SlowGuess/NeurIPS_2023_OCR huggingface dataset:
Results. In Fig. 4, we present identifiability scores and model fits for both well-specified and misspecified models (corresponding to different intervention target choices). Notably, we observe that the well-specified model (in blue) or its equivalent (in yellow) yield the highest log-likelihood in the majority of cases, as depicted in Fig. 4 (Right). This demonstrates that, even in this nonparametric setting without fully specified graph, the log-likelihood remains a reliable criterion for selecting the correct intervention targets. Fig. 4 (Left) shows that the selected models (blue or yellow) approximately identify the ground-truth latent variables up to element-wise rescaling, whereas other choices lead to much lower MCCs.
It is worth noting that, compared to the parametric setting investigated in § 6 and Fig. 3, the nonparametric setting appears to be more challenging (as expected), as there is a less pronounced distinction between well-specified and misspecified models, both in terms of identifiability scores and model fits. Moreover, future work is needed to parse the implicitly learned causal relationships in the transformation  $g^{\mathrm{SCM}}$  in (D.11): since only the (pre-imposed) causal order is specified, in practice,  $g^{\mathrm{SCM}}$  may learn to use additional or fewer edges than in the true graph  $G$ .
E Discussion of the Role of Our Assumptions
Below, we summarize the rationale and intuition behind each assumption:
- Asm. 2.2 helps rule out degenerate cases (cancellation along different paths) in which variables are (conditionally) independent despite being causally related. It is a standard assumption in classical causal discovery from observational data, and therefore also helps in CRL to recover the true causal graph.
- Asm. 2.3 is required to know how many latent variables we are looking for. It is a standard assumption in identifiable representation learning (that is often made implicitly). However, it may be dropped when suitable techniques for estimating the intrinsic dimensionality of  $\mathcal{X}$  can be employed.
- Asm. 2.5 is needed for the mapping between latents and observations to be invertible in the first place. Without it, full recovery of the causal variables (up to CRL equivalence) is infeasible. This assumption is also standard for the simpler problem of nonlinear ICA.
- Asm. 2.8 is a characterisation of our generative setup. Sharing of some mechanisms and the mixing function is needed for the multi-environment setting to provide useful additional information: if everything may change across environments, the datasets can only be analysed in isolation, running into the non-identifiability of CRL from iid data.
- Asm. 2.9 and (A2) / (A2') are needed since with imperfect interventions or interventions not on all nodes, identifiability is not achievable even in the linear setting as shown by Squires et al. [117].
- Asm. (A1) is a technical assumption needed for our analysis. It is not strictly necessary (it can also be relaxed to fully supported on a Cartesian product of intervals) but substantially eases the readability and accessibility of the proof, without a major impact on the main causal aspects of the problem setup.
- Asm. (A3) / (A3') is needed to avoid spurious solutions based on applying a measure preserving transformation on a part of the domain unaffected by the intervention.
- Asm. (A4) is needed to rule out a fine-tuning of the ground-truth generating process that are possible due to fully non-parametric nature of the setup, see also Remark 4.2 and the following paragraph.