Table 1

High-level description of the tested causal orientation methods.

Method

Reference

Key principles

Sufficient assumptions for causally orienting X → Y

Sound


ANM

[14]

Assuming X → Y with Y = f(X) + e1, where X and e1 are independent, there will be no such additive noise model in the opposite direction X ← Y, X = g(Y) + e2, with Y and e2 independent.

• Y = f(X) + e1;

• X and e1 are independent;

• f is non-linear, or one of X and e is non-Gaussian;

• Probability densities are strictly positive;

• All functions (including densities) are 3 times differentiable.

Yes

PNL

[15]

Assuming X → Y with Y = f2(f1(X) + e1), there will be no such model in the opposite direction X←Y, X = g2(g1(Y) + e2) with Y and e2 independent.

• Y = f2(f1(X) + e1);

• X and e1 are independent;

• Either f1 or e1 is Gaussian;

• Both f1 and f2 are continuous and invertible.

Yes

IGCI

[16,17]

Assuming X→Y with Y = f(X), one can show that the KL-divergence (a measure of the difference between two probability distributions) between P(Y) and a reference distribution (e.g., Gaussian or uniform) is greater than the KL-divergence between P(X) and the same reference distribution.

• Y = f(X) (i.e., there is no noise in the model);

• f is continuous and invertible;

• Logarithm of the derivative of f and P(X) are not correlated.

Yes

GPI-MML

[18]

Assuming X→Y, the least complex description of P(X, Y) is given by separate descriptions of P(X) and P(Y|X). By estimating the latter two quantities using methods that favor functions and distributions of low complexity, the likelihood of the observed data given X→Y is inversely related to the complexity of P(X) and P(Y | X).

• Y = f(X, e);

• X and e are independent;

• e is Gaussian;

• The prior on f and P(X) factorizes.

No

ANM-MML

[18]

Same as for GPI-MML, except for a different way of estimating P(Y | X) and P(X | Y).

• Y = f(X) + e;

• X and e are independent;

• e is Gaussian.

• The prior on f and P(X) factorizes.

No

GPI

[18]

Assuming X→Y with Y = f(X,e1), where X and e1 are independent and f is "sufficiently simple", there will be no such model in the opposite direction X←Y, X = g(Y,e2) with Y and e2 independent and g "sufficiently simple".

Same as for GPI-MML.

No

ANM-GAUSS

[18]

Same as for ANM-MML, except for the different way of estimating P(X) and P(Y).

Same as for ANM-MML.

No

LINGAM

[13]

Assuming X→Y, if we fit linear models Y = b2X+e1 and X = b1Y+e2 with e1 and e2 independent, then we will have b1 < b2.

• Y = b2X+e1;

• X and e1 are independent;

• e1 is non-Gaussian.

Yes


The last column indicates whether a method is sound, i.e. it can provably orient a causal structure under its sufficient assumptions. Because causal orientation methodologies are fairly new and not completely characterized, it is possible that proofs of correctness will become available for GPI-MML, ANM-MML, GPI, and ANM-GAUSS. All methods implicitly assume that there are no feedback loops. The noise term in the models is denoted by small "e".

Statnikov et al. BMC Genomics 2012 13(Suppl 8):S22   doi:10.1186/1471-2164-13-S8-S22

Open Data