Fig. 1

Inference of v-structures versus non-v-structures by 3-point information from observational data. a Isolated v-structures are predicted for I(x;y;z) < 0, and (b–d) isolated non-v-structures for I(x;y;z) > 0. e Generalized v-structures are predicted for I(x;y;z|{u _i }) < 0 and (f–h) generalized non-v-structures for I(x;y;z|{u _i }) > 0. In addition, as I(x;y;z|{u _i }) are invariant upon xyz permutations, the global orientation of v-structures and non-v-structures also requires to find the most likely base of the xyz triple. Choosing the base xy with the lowest conditional mutual information, i.e., I(x;y|{u _i })= minxyz(I(s;t|{u _i })), is found to be consistent with the Data Processing Inequality expected for (generalized) non-v-structures in the limit of infinite dataset, see main text. In practice, given a finite dataset, the inference of (generalized) v-structures versus non-v-structures can be obtained by replacing 3-point and 2-point information terms I(x;y|{u _i }) and I(x;y;z|{u _i }) by shifted equivalents, I ^′(x;y|{u _i }) and I ^′(x;y;z|{u _i }), including finite size corrections, see text (Eqs. 23 & 24)