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Introduction
CG Interpretations
On the Properties of MVR CGs
Separators in CGs
Learning CGs
Chain Graphs: Properties and Learning
Algorithms
Mohammad Ali Javidian
Department of Computer Science and Engineering
University of South Carolina
October 23, 2019
Mohammad Ali Javidian Chain Graphs: Properties and Learning Algorithms 1 / 44

Introduction
CG Interpretations
Separators in CGs
Learning CGs
Outline
Introduction
Probabilistic Graphical Models
A Motivational Example
CG Interpretations
LWF Chain Graphs
MVR Chain Graphs
AMP Chain Graphs
CGs and Symmetric Associations
Separators in CGs
Learning CGs
Order-Independent PC-like Algorithm for Learning MVR CGs
Order-dependent PC-like Algorithm for Learning MVR CGs
Learning MVR CGs via Decomposition

Introduction
CG Interpretations
Separators in CGs
Learning CGs
Outline
Introduction
CG Interpretations
LWF Chain Graphs
MVR Chain Graphs
AMP Chain Graphs
Separators in CGs
Learning CGs

Introduction
CG Interpretations
Separators in CGs
Learning CGs
Probabilistic Graphical Models (PGMs)
Any PGM consists of:
a graph,
a joint distribution
Applications: probabilistic reasoning and decision making

Introduction
CG Interpretations
Separators in CGs
Learning CGs
Commonly Used PGMs
Markov networks
Bayesian networks

Introduction
CG Interpretations
Separators in CGs
Learning CGs
Creating a Graphical Model from Expert Knowledge
Figure: Lappenschaar et. al. Qualitative chain graphs and their
application. IJAR (2014)

Introduction
CG Interpretations
Separators in CGs
Learning CGs
LWF Chain Graphs
MVR Chain Graphs
AMP Chain Graphs
Outline
Introduction
CG Interpretations
LWF Chain Graphs
MVR Chain Graphs
AMP Chain Graphs
Separators in CGs
Learning CGs

Introduction
CG Interpretations
Separators in CGs
Learning CGs
LWF Chain Graphs
MVR Chain Graphs
AMP Chain Graphs
Chain Graphs (CGs)
Chain graphs:
admit both directed and undirected edges,
there are no partially directed cycles.
A Partially directed cycle: is a sequence of n distinct vertices
v1, v2, . . . , vn(n ≥ 3), and vn+1 ≡ v1, s.t.
for all i (1 ≤ i ≤ n) either vi − vi+1 or vi → vi+1, and
there exists a j (1 ≤ j ≤ n) such that vj ← vj+1.
Example:

Introduction
CG Interpretations
Separators in CGs
Learning CGs
LWF Chain Graphs
MVR Chain Graphs
AMP Chain Graphs
LWF Chain Graphs
Global Markov property: if for all A, B, C ⊆ V such that C
separates A and B in (GAn(A∪B∪C))m
, we have
A B | C.
Example: a b, a d | {b, c}, and b c | {a, d}.
Figure: The LWF CG G with chain components:
T = {T1 = {a}, T2 = {b}, T3 = {c, d}}.

Introduction
CG Interpretations
Separators in CGs
Learning CGs
LWF Chain Graphs
MVR Chain Graphs
AMP Chain Graphs
Multivariate Regression CGs (MVR CGs)
separates A and B in (GAn(A∪B∪C))a
, we have
A B | C.
Example: a b, a d, and b c.
Figure: The MVR CG G with chain components:
T = {T1 = {a}, T2 = {b}, T3 = {c, d}}.

Introduction
CG Interpretations
Separators in CGs
Learning CGs
LWF Chain Graphs
MVR Chain Graphs
AMP Chain Graphs
AMP Chain Graphs
separates A and B in (G[A ∪ B ∪ C])a
, we have
A B | C.
Example: a b, a b | c, a b | d, a d, a d | b, b c and
b c | a.
Figure: The AMP CG G with chain components:
T = {T1 = {a}, T2 = {b}, T3 = {c, d}}.

