Table 1:

Summary of the assumptions for our mixed-LiNGAM model

$\text{Model:}{x}_l^{(i)}={\mu }_l+{\tilde{\mu }}_l^{(i)}+{\sum }_{k(m)<k(l)}{b}_{lm}{x}_m^{(i)}+e_l^{(i)}(l,m=1,2;l\ne m),$ where blm are non-zero.

$e_l^{(i)}$ (l = 1, 2; i = 1, ..., n) are i.i.d..

$e_l$ (l = 1, 2) are mutually independent.

$e_l$ (l = 1, 2) follow Laplace distributions with zero mean and standard deviations $|h_l|$

Prior distributions:

μl, blm and hl (l = 1, 2; m = 1, 2; l ≠ m) follow Gaussian distributions with zero mean and variance ${\tau }_{\mu l}^{cmmn},{\tau }_{b_{lm}}^{\mathrm{cmmn}},\text{and}{\tau }_{h_l}^{cmmn}$

${\tilde{\mu }}_l^{(i)}(l=1,2;i=1,\cdots ,n)$ are the sum of latent confounders $f_q^{(i)}:\sum _{q=1}^Q{\lambda }_{lq}{f}_q^{(i)}$ and are independent of $e_l^{(i)}$.

${\tilde{\mu }}_l^{(i)}$ (l = 1, 2; i = 1, ..., n) are i.i.d..

μl (l = 1, 2) follow multivariate t-distributions with v degrees of freedom, zero mean, variances ${\tau }_l^{indvdl}$ and correlation σ12 (here, v = 6).

Hyper-parameters:

${\tau }_{{\mu }_l}^{\mathrm{cmmn}},{\tau }_{b_{lm}}^{cmmn}\text{and}{\tau }_{h_l}^{cmmn}$ (l = 1, 2; m = 1, 2; l ≠ m) are set to be large values so that the priors are not very informative.

${\tau }_l^{indvdl}$ (l = 1, 2) are uniformly varied from zero to large values.

σ12 are uniformly varied in the interval between −0.9 and 0.9.