Recover error terms matrix for binary data using LiM

[marcoBmota8]

Great package, congratulations!

I was loooking at the LiM code and paper and I have not figured out a way to recover the continuous independent error terms for each observed binary variable. Is there a way for LiM to retrieve the matrix E where columns are independent?

Best,

[sshimizu2006]

I think that for discrete variables, E is not uniquely determined, so it might be difficult to recover E.
As shown in the example below, using semopy to check the model fit could be one possible option.
However, this would be an evaluation from the viewpoint of the covariance matrix.

https://github.com/cdt15/lingam/blob/master/examples/Example_to_analyze_clinical_data.ipynb

[marcoBmota8]

Yeah, the binarization of the latent utility is a lossy process and removes individual identifiability. However, I think the error distribution can still be asymptotically aproximated.
@sshimizu2006 what do you think about using data augmentation using Gibbs sampling from a logistic model?
That would entail sample $e_i$ from a truncated logistic distribution $truncated$ $logistic\sim(\mu, 1.0, a, b)$, where $\mu,$ $a,$ & $b$ are chosen based on $x_i$ and $\eta_i$. $\eta$ being the contribution of the parents and $a, b$ the lower and upper limits for the truncated logistic distribution.

[sshimizu2006]

I haven’t tried Gibbs sampling myself, so I’m not sure — but it might work well.
However, since the parameter values in LiM are estimated from data (and from the learned structure), the sampled errors may not end up fully independent. It might be necessary to relax or adjust the independence criterion when checking it. Probably a bit looser would make sense.