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An identifiable nonlinear ANM with a cubic cause-to-effect function

Example

Assume AC. Let $X, N$ be independent standard normal random variables and set $Y = X^{3} + N$ . The structural function is nonconstant and $C^{3}$ , and both noise densities are strictly positive and $C^{3}$ . For $f (x) = x^{3}$ and standard normal input and noise, the triple satisfies Condition 19: at $(x, y) = (1, 0)$ its left side is $0$ and its right side is $28$ . The bivariate direction-identifiability theorem therefore excludes every reverse additive-noise representation with an independent residual and a $C^{3}$ structural function.

Example references

The bivariate additive-noise equation and causal minimality

The Condition 19 equation and its nonzero domain

The bivariate identifiability conclusion under Condition 19

The positive forward joint density

Peters et al., JMLR (2014), Condition 19 and Theorem 20

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

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

Facts & Assumptions

Given: Assume AC and take independent $X, N \sim N (0, 1)$ with standard normal density $ϕ (t) = (2 π)^{- 1 / 2} e^{- t^{2} / 2}$ .

[A1]

AC is the Axiom of Choice and implies countable choice (AC $_{ω}$ ), the additional axiom interface used by the standard-normal suppliers. The Axiom of Choice, The Axiom of Countable Choice ( ${A C}_{ω}$ ), AC implies DC implies countable choice

[F1]

The standard normal law has density $ϕ$ , and this density is positive and normalized. Standard normal and normal laws, The standard normal density has total mass one

[F2]

For a smooth bivariate ANM, Condition 19 fails when its displayed equation holds at every point of the nonzero domain. Condition 19 and restricted additive-noise models

[F3]

If Condition 19 holds, no reverse ANM in the stated smooth class represents the same observational law. Bivariate additive-noise direction identifiability under Condition 19

[F4]

The bivariate forward joint density is the product of the input density and shifted noise density. Positive joint density of a finite additive-noise model

Verification

Given: The independent standard normal variables $X, N$ and $Y = X^{3} + N$ .

1.1givenA1F1F4construct

Under AC [A1], the equation is a bivariate additive-noise model with $f (x) = x^{3}$ , independent noise, strictly positive $C^{3}$ input and noise densities, and nonconstant $C^{3}$ function. By [F1] and [F4], its joint density is $p (x, y) = ϕ (x) ϕ (y - x^{3})$ .

1.2givenF1algebra

Put $z = y - f (x)$ . The standard normal log densities give $ξ^{''} (x) = - 1$ , $ξ^{'''} (x) = 0$ , $ν^{'} (z) = - z$ , $ν^{''} (z) = - 1$ , and $ν^{'''} (z) = 0$ . At $(x, y) = (1, 0)$ , $z = - 1$ , $f^{'} = 3$ , $f^{''} = 6$ , and $f^{'''} = 6$ , so $ν^{''} (z) f^{'} = - 3 \neq 0$ .

2.1F2F3step 1.2algebra

Substitution into Condition 19's right side gives $(- 1) (\frac{6}{3}) - 2 (- 1) (6) (3) + (1) (6) + 0 - (1) \frac{6^{2}}{3} = - 2 + 36 + 6 - 12 = 28,$ whereas $ξ^{'''} (1) = 0$ . Hence Condition 19 holds. The theorem [F3] now excludes a reverse ANM in its stated class, proving the Example's identifiability assertion.

3.1step 1.1step 1.2step 2.1∎

The input and noise densities, and the cubic function, satisfy the hypotheses in the Statement; the nonzero-domain witness in step 1.2 satisfies the Condition 19 inequality in step 2.1.

References

No reverse representation under Condition 19

Depends on

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Sources

Jonas Peters, Joris Mooij, Dominik Janzing, and Bernhard Schölkopf, Causal Discovery with Continuous Additive Noise Models, Condition 19 and Theorem 20 (standard reference, not scraped)
Jonas Peters, Dominik Janzing, and Bernhard Schölkopf, Elements of Causal Inference, §4.1.4, 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)