficient if
there are no variables not in S that are direct
causes of more than one variable in S.
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Causal Sufficiency
ng
bl
bd
S = {ng, ns} is causally
sufficient.
S = {ng, ns, bl} is not
causally sufficient.
ns
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Causal Markov Assumption
In a population Pop with causally sufficient
graph G and distribution P, each variable is
independent of its non-descendants (noneffects) conditional on its parents
(immediate causes) in P.
Equivalently: G is an I-map of P.
Causal Faithfulness Assumption
In a population Pop with causally sufficient
graph G and distribution P, I(X,Y|Z) in P
only if X is entailed to be independent of Y
conditional on Z by G.
Causal Faithfulness Assumption
As currently used, it is a substantive
simplicity assumption, not a methodological
simplicity assumption. It says to prefer the
simplest explanation: if a more complicated
explanation is true, it leads astray.
Causal Faithfulness Assumption
It serves two roles in practice:
Aiding model choice (in which case weaker
versions of the assumption will do)
Simplifying search
Causal Minimality Assumption
In a population Pop with causally sufficient
graph G and distribution P, G is a minimal
I-map of P.
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Causal Minimality Assumption
This is a strictly weaker simplicity
assumption than Causal Faithfulness.
Given the manipulation interpretation of the
causal graph, and an everywhere positive
distribution, the Causal Minimality
Assumption is entailed by the Causal
Markov Assumption.
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Outline
Bayesian Networks
Simplicity and Bayesian Networks
Causal Interpretation of Bayesian Networks
Greedy Equivalence Search
37
Greedy Equivalence Search
Inputs: Sample from a probability
distribution over a causally sufficient set of
variables from a population with causal
graph G.
Output: A pattern that represents G.
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Assumptions
Causally sufficient set of varaibles
No feedback
The true causal structure can be represented
by a DAG.
Causal Markov Assumption
Causal Faithfulness Assumption
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Consistent scores
If model M1 contains the true
distribution, while model M2 doesn’t,
then in the large sample limit M1 gets
the higher score.
If both M1 and M2 contain the true
distribution, and M1 has fewer
parameters than M2 does, then in the
large sample limit M1 gets the higher
score.
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Consistent scores
So, assuming the Causal Markov and
Faithfulness conditions, a consistent
score assigns the true model (and its
Markov equivalent models) the highest
score in the limit.
Examples: Bayesian Information
Criterion, posterior of Bde or Bge prior
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Score Equivalence
Under BIC, Bde, and Bge, Markov
equivalent DAGs are also score equivalent,
i.e., always receive the same score. (Could
also use posterior probabilities under a wide
variety of priors).
This allows GES to search over the space of
patterns, instead of the space of DAGs.
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Two Phases of GES
Forward Greedy Search (FGS): Starting with any
pattern Q, evaluate all patterns that are I-maps of
Q with one more edge, and move to the one with
the best increase of score. Iterate until local
maximum.
Backward Greedy Search (BGS): Starting with a
pattern Q that contains the true distribution,
evaluate all patterns with one fewer edge of which
Q is an I-map, and move to the one with the best
increase of score. Iterate until local maximum.
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Asymptotic Optimality
In the large sample limit, GES always returns a
pattern that is minimal and Markov to P.
Proof. Because the score is consistent, the
forward phase continues until it reaches a
pattern G such that P is Markov to G. The
backward phase preserves Markov, and
continues until there is no pattern that is both
Markov to P and entails more independencies.
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Asymptotic Optimality
If P is faithful to the causal pattern G, then
BGS returns G.
Proof Sketch: G is the unique pattern that is
minimal and Markov to P. By the previous
theorem, GES returns G.
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When GES Fails to Find the Right
Causal Pattern
In the large sample limit, GES always finds
a Markov Minimal pattern
Some Markov Minimal pattern may not be
the true causal pattern if
Causal Minimality Assumption is violated; or
There are multiple Markov Minimal patterns,
and GES outputs the wrong one
One Markov Minimal Pattern for
P, but Not the Causal Pattern
X
Y
Z
+ I(X,Z| 
X
Y
Z
I(X,Z| 
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Two Markov Minimal Graphs
for P
X
Y
G
Z
I(X,Z|Y) + I(X,Z| 
X
Y
G’
Z
I(X,Z|Y) + I(X,Z| 
For some parametric families G and G’ have
the same score and the same dimension.
For other parametric families, G’ has lower
dimension and higher score than G.
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Multiple Minimal I-maps?
Lower Standard of Success
Output all Markov Minimal
Output any Markov Minimal
More Data
Experiments
More Background Knowledge
Time order, etc.
References
Chickering, M. (2002) Optimal Structure
Identification With Greedy Search, Journal
of Machine Learning Research 3, pp. 507554.
Spirtes, P., Glymour, C., and Scheines, R.,
(2000) Causation, Prediction, and Search,
2nd edition. MIT Press, Cambridge, MA.
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