1 day 2: search june 9, 2015 carnegie mellon university center for causal discovery
TRANSCRIPT
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Day 2: Search
June 9, 2015
Carnegie Mellon University
Center for Causal Discovery
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Outline
Day 2: Search
1. Bridge Principles: Causation Probability
2. D-separation
3. Model Equivalence
4. Search Basics (PC, GES)
5. Latent Variable Model Search (FCI)
6. Examples
2
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Tetrad Demo and Hands On
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Tetrad Demo and Hands-on
1) Go to “estimation2”
2) Add Search node (from Data1)
- Choose and execute one of the
“Pattern searches”
3) Add a “Graph Manipulation” node to search
result: “choose Dag in Pattern”
4) Add a PM to GraphManip
5) Estimate the PM on the data
6) Compare model-fit to model fit for true model
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Backround KnowledgeTetrad Demo and Hands-on
1) Create new session
2) Select “Search from Simulated Data” from Template menu
3) Build graph below, PM, IM, and generate sample data N=1,000.
4) Execute PC search, a = .05
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Backround KnowledgeTetrad Demo and Hands-on
1) Add “Knowledge” node – as below
2) Create “Required edge X3 X1 as shown below.
3) Execute PC search again, a = .05
4) Compare results (Search2) to previous search (Search1)
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Backround KnowledgeTetrad Demo and Hands-on
1) Add new “Knowledge” node
2) Create “Tiers” as shown below.
3) Execute PC search again, a = .05
4) Compare results (Search2) to previous search (Search1)
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Backround KnowledgeDirect and Indirect Consequences
True Graph
PC Output
Background Knowledge
PC Output
No Background Knowledge
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Backround KnowledgeDirect and Indirect Consequences
True Graph
PC Output
Background Knowledge
PC Output
No Background Knowledge
Direct Consequence
Of Background Knowledge
Indirect Consequence
Of Background Knowledge
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Charitable Giving (Search)
1) Load in charity data
2) Add search node
3) Enter Background Knowledge:
• Tangibility is exogenous
• Amount Donated is endogenous only
• Tangibility Imaginability is required
4) Choose and execute one of the
“Pattern searches”
5) Add a “Graph Manipulation” node to
search result: “choose Dag in Pattern”
6) Add a PM to GraphManip
7) Estimate the PM on the data
8) Compare model-fit to hypothetical model
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1) Adjacency phase
2) Orientation phase
Constraint-based Search for Patterns
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Constraint-based Search for Patterns: Adjacency phase
X and Y are not adjacent if they are independent conditional on any subset that doesn’t X and Y
1) Adjacency
• Begin with a fully connected undirected graph
• Remove adjacency X-Y if X _||_ Y | any set S
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X1
X2
X3 X4
Causal Graph
Independcies
Begin with:
From
X1
X2
X3 X4
X1 X2
X1 X4 {X3}
X2 X4 {X3}
X1
X2
X3 X4
X1
X2
X3 X4
X1
X2
X3 X4
From
From
X1 X2
X1 X4 {X3}
X2 X4 {X3}
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2) Orientation • Collider test:
Find triples X – Y – Z, orient according to whether the set that separated X-Z contains Y
• Away from collider test: Find triples X Y – Z, orient Y – Z connection via collider test
• Repeat until no further orientations
• Apply Meek Rules
Constraint-based Search for Patterns: Orientation phase
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Search: Orientation
Patterns
Y Unshielded
X Y Z
Test: X _||_ Z | S, is Y S
Yes
Non-Collider
X Y Z
X Y Z
X Y Z
X Y Z
Collider
X Y Z
No
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Search: Orientation
X1 _||_ X3 | S, X2 S
No
X3
X2
X1
Test
Away from Collider
X3
X2
X1
1) X1 - X2 adjacent, and into X2. 2) X2 - X3 adjacent 3) X1 - X3 not adjacent
Test Conditions
Yes
X3
X2
X1
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Search: Orientation
X4 X3
X2
X1
X1 _||_ X4 | X3
X2 _||_ X4 | X3
After Adjacency Phase
X1 _||_ X2
Collider Test: X1 – X3 – X2
Away from Collider Test:
X1 X3 – X4 X2 X3 – X4
X4 X3
X2
X1
X4 X3
X2
X1
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Away from Collider Power!
X1 _||_ X3 | S, X2 S
X3 X2 X1
X3 X2 X1
X2 – X3 oriented as X2 X3
Why does this test also show that X2 and X3 are not confounded?
