Recent Advanced in Causal Modelling Using Directed Graphs

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Causal Inference with Latent Variables
Richard Scheines David Danks (Carnegie
Mellon) Frederick Eberhardt (Carnegie
Mellon) Ricardo Silva (Gatsby, London)
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Graphical Models

• Non-parametric Representation of Causation
• X direct cause of Y relative to Z ---- X ?
  Y
• Intervention and manipulation represented
  explicitly
• Causal Inference
• From data and background knowledge -- find the
  set of indistinguishable causal structures
• Parametric Causal Models Structural Equation
  Models, Bayes Networks

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Representing Causal Systems Causal Graphs

• Causal Graph G V,E
• Each edge X ? Y represents a direct causal
  claim
• X is a direct cause of Y relative to V

Chicken Pox
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Interventions Causal Graphs

• Model an ideal intervention by adding an
  intervention variable outside the original
  system as a direct cause of its target.

Intervene to treat Infection
Post-intervention graph
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Causation Semantics vs. Epistemology

• Semantics what do we mean by X ? Y, or X Y
• Epistemology when do we know that X ? Y, or X
  Y

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Semantics of Causation

• Choice 1 Causal systems over V ? Probabilistic
  Independence Relations in P(V)
• Choice 2 Define X ? Y, or X Y in terms of
  intervention, i.e., (hypothetical) treatment)

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Choice 1 Causal Markov Axiom

• If G is a causal graph, and P a probability
  distribution over the variables in G, then in P
• every variable V is independent of its
  non-effects, conditional on its immediate causes.

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Choice 2 Define Causation from Manipulation

• X is a cause of Y iff
• ?x1 ? x2 P(Y X set x1) ? P(Y X set x2)
• Causation is asymmetric X Y ? Y X

X and Y are associated (X __ Y) iff ?x1 ? x2
P(Y X x1) ? P(Y X x2) Association is
symmetric X __ Y ? Y __ X
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Choice 2 Define Direct Causation from
Intervention

• X is a direct cause of Y relative to S, iff
• ?z,x1 ? x2 P(Y X set x1 , Z set z)
• ? P(Y X set x2 , Z set z)
• where Z S - X,Y

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Epistemology of Causal DiscoveryStatistical Data
? Causal Structure
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Equivalence Classes

• Patterns No unmeasured confounders, no
  feedback
• Partial Ancestral Graphs (PAGs) unmeasured
  confounders allowed
• Psychometric Models

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Patterns
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PAGs Partial Ancestral Graph
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Causal Epistemology

• Causal knowledge is
• Features of a causal structure shared by every
  member of the equivalence class

• Examples

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Three Problems for Causal Discovery with latents

• How can we use sequences of interventions
  optimally when there might be latents?
• How can we patch together knowledge from
  disparate datasets that overlap?
• How can we search over SEMs, in particular
• search for an appropriate measurement model, and
  
• search for the structural causal model

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Experimental vs. Statistical Control
Does X1 directly influence X3?
Experimental Control
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Searching for Causal SEMs
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Latents and Measurement Models
Does Lead harm Cognitive Function?
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Latents and Scales
ScalesLeadScale (X1 X2 X3 ) / 3
CogScale (X4 X5 X6) / 3HomeScale X7
X8 X9 ) / 3
Even without T, If rHome,HomeScale lt 1.0,
thenLeadScale __ CogScale HomeScale

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References

• Causation, Prediction, and Search, 2nd Edition,
  (2000), by P. Spirtes, C. Glymour, and R.
  Scheines ( MIT Press)
• Causality Models, Reasoning, and Inference,
  (2000), Judea Pearl, Cambridge Univ. Press
• Computation, Causation, Discovery (1999),
  edited by C. Glymour and G. Cooper, MIT Press
• Causality in Crisis?, (1997) V. McKim and S.
  Turner (eds.), Univ. of Notre Dame Press.
• TETRAD IV www.phil.cmu.edu/projects/tetrad