Introduction
CG Interpretations
Separators in CGs
Learning CGs
LWF Chain Graphs
MVR Chain Graphs
AMP Chain Graphs
Chain Graphs and Symmetric Associations
MVR CGs:
there is an unmeasured common cause of both X and Y;
LWF CGs:
there is feedback between X and Y. They arrive at a
stochastic equilibrium as time goes to inﬁnity.
AMP

Introduction
CG Interpretations
Separators in CGs
Learning CGs
LWF Chain Graphs
MVR Chain Graphs
AMP Chain Graphs
Creating a CG from Expert Knowledge
Figure: Lappenschaar et. al. Qualitative chain graphs and their
application. IJAR (2014)

Introduction
CG Interpretations
Separators in CGs
Learning CGs
Outline
Introduction
CG Interpretations
LWF Chain Graphs
MVR Chain Graphs
AMP Chain Graphs
Separators in CGs
Learning CGs

Introduction
CG Interpretations
Separators in CGs
Learning CGs
Are the Proposed Markov Properties of MVR CGs
Equivalent?1
Theorem
Let G be an MVR chain graph. For an independence model over the node set
of G, the following conditions are equivalent:
(i) satisfies the global Markov property (based on the pathwise
m-separation criterion) w.r.t. G in (Richardson & Spirtes, 2002);
(ii) satisfies the global Markov property (based on the augmentation
separation criterion) w.r.t. G in (Richardson & Spirtes, 2002);
(iii) satisfies the block recursive Markov property w.r.t. G in (Drton, 2009);
(iv) satisfies the MR Markov property w.r.t. G in (Marchetti & Lupparelli,
2011).
(v) satisfies the ordered local Markov property w.r.t. G in (Richardson, 2003).
1
(Javidian & Valtorta, WPGM 2018)

Introduction
CG Interpretations
Separators in CGs
Learning CGs
An Alternative Factorization for MVR Chain Graphs
Factorization: [(Drton, 2009) and (Marchetti & Lupparelli, 2011)]
p(x) = T∈T p(xT |xpaD (T)).
Theorem ((Javidian & Valtorta, WPGM 2018))
Let G be an MVR chain graph with chain components (T|T ∈ T ). If
a probability distribution P obeys the global Markov property for G
then p(x) = T∈T p(xT |xpaG(T)).
Example: p = p1234|56p56|7p7 vs p = p1234|5p56|7p7
Figure: an MVR CG with: T = {T1 = {1, 2, 3, 4}, T2 = {5, 6}, T3 = {7}}.

Introduction
CG Interpretations
Separators in CGs
Learning CGs
Outline
Introduction
CG Interpretations
LWF Chain Graphs
MVR Chain Graphs
AMP Chain Graphs
Separators in CGs
Learning CGs

Introduction
CG Interpretations
Separators in CGs
Learning CGs
Main Theorems: Theorem 11
Theorem
The problem of ﬁnding a minimal separating set for X and Y in an
LWF/MVR/AMP chain graph G is equivalent to the problem of
ﬁnding a minimal separating set forX and Y in the undirected
graph (GAn(X∪Y))m
/(GAn(X∪Y))a
/(Gant(X∪Y))a
.
1
(Javidian & Valtorta, PGM 2018, WUAI 2018, & Jamshidi JMLR 2019)

Introduction
CG Interpretations
Separators in CGs
Learning CGs
Main Theorems: Theorem 2
Theorem
Given two nodes X and Y in an LWF/MVR chain graph G and a
set S of nodes not containing X and Y, there exists some subset of
S which separates X and Y if only if the set S = S ∩ An(X ∪ Y)
separates X and Y.
Note that in the case of AMP CGs, S = S ∩ ant(X ∪ Y).

