Statistical Decidability in Confounded, Linear Non-Gaussian Models

Since Spirtes et al. (2000), it is well known that if causal relationships are linear and noise terms are independent and Gaussian, causal orientation is not identified from observational data — even if causal faithfulness is satisfied. Shimizu et al. (2006) showed that linear, non-Gaussian (LiNGAM) causal models are identified from observational data, so long as no latent confounders are present. That holds even when faithfulness fails. Genin and Mayo-Wilson (2020) refine that identifiability result: not only are causal relationships identified, but causal orientation is statistically decidable. That means that for every α > 0, there is a method that converges in probability to the correct orientation and, at every sample size, outputs an incorrect orientation with probability less than α.These results naturally raise questions about what happens in the presence of latent confounders. Hoyer et al. (2008) and Salehkaleybar et al. (2020) show that, although the causal model is not uniquely identified, causal orientation among observed variables is identified in the presence of latent confounders, so long as faithfulness is satisfied. This paper refines these results. When we allow for the presence of latent confounders, causal orientation is no longer statistically decidable. Although it is possible to converge in probability to the correct orientation, it is not possible to do so with finite-sample bounds on the probability of orientation errors. That is true even if causal faithfulness is satisfied.