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Condition 19 and restricted additive-noise models

Definition

For a smooth bivariate additive-noise model $Y = f (X) + N$ , let $f$ be three times continuously differentiable, and let $X$ and $N$ have strictly positive three-times continuously differentiable densities $p_{X}$ and $p_{N}$ . Put $ξ (x) = \log p_{X} (x)$ and $ν (z) = \log p_{N} (z)$ . The triple $(f, p_{X}, p_{N})$ satisfies Condition 19 when there are $x, y \in R$ with $ν^{''} (y - f (x)) f^{'} (x) \neq 0$ at which $ξ^{'''} (x) \neq ξ^{''} (x) (- \frac{ν^{'''} (y - f (x)) f^{'} (x)}{ν^{''} (y - f (x))} + \frac{f^{''} (x)}{f^{'} (x)}) - 2 ν^{''} (y - f (x)) f^{''} (x) f^{'} (x) + ν^{'} (y - f (x)) f^{'''} (x) + \frac{ν^{'} (y - f (x)) ν^{'''} (y - f (x)) f^{''} (x) f^{'} (x)}{ν^{''} (y - f (x))} - \frac{ν^{'} (y - f (x)) (f^{''} (x))^{2}}{f^{'} (x)} .$ Here every derivative of $ν$ is evaluated at $y - f (x)$ and every derivative of $f$ at $x$ . The nonzero-domain condition makes both denominators in this formula well defined.

A finite additive-noise model is a restricted additive-noise model when every structural function $f_{j}$ is $C^{3}$ and every noise has a strictly positive $C^{3}$ Lebesgue density, and when for every vertex $j$ , every parent $i \in pa (j)$ , and every set $S$ satisfying $pa (j) ∖ {i} \subseteq S \subseteq nd (j) ∖ {i},$ there is a value $s$ with $0 < p_{S} (s) < \infty$ such that the conditional law of $X_{i}$ given $X_{S} = s$ , together with the structural slice $u \mapsto f_{j} (s_{pa (j) ∖ {i}}, u)$ and the noise $N_{j}$ , is a smooth bivariate additive-noise model satisfying Condition 19. The conditional density is the ratio $p_{X_{i}, X_{S}} (u, s) / p_{S} (s)$ for the finite positive-density version supplied by the joint-density formula. The structural slice and conditional input density have the smoothness and strict positivity required in the preceding Condition 19 definition. If the model has no edges, the restricted condition is vacuous.

Definition references

Finite continuous additive-noise models and causal minimality

Strictly positive joint density and its finite conditional-density versions

Peters et al., JMLR (2014), Definitions 18/27 and Condition 19

Peters, Janzing, and Schölkopf, Elements of Causal Inference, §4.1.4 and Theorem 4.5

Assaad, Devijver, and Gaussier, course notes (2023), slides 22–23

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Sources

Jonas Peters, Joris Mooij, Dominik Janzing, and Bernhard Schölkopf, Causal Discovery with Continuous Additive Noise Models, Definitions 18/27 and Condition 19 (standard reference, not scraped)
Jonas Peters, Dominik Janzing, and Bernhard Schölkopf, Elements of Causal Inference, §4.1.4 and Theorem 4.5 (standard reference, not scraped)
K. Assaad, E. Devijver, and G. Gaussier, Causal discovery: noise-based methods, ENS Lyon course notes (2023), slides 22–23 (standard reference, not scraped)