Introduction
CG Interpretations
Separators in CGs
Learning CGs
Algorithm for Finding Minimal Separator
Algorithm 1: Restricted separation
Input: A set S of nodes not containing X and Y in the LWF chain graph G.
Output: If there is a subset of S that separates X from Y then the algorithm
returns Z ⊆ S that separates X from Y otherwise, returns FALSE.
1 Construct GAn(X∪Y);
2 Construct (GAn(X∪Y))m
;
3 Set S = S ∩ An(X ∪ Y);
4 Remove S from (GAn(X∪Y))m
;
5 Starting from X, run BFS;
6 if Y is met then
7 return FALSE
8 else
9 return Z = S
10 end

Introduction
CG Interpretations
Separators in CGs
Learning CGs
Outline
Introduction
CG Interpretations
LWF Chain Graphs
MVR Chain Graphs
AMP Chain Graphs
Separators in CGs
Learning CGs

Introduction
CG Interpretations
Separators in CGs
Learning CGs
PC-Like Algorithm for Learning MVR CGs (Sonntag &
Pe˜na, 2012)
X
Y
Observational Data
a
b
c
d
e
Skeleton Recovery
a
b
c
d
e
v-StructureRecovery
a
b
c
d
e
Essential Recovery
Figure: The procedure of learning the structure of an essential MVR
chain graph from a faithful distribution.

Introduction
CG Interpretations
Separators in CGs
Learning CGs
Markov Equivalence Criterion
The skeleton (underlying graph) of an MVR CG G is obtained
from G by changing all directed/bidirected edges of G into
undirected edges.
A v-structure in a chain graph is an induced subgraph of the
form a → c ← b.
Theorem
Two MVR chain graphs G and H are Markov equivalent if and only
if they have the same skeletons and the same v-structures
(Wermuth & Sadeghi, 2012).

Introduction
CG Interpretations
Separators in CGs
Learning CGs
Skeleton Recovery
Algorithm 2: Skeleton Recovery of MVR chain graphs
Input: a set V of nodes and a probability distribution p faithful to an unknown
MVR CG G.
Output: The skeleton of the corresponding MVR CG G.
1 Let H denote the complete undirected graph over V;
2 for i ← 0 to |VH| − 2 do
3 while possible do
4 Select any ordered pair of nodes A and B in H such that A ∈ adH(B)
and |ad(A) B| ≥ i;
5 if there exists S ⊆ (adH(A) B) s.t. |S| = i and A ⊥⊥p B|S (i.e., A is
independent of B given S in the probability distribution p) then
6 Set SAB = SBA = S;
7 Remove the edge A − B from H;
8 end
9 end
10 end

Introduction
CG Interpretations
Separators in CGs
Learning CGs
Skeleton Recovery: Example
Given order: a, b, c, d, e.
a
b
c
d
e
MVR CG G
a ⊥⊥d d|{b}
a ⊥⊥d e|{c}
b ⊥⊥d e|{c}
d ⊥⊥d e|{c}
a
b
c
d
e
Start with
H = K5,
and after
i = 0
a
b
c
d
e
After i = 1
a ⊥⊥d d|{b}
a ⊥⊥d e|{c}
b ⊥⊥d e|{c}
d ⊥⊥d e|{c}

Introduction
CG Interpretations
Separators in CGs
Learning CGs
v-Structure Recovery
Algorithm 3: Pattern recovery of MVR chain graphs
Input: H, the skeleton of the MVR CG G.
Output: The pattern of the MVR CG G.
1 for each m-separator Suv do
2 if u w v appears in the skeleton and w is not in Suv then
/* u w means u w or u w. Also, w v means
w v or w v. */
3 Determine a v-structure u w v;
4 end
5 end

Introduction
CG Interpretations
Separators in CGs
Learning CGs
v-Structure Recovery: Example
Given order: a, b, c, d, e.
a
b
c
d
e
MVR CG G
a ⊥⊥d d|{b}
a ⊥⊥d e|{c}
b ⊥⊥d e|{c}
d ⊥⊥d e|{c}
a
b
c
d
e
Start with
H = K5.
After
i = 0
a
b
c
d
e
After i = 1
a ⊥⊥d d|{b}
a ⊥⊥d e|{c}
b ⊥⊥d e|{c}
d ⊥⊥d e|{c}
a
b
c
d
e
v-StructureRecovery
a ⊥⊥d d|{c}
So, c Sad
i.e., a, c, d
is v-structure