X3 X2 X1
X3 X2 X1
C
X1 _||_ X3 | S, X2 S
X1 _||_ X3 | S, X2 S, C S
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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
• PAGs: (Richardson 1994) graphical representation of a d-separation equivalence class that includes models with latent common causes and sample selection bias that are d-separation equivalent over a set of measured variables X
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PAGs: Partial Ancestral Graphs
X2
X3
X1
X2
X3
Represents
PAG
X1 X2
X3
X1
X2
X3
T1
X1
X2
X3
X1
etc.
T1
T1 T2
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PAGs: Partial Ancestral Graphs
Z2
X
Z1
Z2
X3
Represents
PAG
Z1 Z2
X3
Z1
etc.
T1
Y
Y Y
Z2
X3
Z1 Z2
X3
Z1
T2
Y Y
T1
T1
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PAGs: Partial Ancestral Graphs
X2 X1
X2 X1
X2 X1
X2 There is a latent commoncause of X1 and X2
No set d-separates X2 and X1
X1 is a cause of X2
X2 is not an ancestor of X1
X1
X2 X1 X1 and X2 are not adjacent
What PAG edges mean.
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PAG Search: Orientation
PAGs
Y Unshielded
X Y Z
X _||_ Z | YX _||_ Z | Y
Collider Non-Collider
X Y Z X Y Z
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PAG Search: Orientation
X4 X3
X2
X1
X1 _||_ X4 | X3
X2 _||_ X4 | X3
After Adjacency Phase
X1 _||_ X2
Collider Test: X1 – X3 – X2
Away from Collider Test:
X1 X3 – X4 X2 X3 – X4
X4 X3
X2
X1
X4 X3
X2
X1
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Interesting Cases
X Y Z
L
X
Y
Z2
L1
M1M2
M4
Z1L2
X1
Y2
L1
Y1
X2
X Y
Z1
M3Z2
L
L
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Tetrad Demo and Hands-on
1) Create new session
2) Select “Search from Simulated Data” from Template menu
3) Build graphs for M1, M2, M3 “interesting cases”, parameterize,
instantiate, and generate sample data N=1,000.
4) Execute PC search, a = .05
5) Execute FCI search, a = .05X Y Z
LM1
M2
X1
Y2
L1
Y1
X2L
X Y
Z1M3
Z2
L
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Regression &
Causal Inference
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Regression & Causal Inference
2. So, identifiy and measure potential confounders Z:
a) prior to X,
b) associated with X,
c) associated with Y
Typical (non-experimental) strategy:1. Establish a prima facie case (X associated with Y)
3. Statistically adjust for Z (multiple regression)
X Y
Z
But, omitted variable bias
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Regression & Causal Inference
Multiple regression or any similar strategy is provably unreliable for causal inference regarding X Y, with covariates Z, unless:
• X prior to Y
• X, Z, and Y are causally sufficient (no confounding)
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Tetrad Demo and Hands-on
1) Create new session
2) Select “Search from Simulated Data” from Template menu
3) Build a graph for M4 “interesting cases”, parameterize as SEM, instantiate,
and generate sample data N=1,000.
4) Execute PC search, a = .05
5) Execute FCI search, a = .05
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Summary of Search
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Causal Searchfrom Passive Observation
• PC, GES Patterns (Markov equivalence class - no latent confounding)
• FCI PAGs (Markov equivalence - including confounders and selection bias)
• CCD Linear cyclic models (no confounding)
• BPC, FOFC, FTFC (Equivalence class of linear latent variable models)
• Lingam unique DAG (no confounding – linear non-Gaussian – faithfulness not
needed)
• LVLingam set of DAGs (confounders allowed)
• CyclicLingam set of DGs (cyclic models, no confounding)
• Non-linear additive noise models unique DAG
• Most of these algorithms are pointwise consistent – uniform consistent algorithms
require stronger assumptions
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Causal Searchfrom Manipulations/Interventions
• Do(X=x) : replace P(X | parents(X)) with P(X=x) = 1.0
• Randomize(X): (replace P(X | parents(X)) with PM(X), e.g., uniform)
• Soft interventions (replace P(X | parents(X)) with PM(X | parents(X), I), PM(I))
• Simultaneous interventions (reduces the number of experiments required to be
guaranteed to find the truth with an independence oracle from N-1 to 2 log(N)
• Sequential interventions
• Sequential, conditional interventions
• Time sensitive interventions
What sorts of manipulation/interventions have been studied?
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Tetrad Demo and Hands-on
1) Search for models of Charitable Giving
2) Search for models of Foreign Investment
3) Search for models of Lead and IQ