From: New methods for separating causes from effects in genomics data
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 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 = 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 | [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,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 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 = 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 non-Gaussian. | Yes |