Introduction
CG Interpretations
Separators in CGs
Learning CGs
Essential Graph Recovery
Algorithm 4: Essential CG recovery of MVR chain graphs
Input: The pattern of the MVR CG G.
Output: The Essential of the MVR CG G.
1 Apply rules 1-3 (Sonntag & Pe˜na, 2012) in the following ﬁgure while possible;

Introduction
CG Interpretations
Separators in CGs
Learning CGs
Essential Graph Recovery: Example
a
b
c
d
e
v-StructureRecovery
Learned
Pattern
a
b
c
d
e
Essential Recovery
Apply
Rules 1&3

Introduction
CG Interpretations
Separators in CGs
Learning CGs
Ex. Order-Dependent Skeleton of the PC-like Algorithm
e
d
a
b c
(a)
e
d
a
b c
(b)
Figure: (a) DAG G with minimal sep. sets: a ⊥⊥ d|{b, c} & a ⊥⊥ e|{b, c},
(b) the skeleton returned by OPC with order1(V) = (d, e, a, c, b).
Table: The trace table of OPC for i = 3 and order1(V) = (d, e, a, c, b).
Ordered Is Suv ⊆ Is u v
Pair (u, v) adH(u) Suv adH(u) {v}? removed?
(e, a) {a, b, c, d} {b, c, d} Yes Yes
(e, c) {b, c, d} {a, b, d} No No
(c, e) {a, b, d, e} {a, b, d} Yes Yes

Introduction
CG Interpretations
Separators in CGs
Learning CGs
e
d
a
b c
(a)
e
d
a
b c
(c)
Figure: (a) DAG G with minimal sep. sets: a ⊥⊥ d|{b, c} & a ⊥⊥ e|{b, c}, (c)
the skeleton returned by OPC with order2(V) = (d, c, e, a, b).
Table: The trace table of OPC for i = 3 and order2(V) = (d, c, e, a, b).
Ordered Is Suv ⊆ Is u v
Pair (u, v) adH(u) Suv adH(u) {v}? removed?
(c, e) {a, b, d, e} {a, b, d} Yes Yes
(e, a) {a, b, d} {b, c, d} No No
(a, e) {b, c, e} {b, c, d} No No

Introduction
CG Interpretations
Separators in CGs
Learning CGs
e
d
a
b c
(a)
e
d
a
b c
(b)
e
d
a
b c
(c)
(b) the skeleton returned by OPC with order1(V) = (d, e, a, c, b), (c) the
skeleton returned by OPC with order2(V) = (d, c, e, a, b).

Introduction
CG Interpretations
Separators in CGs
Learning CGs
Algorithm 5: The order-independent (stable) skeleton recovery of
MVR CGs (Javidian et al., SUM 2019).
Input: A set V of nodes and a probability distribution p faithful to an unknown MVR CG G
and an ordering order(V) on the variables.
Output: The skeleton of the corresponding MVR CG G.
1 Let H denote the complete undirected graph over V = {v1, . . . , vn};
2 for i ← 0 to |VH| − 2 do
3 for j ← 1 to |VH| do
4 Set aH(vi) = adH(vi);
5 end
6 while possible do
7 Select any ordered pair of nodes u and v in H such that u ∈ aH(v) and
|aH(u) v| ≥ i using order(V);
8 if there exists S ⊆ (aH(u) v) s.t. |S| = i and u ⊥⊥p v|S (i.e., u is independent of v
given S in the probability distribution p) then
9 Set Suv = Svu = S;
10 Remove the edge u v from H;
11 end
12 end
13 end

Introduction
CG Interpretations
Separators in CGs
Learning CGs
Ex. Order-Independ. Skeleton of the PC-like Algorithm
After CI tests at some signiﬁcance level α: a ⊥⊥ d|{b, c},
a ⊥⊥ e|{b, c, d}, and c ⊥⊥ e|{a, b, d}. When i = 3:
aH(a) = aH(d) = {b, c, e} and aH(v) = V {v}, for v = b, c, e.
e
d
a
b c
(a)
e
d
a
b c
(b)
(b) the skeleton returned by OPC with both ordering
order1(V) = (d, e, a, c, b) and order2(V) = (d, c, e, a, b).

