Transcript Session2
Outline
1) Motivation
2) Representing/Modeling Causal Systems
3) Estimation and Updating
4) Model Search
5) Linear Latent Variable Models
6) Case Study: fMRI
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Outline
Search I: Causal Bayes Nets
1) Bridge Principles:
Causal Structure Testable Statistical Constraints
2) Equivalence Classes
3) Pattern Search
4) PAG Search
5) Variants
6) Simulation Studies on the Tetrad workbench
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Bridge Principles:
Acyclic Causal Graph over V Constraints on P(V)
Weak Causal Markov Assumption
V1,V2 causally disconnected V1 _||_ V2
V1 _||_ V2 v1,v2 P(V1=v1 | V2 = v2) = P(V1=v1)
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Bridge Principles:
Acyclic Causal Graph over V Constraints on P(V)
Weak Causal Markov Assumption
Determinism
V1,V2 causally disconnected V1 _||_ V2
(Structural Equations)
Causal Markov Axiom
If G is a causal graph, and P a probability distribution over the variables in
G, then in <G,P> satisfy the Markov Axiom iff:
every variable V is independent of its non-effects,
conditional on its immediate causes.
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Bridge Principles:
Acyclic Causal Graph over V Constraints on P(V)
Causal Markov Axiom
Acyclicity
d-separation criterion
Causal Graph
Z
X
Independence Oracle
Y1
Y2
Z _||_ Y1 | X
Z _||_ Y2 | X
Z _||_ Y1 | X,Y2
Z _||_ Y2 | X,Y1
Y1 _||_ Y2 | X
Y1 _||_ Y2 | X,Z
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Faithfulness
Constraints on a probability distribution P generated by a
causal structure G hold for all parameterizations of G.
Tax Rate
b3
b1
Tax
Revenues
Economy
Revenues := b1Rate + b2Economy + eRev
Economy := b3Rate + eEcon
b2
Faithfulness:
b 1 ≠ - b 3b 2
b 2 ≠ - b 3b 1
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Equivalence Classes
Equivalence:
• Independence Equivalence: M1 ╞ (X _||_ Y | Z) M2 ╞ (X _||_ Y | Z)
• Distribution Equivalence: q1 q2 M1(q1) = M2(q2), and vice versa)
• Independence (d-separation equivalence)
• DAGs : Patterns
• PAGs : Partial Ancestral Graphs
• Intervention Equivalence Classes
• Measurement Model Equivalence Classes
• Linear Non-Gaussian Model Equivalence Classes
• Etc.
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d-separation/Independence Equivalence
D-separation Equivalence Theorem (Verma and Pearl, 1988)
Two acyclic graphs over the same set of variables are
d-separation equivalent iff they have:
• the same adjacencies
• the same unshielded colliders
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Colliders
Y: Collider
X
Y: Non-Collider
Z
X
Z
Shielded
X
Unshielded
Z
Y
Z
X
Y
Y
Y
Y
Z
X
X
Z
Y
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D-separation
X is d-separated from Y by Z in G iff
Every undirected path between X and Y in G is inactive relative to Z
An undirected path is inactive relative to Z iff
any node on the path is inactive relative to Z
A node N is inactive relative to Z iff
a) N is a non-collider in Z, or
b) N is a collider that is not in Z,
and has no descendant in Z
Z1
1) X --> Z1 <-- W --> Y
2) X <-- V --> Y
V
X
Undirected Paths between X , Y:
W
Y
Z2
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D-separation
X is d-separated from Y by Z in G iff
Every undirected path between X and Y in G is inactive relative to Z
An undirected path is inactive relative to Z iff
any node on the path is inactive relative to Z
A node N is inactive relative to Z iff
a) N is a non-collider in Z, or
b) N is a collider that is not in Z,
and has no descendant in Z
Z1
Z2
1) X --> Z1 <-- W --> Y
2) X <-- V --> Y
V
X
Undirected Paths between X , Y:
W
Y
X d-sep Y relative to Z = ?
