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Probabilities of causation: Bounds and identification
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The expression in (54) provides a lower bound for PN in the general, nonmonotonic case.
“It can also be shown (Tian and Pearl, 2000) that the expression in (54) provides a lower bound for PN in the general, nonmonotonic case.”
From Causal inference in statistics: An overview (2009) · cites this paper as Tian and Pearl, 2000
The tight upper and lower bounds on PN are given by Eq. (55).
“In particular, the tight upper and lower bounds on PN are given by:”
From Causal inference in statistics: An overview (2009) · cites this paper as Tian and Pearl, 2000
The algebraic machinery of counterfactual notation can be powerful in refining assumptions, deriving consistent estimands, bounding probabilities of necessary and sufficient causation, and combining experimental and nonexperimental data.
“the algebraic machinery offered by the counterfactual notation, Y x (u), once a problem is properly formalized, can be extremely powerful in refining assumptions (Angrist et al., 1996, Heckman andVytlacil, 2005), deriving consistent estimands (Robins, 1986), bounding probabilities of necessary and sufficient causation (Tian and Pearl, 2000), and combining data from experimental and nonexperimental”
From An Introduction to Causal Inference (2010) · cites this paper as Tian and Pearl, 2000
The tight upper and lower bounds on PN are given by max(0, (P(y)-P(y|do(x0)))/P(x,y)) ≤ PN ≤ min(1, (P(y|do(x1))-P(x1,y))/P(x,y)).
“In particular, the tight upper and lower bounds on PN are given by: max 0, P(y)-P(y|do(x )) P(x,y) ≤PN ≤min 1, P(y |do(x ))-P(x ,y ) P(x,y)”
From An Introduction to Causal Inference (2010) · cites this paper as Tian and Pearl, 2000
The expression in (61) provides a lower bound for PN in the general, nonmonotonic case.
“It can also be shown (Tian and Pearl, 2000) that the expression in (61) provides a lower bound for PN in the general, nonmonotonic case.”
From An Introduction to Causal Inference (2010) · cites this paper as Tian and Pearl, 2000
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Retrospective causal inference with multiple effect variables
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Bounds on controlled direct effects under monotonic assumptions about mediators and confounders
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Metadata
- Authors
- Tian, J., & Pearl, J.
- Affiliations
- University of California, Los Angeles (US)
- ORCID
- Tian, Jin
- Venue
- Annals of Mathematics and Artificial Intelligence
- Published
- 2000-10
- Volume
- 28
- Issue
- 1-4
- Pages
- 287-313
- ISSN
- 1012-2443 (ISSN-L)
- 1573-7470
- OpenAlex
- W1498383682
- Cited by
- 179
- References
- 50
- Citations by year
- 2026: 12 · 2025: 18 · 2024: 18 · 2023: 18 · 2022: 18 · 2021: 15 · 2020: 6 · 2019: 10 · 2018: 4 · 2017: 4 · 2016: 8 · 2015: 3 · 2014: 8 · 2013: 7 · 2012: 3
- FWCI
- 1.66
- Type
- journal article
- Language
- en
- Open access
- green
- arxiv.org
- Licence
- Locations
- Annals of Mathematics and Artificial Intelligence (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- eScholarship (California Digital Library) (submittedVersion)
- arXiv (Cornell University)
- Department of Statistics, UCLA
- Funders
- National Science Foundation
- Office of Naval Research
- Air Force Office of Scientific Research
- Topics
- Physical Sciences › Mathematics › Statistics and Probability › Advanced Causal Inference Techniques
- Physical Sciences › Computer Science › Artificial Intelligence › Bayesian Modeling and Causal Inference
- Social Sciences › Arts and Humanities › History and Philosophy of Science › Philosophy of Biology and Scientific Explanation
- Keywords
- probability of causation, sharp bounds, causal inference, exogeneity, monotonicity, attribution, structural causal models
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
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