

![0_image_0.png](0_image_0.png)

(2) x = Bx + e, where B **can be permuted by simultaneous equal row and column permutations to** be lower block-triangular due to the acyclicity of disjoint subsets in chain graphs [14, 2]. Moreover, if we represent the model (2) as

## (3) X = Ae,

the matrix A **(:= (**I − B)
−1**) (called a mixing matrix) also becomes lower blocktriangular (and with all unities in the diagonal). Note that, in the case** of m = p, i.e., the DAG case, the model (3) defines the independent component **analysis**
(ICA) model [5] since the components of e **are independent and non-Gaussian.**
Since the ICA model is identifiable, the model (3) in this case (m = p**) is also**
identifiable, which is the key idea of the original LiNGAM algorithm [11] (we call it ICA-LiNGAM in the later part of this paper).

Now, let us consider an illustrative example in which the model is represented by (cf. Figure 1 (a))
x1 = e1,

$$\begin{array}{l}{{x_{2}=b_{21}x_{1}+e_{2},}}\\ {{x_{3}=b_{32}x_{2}+e_{3},}}\\ {{x_{4}=b_{42}x_{2}+b_{43}x_{3}+e_{4},}}\\ {{x_{5}=b_{51}x_{1}+b_{54}x_{4}+e_{5},}}\end{array}$$

where unobserved confounders f and g exist between e1 and e2 and between e4 and e5**, respectively, as**
e1 = c1f + d1 and e2 = c2f + d2,

## And

e4 = c4g + d4 and e5 = c5g + d5.

d1, d2, d4 and d5 are independent of each other. Note that, in this case, the assumption for the LiNGAM algorithms, i.e.**, exogenous influences are independent of each**
other, is not satisfied. In fact, x1 and x4 depend on x2 and x5**, respectively, because**
f and g **are not observed, and a DAG representation is no longer appropriate to** apply. The ordered blocks for the example (4) are K(1) = {1, 2}, K(2) = {3} and K(3) = {4, 5} (cf. Figure 1 (b)).

(4)