No
X d-sep Y relative to Z = {V} ?
Yes
X d-sep Y relative to Z = {V, Z1 } ?
No
X d-sep Y relative to Z = {W, Z2 } ?
Yes
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D-separation
X3 and X1 d-sep by X2?
X1
X3
X2
T
X1
X2
Yes: X3 _||_ X1 | X2
X3 and X1 d-sep by X2?
X3
No: X3 _||_ X1 | X2
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Statistical Control ≠ Experimental Control
T
X1
X3
X2
X3 _||_ X1 | X2
T
X1
X2
X3
X3 _||_ X1 | X2(set)
I
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Independence Equivalence Classes:
Patterns & PAGs
• Patterns (Verma and Pearl, 1990): graphical
representation of d-separation equivalence among
models with no latent common causes (i.e., causally
sufficient models)
• PAGs: (Richardson 1994) graphical representation of a
d-separation equivalence class that includes models with
latent common causes and sample selection bias that are
Markov equivalent over a set of measured variables X
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Patterns
E xam ple
Possible E dges
X1
X2
X1
X2
X1
X3
X1
X2
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X2
X4
Patterns: What the Edges Mean
X1
X1
X1
X2
X2
X2
X 1 a n d X 2 a re n o t a d ja c e n t in
a n y m e m b e r o f th e
e q u iv a le n c e c la s s
X 1 X 2 (X 1 is a cau se o f X 2 ) in
ev ery m em b er o f th e eq u iv alen ce
class.
X 1 X 2 in so m e m em b ers o f th e
eq u iv alen ce class, an d X 2 X 1 in
o th ers.
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Patterns
X2
X1
P attern
X4
X3
R ep resen ts
X3
X2
X1
X2
X1
X4
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X3
X4
Tetrad Demo
1) Load Session: patterns1.tet
2) Change Graph3 minimally to reduce number of equivalent
DAGs maximally
3) Compute the DAGs that are equivalent to your original 3
variable DAG
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Constraint Based Search
D ata
E quivalence C lass of
C ausal G raphs
X1
X1
X1
X2
X2
X2
X3
X3
X3
Statistical
Inference
Causal Markov Axiom
(D-separation)
D iscovery A lgorithm
Statistical
C onstraints
X1
Background Knowledge
e.g., X2 prior in time to X3
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X3 | X2
Score Based Search
Equivalence C lass of
C ausal G raphs
X1
X2
X3
D a ta
X1
X1
X2
X3
X2
X3
Equivalence C lass of
C ausal G raphs
X1
X2
X3
Model Score
X1
X2
X3
X1
X2
X3
Equivalence C lass of
C ausal G raphs
X1
X2
X3
Background Knowledge
e.g., X2 prior in time to X3
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Overview of Search Methods
Constraint Based Searches
• TETRAD (PC, FCI)
• Very fast – capable of handling 1,000 variables
• Pointwise, but not uniformly consistent
Scoring Searches
•
•
•
•
•
•
Scores: BIC, AIC, etc.
Search: Hill Climb, Genetic Alg., Simulated Annealing
Difficult to extend to latent variable models
Meek and Chickering Greedy Equivalence Class (GES)
Very slow – max N ~ 30-40
Pointwise, but not uniformly consistent
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Tetrad Demo
1) Open new session
2) Template: Search from Simulated Data
3) Create Graph, parameterize, instantiate, generate data N=50
4) Choose PC search, execute
5) Attach new search node, choose GES, execute
6) Play (sample size, parameters, alpha value, etc.)
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Tetrad Demo
1) Open new session
2) Load Charity.txt
3) Create Knowledge:
a. Tangibility is exogenous
b. AmountDonate is Last
c.
Tangibility direct cause of Imaginability
4) Perform Search
5) Estimate output
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PAGs: Partial Ancestral Graphs
X2
X1
PAG
X3
R e p r e s e n ts
X2
X1
X2
X1
T1
X3
X3
e tc .
