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Causal discovery for linear cyclic models with latent variables

Remark. At the end of section 2 we mention the existence of special circumstances when it is possible to identify some direct effects b(• → x j) even when it is not the case that all pair conditions of the form (•, x j) are satisfied. Here is an example. Consider a model for V = {x1, x2, x3}where a single experiment with Jm = {x1} has been performed. The pair condition is satisfied by this experiment for the pairs (x1, x2) and (x1, x3). One of the equations the experiment yields is t(x1 x3||Jm) = t(x1 x2||Jm) b(x2 → x3) + b(x1 → x3). If it now turns out that the measured experimental effect t(x1 x2||Jm) = 0, then the direct effect b(x1 → x3) is determined although the pair condition for the pair (x2, x3) is not satisfied. Lemma 1. Let the true model generating the data be (B,Σe). For each of the experiments (Em)m=1,...,M the obtained data covariance matrix is Σx . If there is a direct effects matrix B̂ , B such that for all (Em)m=1,...,M and all xi ∈ Jm and x j ∈ Um it produces the same experimental effects t(xi x j || Jm), then the model (B̂, Σ̂e) with Σ̂e = (I − B̂)(I − B)−1Σe(I − B)−T (I − B̂)T has data covariance matrices Σ̂x = Σx for all m = 1, ...,M.