Introduction
CG Interpretations
Separators in CGs
Learning CGs
Evaluation: OPC vs SPC
500 1000 5000 10000
0.9750.9800.9850.990
q
q
q
q
P= 50 , N= 2 , OPC vs SPC
Sample size
TDR
q OPC
SPC
500 1000 5000 10000
0.800.850.900.95
q
q
q
q
Sample size
TPR
q OPC
SPC
500 1000 5000 10000
8101418
q
q
q
q
Sample size
SHD
q OPC
SPC
0.05 0.01 0.005 0.001
0.60.70.80.9
q
q
q
q
P= 1000 , N= 2 , OPC vs SPC
p values
TDR
q OPC
SPC
0.05 0.01 0.005 0.001
0.400.440.480.52
q
q
q
q
P= 1000 , N= 2 , OPC vs SPC
p values
TPR
q OPC
SPC
0.05 0.01 0.005 0.001
8009001000
q
q
q
P= 1000 , N= 2 , OPC vs SPC
p values
SHD
q OPC
SPC

Introduction
CG Interpretations
Separators in CGs
Learning CGs
A Decomposition-Based Algorithm for Learning MVR
CGs (Javidian & Valtorta, submitted to IJAR, 2019)

Introduction
CG Interpretations
Separators in CGs
Learning CGs
Local Skeleton Recovery from an m-Separation Tree
Theorem
Let T be an m-separation tree for CG G. Vertices u and v are
m-separated by S ⊆ V in G if and only if (i) u and v are not
contained together in any node C of T or (ii) there exists a node C
that contains both u and v such that a subset S of C, m-separates
u and v.

Introduction
CG Interpretations
Separators in CGs
Learning CGs
Algorithm 6: Pattern recovery algorithm for MVR chain graphs
Input: an m-separation tree T with a node set C = {C1, . . . , CH}
Output: the pattern of MVR CG G.
1 Set S = ∅;
2 for h ← 1 to H do
3 Start from a complete undirected graph ¯Gh with vertex set Ch ;
4 for each vertex pair {u, v} ⊆ Ch do
5 if ∃Suv ⊆ Ch such that u ⊥⊥ v|Suv then
6 Delete the edge (u, v) in ¯Gh ;
7 Add Suv to S;
8 end
9 end
10 end
11 Initialize the edge set ¯EV of ¯GV as the union of all edge sets of ¯Gh , h = 1, . . . , H;
12 for each Vertex pair {u, v} contained in more than one tree node and (u, v) ∈ ¯GV do
13 if ∃Ch such that {u, v} ⊆ Ch and {u, v} ¯Eh then
14 Delete the edge (u, v) in ¯GV ;
15 end
16 end
17 for each m-separator Suv in the list S do
18 if u ◦− w −◦ v appears in the global skeleton and w is not in Suv then
/* u ◦− w means u ← w or u − w. Also, w −◦ v means w → v or w − v. */
19 Determine a v-structure u ◦→ w ←◦ v;
20 end
21 end

Introduction
CG Interpretations
Separators in CGs
Learning CGs
Essential MVR CG Recovery
Apply rules 1-3 in the following ﬁgure while possible:
Figure: The Rules for converting a pattern to an essential MVR CG
(Sonntag and Pea, 2012)

Introduction
CG Interpretations
Separators in CGs
Learning CGs
Evaluation: PC vs LCD
q
q
0.76
0.80
0.84
0.88
0.92
LCD
PC
TPR
sample size
s=1000
s=10000
s=300
s=3000
alpha = 0.05
q
q
0.75
0.80
0.85
0.90
alpha = 0.01
q
q
0.70
0.75
0.80
0.85
0.90
alpha = 0.005
PC
PC
LCD
LCD q
q
0.0040
0.0045
0.0050
LCD
PC
FPR
sample size
s=1000
s=10000
s=300
s=3000
alpha = 0.05
q
q
0.0010
0.0015
0.0020
alpha = 0.01
q
q
0.0004
0.0008
0.0012
0.0016
alpha = 0.005
LCD
LCD
PC
PC
q
q
0.9800
0.9825
0.9850
0.9875
0.9900
LCD
PC
ACC
sample size
s=1000
s=10000
s=300
s=3000
alpha = 0.05
q
q
0.980
0.985
0.990
alpha = 0.01
q
q
0.980
0.985
0.990
alpha = 0.005
PC
PC
LCD
LCD
q
q
30
40
50
LCD
PC
#Differences
sample size
s=1000
s=10000
s=300
s=3000
alpha = 0.05
q
q
20
30
40
50
SHD
alpha = 0.01
q
q
20
30
40
50
alpha = 0.005
PC
LCD
LCD
PC

