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Learning Causal Structure from
Undersampled Time Series
David DanksPhilosophy (& Psychology)
Carnegie Mellon
Sergey PlisMind Research Network
The Situation
Causal timescale Measurement timescale≠
~100 ms ~2 sec????
Unknown extent!
What causal inferences can be made in this situation?
Two Challenges
1. Forwards inference: Given a causal structure at causal timescale, what is implied structure at (undersampled) measurement timescale?
2. Backwards inference: Given inferred causal structure at measurement timescale (with unknown undersampling), what structures at causal timescale are possible?
Representation
Causal timescale Measurement timescale
undersample by 2
(Alternate, Better) Representation
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2 3
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(Alternate, Better) Representation
Causal timescale Measurement timescale
Forwards Inference
• Restatement: Given G1 and undersample rate u, what is Gu?
• Note: X → Y in Gu iff X → ... → Y of length u in G1
• Forwards inference = finding paths of particular lengths
• ⇒ Easy “black box” for forwards inference
• Special case Q: What is true about Gu for many different u?
Forwards Inference
• Undersampling destroys information:
Forwards Inference
• Strongly Connected Component (SCC): Maximal set of nodes S s.t. ∀X,Y ∃ path from X to Y
• SCCs are obviously cyclic• SCC ! set LS of simple loops
(i.e., no repeat nodes)• gcd(LS) := greatest common
divisor of simple loop lengths
Key notion!
3 Forward Theorems(importance of gcd...)
• When does Gu stabilize as u → ∞?
• When is SCC structure stable across all u?
• To what do SCCs converge (when they do)?
Backwards Inference
• Restatement: Given Gu, what are the possible <G1, u> pairs?
• Massive underdetermination for large u
• SCC-graph GS over nodes for SCCs := Si → Sj iff ∃Xi∈Si,Xj∈Sj Xi → Xj
• Encodes high-level between-SCC structure• Ignores where & how the SCCs connect
• Provably, always a DAG
Backwards Inference
• Constancy of SCC-graph:
• ⇒ Given G, we can efficiently recover SCCs & between-SCC structure in G1!
• Polynomial SCC identification algorithms
Backwards Inference
• What about within-SCC structure?• Super-clique ⇒ No internal information• Not-yet-super-clique ⇒ ????
• Reason for hope:
uniquely discoverable!
Fully-general learning algorithm still in development...
Thanks!
Research partially supported by: National Science Foundation
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