Table 1 

Highlevel 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) + e_{1}, where X and e_{1 }are independent, there will be no such additive noise model in the opposite direction X ← Y, X = g(Y) + e_{2}, with Y and e_{2 }independent. 
• Y = f(X) + e_{1}; • X and e_{1 }are independent; • f is nonlinear, or one of X and e is nonGaussian; • Probability densities are strictly positive; • All functions (including densities) are 3 times differentiable. 
Yes 
PNL 
[15] 
Assuming X → Y with Y = f_{2}(f_{1}(X) + e_{1}), there will be no such model in the opposite direction X←Y, X = g_{2}(g_{1}(Y) + e_{2}) with Y and e_{2 }independent. 
• Y = f_{2}(f_{1}(X) + e_{1}); • X and e_{1 }are independent; • Either f_{1 }or e_{1 }is Gaussian; • Both f_{1 }and f_{2 }are continuous and invertible. 
Yes 
IGCI 
Assuming X→Y with Y = f(X), one can show that the KLdivergence (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 KLdivergence 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 

GPIMML 
[18] 
Assuming X→Y, the least complex description of P(X, Y) is given by separate descriptions of P(X) and P(YX). 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 
ANMMML 
[18] 
Same as for GPIMML, 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,e_{1}), where X and e_{1 }are independent and f is "sufficiently simple", there will be no such model in the opposite direction X←Y, X = g(Y,e_{2}) with Y and e_{2 }independent and g "sufficiently simple". 
Same as for GPIMML. 
No 
ANMGAUSS 
[18] 
Same as for ANMMML, except for the different way of estimating P(X) and P(Y). 
Same as for ANMMML. 
No 
LINGAM 
[13] 
Assuming X→Y, if we fit linear models Y = b_{2}X+e_{1 }and X = b_{1}Y+e_{2 }with e_{1 }and e_{2 }independent, then we will have b_{1 }< b_{2}. 
• Y = b_{2}X+e_{1}; • X and e_{1 }are independent; • e_{1 }is nonGaussian. 
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 GPIMML, ANMMML, GPI, and ANMGAUSS. 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:10.1186/1471216413S8S22 