Introduction
CG Interpretations
Separators in CGs
Learning CGs
Summary
We studied all Markov properties for MVR chain graphs,
proposed in the literature, and we proved that all of them are
equivalent (except for pairwise Markov properties).
We proposed a new factorization formula for MVR CGs that is
more concise and informative than the one proposed in the
literature.
We proposed an efﬁcient algorithm to address the problem of
ﬁnding minimal separators in chain graphs for all three
different interpretations.
We proposed a PC-like algorithm for structure learning of LWF
CGs. Also, we proposed an order-independent version of
PC-like algorithm for all three different interpretations.
We proposed a decomposition-based algorithm for learning
the structure of MVR/AMP CGs.

Introduction
CG Interpretations
Separators in CGs
Learning CGs
List of publications for each chapter
LWF CGs • Finding Minimal Separators in LWF Chain Graphs
(Javidian & Valtorta, PGM2018) [section 2.3].
• Learning LWF Chain Graphs: An Order-Independent Algorithm
(Javidian & Valtorta & Jamshidi, submitted to AISTATS 2020) [section 2.4].
MVR CGs • On the properties of MVR Chain Graphs (Javidian & Valtorta, WPGM2018) [section 3.2].
• Finding Minimal Separators in MVR Chain Graphs
(Javidian & Valtorta, WUAI2018) [section 3.3].
• Order-Independent Structure Learning of Multivariate Regression Chain Graphs
(Javidian & Valtorta& Jamshidi, SUM 2019) [section 3.4].
• A Decomposition-Based Algorithm for Learning the Structure of MVR Chain Graphs
(Javidian & Valtorta, submitted to IJAR, 11 Apr 2019) [section 3.5].
AMP CGs • AMP CGs: Minimal Separators and Structure Learning Algorithms
(Javidian & Valtorta& Jamshidi, submitted to JMLR, 23 Jul 2019).
Bayesian hypergraphs • On a hypergraph probabilistic graphical model
(Javidian & Wang & Lu & Valtorta, submitted to Ann Math Artif Intel, 19 Apr 2019) [Chapter 5]
• The Causal Interpretations of Bayesian Hypergraphs
(Wang & Javidian & Lu & Valtorta, AAAI Spring Symposium 2019) [section 5.4].
Causal transfer learning • Transfer Learning for Performance Modeling of Conﬁgurable Systems:
A Causal Analysis (Javidian & Jamshidi & Valtorta, AAAI Spring Symposium 2019)
[Chapter 6].

Introduction
CG Interpretations
Separators in CGs
Learning CGs
List of other publications
Mohammad Ali Javidian , Pooyan Jamshidi, and Rasoul
Ramezanian. ”Avoiding Social Disappointment in Elections”
Proceedings of the 18th International Conference on
Autonomous Agents and MultiAgent Systems (AAMAS 2019,
Montreal), May 13-17, 2019, Pages: 2039-2041.
Mohammad Ali Javidian, Marco Valtorta, and Pooyan
Jamshidi. ”Learning Bayesian networks in the presence of
unmeasured confounders: an efﬁcient approach based on
Markov blankets.” (submitted to AAAI 2020).

Introduction
CG Interpretations
Separators in CGs
Learning CGs
AMP CGs and Symmetric Associations
X and Y are both causes of some ’selection’ variable;
Every AMP CG is Markov equivalent to some DAG with error
and selection nodes under marginalization of the error nodes
and conditioning of the selection nodes (Pe˜na, 2014).
Figure: IAMP(G) = [IAMP(G )]∅
= [IAMP(G )]S
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