X1
X2
X2
X1
T1
T1
X3
X3
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T2
PAGs: Partial Ancestral Graphs
Z1
Z2
PAG
X
Y
Represents
Z1
Z2
Z1
Z2
T1
X3
X3
Y
Y
etc.
T1
Z1
Z2
Z1
Z2
T2
T1
X3
X3
Y
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Y
PAGs: Partial Ancestral Graphs
What PAG edges mean.
X1
X2
X 1 an d X 2 are n o t ad jacen t
X1
X2
X 2 is n o t an an cesto r o f X 1
X1
X2
N o set d -sep arates X 2 an d X 1
X1
X2
X 1 is a cau se o f X 2
X1
X2
T h ere is a laten t co m m o n
cau se o f X 1 an d X 2
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Constraint-based Search
1) Adjacency
2) Orientation
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Constraint-based Search: Adjacency
1. X and Y are adjacent if they are dependent
conditional on all subsets that don’t include
them
2. X and Y are not adjacent if they are
independent conditional on any subset that
doesn’t include them
Search: Orientation
Patterns
Y Unshielded
X
Z
Y
X _||_ Z | Y
X _||_ Z | Y
Non-Collider
Collider
X
Y
Z
X
Y
Z
X
Y
Z
X
Y
Z
X
Y
Z
Search: Orientation
PAGs
Y Unshielded
X
Y
Z
X _||_ Z | Y
X _||_ Z | Y
Non-Collider
Collider
X
Y
Z
X
Y
Z
Search: Orientation
Away from Collider
T est C o n d itio n s
X1
X3
*
1 ) X 1 - X 2 ad jacen t, an d in to X 2 .
2 ) X 2 - X 3 ad jacen t
3 ) X 1 - X 3 n o t ad jacen t
X2
T est
X1
X3 | X2
Y es
No
X1
*
X3
X2
X1
*
X3
X2
Caus al
Graph
Independcies
X1
X1
X3
X4
X2
X2
X1
X4
{X3}
X2
X4
{X3}
X1
Begin w ith:
X3
X4
X2
From
X1
X1
X3
X2
X4
X2
From
X1
X4
X1
{X3}
X3
X4
X2
From
X1
X2
X4
{X3}
X3
X2
X4
Search: Orientation
After Orientation
Phase
X1
P a tte rn
X3
X1
X3
X4
X4
X2
X2
X1
X1
X1 || X2
PA G
X3
X4
X2
X3
X4
X2
X1
X1
X1 || X4 | X3
X2 || X4 | X3
X3
X3
X2
X4
X2
X4
Interesting Cases
M1
X1
L
X2
M2
X
Y
Z
Y1
Y2
L1
L1
L2
Z1
X
Z2
Y
M3
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Tetrad Demo
1) Open new session
2) Create graph for M1, M2, M3 on previous slide
3) Search with PC and FCI on each graph, compare results
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Tetrad Demo
1) Open new session
2) Load data: regression_data
3) X is “putative cause”, Y is putative effect,
Z1,Z2 prior to both (potential confounders)
4) Use regression to estimate effect of X on Y
5) Apply FCI search to data
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Variants
1) CPC, CFCI
2) Lingam
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LiNGAM
1. Most of the algorithms included in Tetrad (other than KPC) assume
causal graphs are to be inferred from conditional independence
tests.
2. Usually tests that assume linearity and Gaussianity.
3. LiNGAM uses a different approach.
4. Assumes linearity and non-Gaussianity.
5. Runs Independent Components Analysis (ICA) to estimate the
coefficient matrix.
6. Rearranges the coefficient matrix to get a causal order.
7. Prunes weak coefficients by setting them to zero.
ICA
Although complicated, the basic idea is very simple.
a11 X1 + ... + a1n Xn = e1
...
an1 X1 + ... + ann Xn = en
Assume e1,...,en are i.i.d.
