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Continuous additive-noise models on finite DAGs
Definition
Fix a nonempty finite directed acyclic graph with a specified topological order ; every arrow points from an earlier to a later vertex. For , let be its parent set, and let be the vertices other than that are not descendants of . A continuous additive-noise model on consists of real random variables on a common probability space and continuous Borel functions such that recursively in the topological order. The finite family of noises is mutually independent, and each has a strictly positive Lebesgue density . All vertices of are observed; no latent vertices are omitted from the model. The observational law is the law of the vector .
The model is causally minimal when every parent input is essential: for each edge , there are values of the other parent coordinates and two distinct real values of the -coordinate at which takes different values. A root has no parent-coordinate minimality condition.
Definition references
Recursive structural causal models on finite DAGs
Descendants and nondescendants in a finite DAG
Mutually independent random elements
The law of the observed random vector
Peters et al., JMLR (2014), Definition 16 and Proposition 17
Peters, Janzing, and Schölkopf, Elements of Causal Inference, §4.1.4
Assaad, Devijver, and Gaussier, course notes (2023), slides 22–23 and 29–31
Depends on
Used by
- RESIT: regression with subsequent independence testing Algorithm
- Condition 19 and restricted additive-noise models Definition
- A reversible linear-Gaussian additive-noise model Example
- An identifiable nonlinear ANM with a cubic cause-to-effect function Example
- Latent confounding can mimic an observed ANM direction Example
- An ANM density factorization implies conditional independence under d-separation Lemma
- An ANM noise is independent of every set of nondescendants Lemma
- Descendants of a youngest child are conditionally independent of its parent Lemma
- Distinct positive minimal ANM DAGs contain a conditionally reversible edge Lemma
- Parent dependence persists under nondescendant conditioning Lemma
- Positive joint density of a finite additive-noise model Lemma
- Bivariate additive-noise direction identifiability under Condition 19 Theorem
- Population identifiability and RESIT correctness for restricted ANMs Theorem
Dependency tree · two levels
11 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Jonas Peters, Joris Mooij, Dominik Janzing, and Bernhard Schölkopf, Causal Discovery with Continuous Additive Noise Models, JMLR 15 (2014), Definition 16 and Proposition 17 (standard reference, not scraped)
- Jonas Peters, Dominik Janzing, and Bernhard Schölkopf, Elements of Causal Inference, §4.1.4, Definition 4.4 (standard reference, not scraped)
- K. Assaad, E. Devijver, and G. Gaussier, Causal discovery: noise-based methods, slides 22–23 and 29–31 (standard reference, not scraped)