SupplementaryMaterial: IdentificationofPartially ObservedLinearCausalModels

Letnbe the dimensionality of X. Remeber p is the number of noise terms. In the case wheren = p, MX in (4) is a square matrix, and its identifiability fromXup tocolumn rescaling and permutations has been provided by Matsuoka et al. [1], but we are concerned with the case wheren < p. Lemma 5. Suppose matrixK Rn n has linearly independent columns, i.e.,Rank(K) = n. Let K = K d 1|, where d Rn and 1 is the length-n vector of all 1's. Further assume that eachncolumns ofMX are linearlyindependentandthatp 2n 2. ThenifXadmitsamodel X= MX ε, (16) where εalso follows the above assumption onε, every column of MX must be proportional to a columnofMX andviceversa. Let σ2ti be the variance of εi in the tth domain.