Try to maximize the non-Gaussianity of w1 X1 + ... + wn Xn = ?
There are n ways to do it up to symmetry!
(Cf. Central Limit Theorem, Hyavarinen et al., 2002)
You can use the coefficients for e1, or for e2, or for...
All other linear combinations of e1,...,en are more Gaussian.
ICA
This equation is usually denoted Wx = s
But also X = BX + s where B is the coefficient matrix
So Wx = (I – B)x = e
s is the vector of independent components
x is the vector of variables
Just showed that under strong conditions we can estimate W.
So we can estimate B! (But with unknown row order)
Using assumptions of linearity and non-Gaussianity (of all but one
variable) alone.
More sophisticated analyses allow errors to be non-i.i.d.
LiNGAM
LiNGAM runs ICA to estimate the coefficient matrix B.
The order of the errors is not fixed by ICA, so some rearranging of the
B matrix needs to be done.
Rows of the B matrix are swapped so the it is lower triangular.
a[i][j] should be non-zero (representing an edge) just in case ij
Typically, a cutoff is used to determine if a matrix element is zero.
The rearranged matrix corresponds to the idea of a causal order.
LiNGAM
Once you know which nodes are adjacent in the graph and what
the causal order is, you can infer a complete DAG.
Review:
Use data from a linear non-Gaussian model (all but one variable nonGaussian)
Infer a complete DAG (more than a pattern!)
Hands On
1) Attach a Generalized SEM IM.
2) Attach a data set, simulate 1000 points.
3) Attach a Search box and run LiNGAM.
4) Attach another search box to Data and run PC.
5) Compare PC to LiNGAM.
Special Variants of Algorithms
PC Pattern
PC Pattern enforces the requirement that the output of the algorithm
will be a pattern.
PCD
PCD adds corrective code to PC for the case where some variables
stand in deterministic relationships.
This results in fewer edges being removed from the graph.
For example, if X _||_ Y | Z but Z determines Y, X---Y is not taken out.
Special Variants of Algorithms
CPC
The PC algorithm may jump too quickly to the conclusion that a
collider and noncolliders should be oriented, X->Y<-Z, X---Y---Z
The CPC algorithm uses a much more conservative test for colliders
and noncolliders, double and triple checking to make sure they should
be oriented, against different adjacents to X and to Z.
The result is a graph with fewer but more accurate orientations.
Hands On
1.
1.
2.
3.
4.
5.
6.
7.
8.
9.
Simulate data from a “complicated” DAG using a SEM IM.
Choose the Search from Simulated Data item from the Templates menu.
Make a random 20 node 20 edge DAG.
Parameterize as a linear SEM, accepting defaults.
Run CPC.
Attach another search box to data.
Run PC.
Layout the PC graph using Fruchterman-Reingold.
Copy the layout to the CPC graph.
Open PC and CPC simultaneously and note the differences.
Special Variants of Algorithms
1.
CFCI
1.
Same idea as for CPC but for FCI instead.
2.
KPC
1.
The PC algorithm typically uses independence tests that assume
linearity.
2.
The KPC algorithm makes two changes:
1.
It uses a non-parametric independence test.
2.
It adds some steps to orient edges that are unoriented in the PC
pattern.
Special Variants of Algorithms
1.
PcLiNGAM
1.
If some variables are Gaussian (more than one), others nonGaussian, this algorithm applies.
2.
Runs PC, then orients the unoriented edges (if possible) using nonGaussianity.
2.
LiNG
1.
Extends LiNGAM to orient cycles using non-Gaussianity
Special Variants of Algorithms
1.
JCPC
1.
Uses a Markov blanket style test to add/remove individual edges,
using CPC style orientation.
2.
Allows individual adjacencies in the graph to be revised from the initial
estimate using the PC adjacency search.
Simulation Studies with Tetrad
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