One of the most common mistakes made when performing data analysis is attributing causal meaning to regression coefficients. Formally, a causal effect can only be computed if it is identifiable from a combination of observational data and structural knowledge about the domain under investigation (Pearl, 2000, Ch. 5). Building on the literature of instrumental variables (IVs), a plethora of methods has been developed to identify causal effects in linear systems. Almost invariably, however, the most powerful such methods rely on exponential-time procedures. In this paper, we investigate graphical conditions to allow efficient identification in arbitrary linear structural causal models (SCMs). In particular, we develop a method to efficiently find unconditioned instrumental subsets, which are generalizations of IVs that can be used to tame the complexity of many canonical algorithms found in the literature. Further, we prove that determining whether an effect can be identified with TSID (Weihs et al., 2017), a method more powerful than unconditioned instrumental sets and other efficient identification algorithms, is NP-Complete. Finally, building on the idea of flow constraints, we introduce a new and efficient criterion called Instrumental Cutsets (IC), which is able to solve for parameters missed by all other existing polynomial-time algorithms.

Simulation-based Bayesian inference (SBI) can be used to estimate the parameters of complex mechanistic models given observed model outputs without requiring access to explicit likelihood evaluations. A prime example for the application of SBI in neuroscience involves estimating the parameters governing the response dynamics of Hodgkin-Huxley (HH) models from electrophysiological measurements, by inferring a posterior over the parameters that is consistent with a set of observations. To this end, many SBI methods employ a set of summary statistics or scientifically interpretable features to estimate a surrogate likelihood or posterior. However, currently, there is no way to identify how much each summary statistic or feature contributes to reducing posterior uncertainty. To address this challenge, one could simply compare the posteriors with and without a given feature included in the inference process. However, for large or nested feature sets, this would necessitate repeatedly estimating the posterior, which is computationally expensive or even prohibitive. Here, we provide a more efficient approach based on the SBI method neural likelihood estimation (NLE): We show that one can marginalize the trained surrogate likelihood post-hoc before inferring the posterior to assess the contribution of a feature. We demonstrate the usefulness of our method by identifying the most important features for inferring parameters of an example HH neuron model. Beyond neuroscience, our method is generally applicable to SBI workflows that rely on data features for inference used in other scientific fields.

UPDATE: Thank you for the thoughtful response, those changes should improve the things that were unclear to me. There is a rich recent literature on identification criteria for linear structural causal models, but most of the recently proposed criteria largely ignore the question of efficient computability. This paper answers important questions in this area by given efficient algorithms for some criteria, while showing others to be NP-complete. The paper is original and generally clear and of high quality. Minor comments: l100: double "a" l104: the equation you refer to is in the supplement, which should be mentioned here.

The paper proposes a method to efficiently find instrumental subsets for identification in linear acyclic SCMs. The reviewers think that the method is interesting and relevant. An improvement to its evaluation would be the addition of an experimental section -- the authors indicated that they will add it in the revised version of the paper.

Simulation-based Bayesian inference (SBI) can be used to estimate the parameters of complex mechanistic models given observed model outputs without requiring access to explicit likelihood evaluations. A prime example for the application of SBI in neuroscience involves estimating the parameters governing the response dynamics of Hodgkin-Huxley (HH) models from electrophysiological measurements, by inferring a posterior over the parameters that is consistent with a set of observations. To this end, many SBI methods employ a set of summary statistics or scientifically interpretable features to estimate a surrogate likelihood or posterior. However, currently, there is no way to identify how much each summary statistic or feature contributes to reducing posterior uncertainty. To address this challenge, one could simply compare the posteriors with and without a given feature included in the inference process.

One of the most common mistakes made when performing data analysis is attributing causal meaning to regression coefficients. Formally, a causal effect can only be computed if it is identifiable from a combination of observational data and structural knowledge about the domain under investigation (Pearl, 2000, Ch. 5). Building on the literature of instrumental variables (IVs), a plethora of methods has been developed to identify causal effects in linear systems. Almost invariably, however, the most powerful such methods rely on exponential-time procedures. In this paper, we investigate graphical conditions to allow efficient identification in arbitrary linear structural causal models (SCMs).

Fast transforms correspond to factorizations of the form $\mathbf{Z} = \mathbf{X}^{(1)} \ldots \mathbf{X}^{(J)}$, where each factor $ \mathbf{X}^{(\ell)}$ is sparse and possibly structured. This paper investigates essential uniqueness of such factorizations, i.e., uniqueness up to unavoidable scaling ambiguities. Our main contribution is to prove that any $N \times N$ matrix having the so-called butterfly structure admits an essentially unique factorization into $J$ butterfly factors (where $N = 2^{J}$), and that the factors can be recovered by a hierarchical factorization method, which consists in recursively factorizing the considered matrix into two factors. This hierarchical identifiability property relies on a simple identifiability condition in the two-layer and fixed-support setting. This approach contrasts with existing ones that fit the product of butterfly factors to a given matrix via gradient descent. The proposed method can be applied in particular to retrieve the factorization of the Hadamard or the discrete Fourier transform matrices of size $N=2^J$. Computing such factorizations costs $\mathcal{O}(N^{2})$, which is of the order of dense matrix-vector multiplication, while the obtained factorizations enable fast $\mathcal{O}(N \log N)$ matrix-vector multiplications and have the potential to be applied to compress deep neural networks.

One of the most common mistakes made when performing data analysis is attributing causal meaning to regression coefficients. Formally, a causal effect can only be computed if it is identifiable from a combination of observational data and structural knowledge about the domain under investigation (Pearl, 2000, Ch. 5). Building on the literature of instrumental variables (IVs), a plethora of methods has been developed to identify causal effects in linear systems. Almost invariably, however, the most powerful such methods rely on exponential-time procedures. In this paper, we investigate graphical conditions to allow efficient identification in arbitrary linear structural causal models (SCMs).