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We analyze identifiability as a possible explanation for the ubiquity of linear properties across language models, such as the vector difference between the representations of "easy" and "easiest" being parallel to that between "lucky" and "luckiest". For this, we ask whether finding a linear property in one model implies that any model that induces the same distribution has that property, too. To answer that, we first prove an identifiability result to characterize distribution-equivalent next-token predictors, lifting a diversity requirement of previous results. Second, based on a refinement of relational linearity [Paccanaro and Hinton, 2001; Hernandez et al., 2024], we show how many notions of linearity are amenable to our analysis. Finally, we show that under suitable conditions, these linear properties either hold in all or none distribution-equivalent next-token predictors.

Independent Mechanism Analysis (IMA) seeks to address non-identifiability in nonlinear Independent Component Analysis (ICA) by assuming that the Jacobian of the mixing function has orthogonal columns. As typical in ICA, previous work focused on the case with an equal number of latent components and observed mixtures. Here, we extend IMA to settings with a larger number of mixtures that reside on a manifold embedded in a higher-dimensional than the latent space -- in line with the manifold hypothesis in representation learning. For this setting, we show that IMA still circumvents several non-identifiability issues, suggesting that it can also be a beneficial principle for higher-dimensional observations when the manifold hypothesis holds. Further, we prove that the IMA principle is approximately satisfied with high probability (increasing with the number of observed mixtures) when the directions along which the latent components influence the observations are chosen independently at random. This provides a new and rigorous statistical interpretation of IMA., Comment: 6 pages, Accepted at Neurips Causal Representation Learning 2023

The ability to perform causal reasoning is widely considered a core feature of intelligence. In this work, we investigate whether large language models (LLMs) can coherently reason about causality. Much of the existing work in natural language processing (NLP) focuses on evaluating commonsense causal reasoning in LLMs, thus failing to assess whether a model can perform causal inference in accordance with a set of well-defined formal rules. To address this, we propose a new NLP task, causal inference in natural language, inspired by the "causal inference engine" postulated by Judea Pearl et al. We compose a large dataset, CLadder, with 10K samples: based on a collection of causal graphs and queries (associational, interventional, and counterfactual), we obtain symbolic questions and ground-truth answers, through an oracle causal inference engine. These are then translated into natural language. We evaluate multiple LLMs on our dataset, and we introduce and evaluate a bespoke chain-of-thought prompting strategy, CausalCoT. We show that our task is highly challenging for LLMs, and we conduct an in-depth analysis to gain deeper insights into the causal reasoning abilities of LLMs. Our data is open-sourced at https://huggingface.co/datasets/causalNLP/cladder, and our code can be found at https://github.com/causalNLP/cladder., Comment: NeurIPS 2023; updated with CLadder dataset v1.5

We study causal representation learning, the task of inferring latent causal variables and their causal relations from high-dimensional mixtures of the variables. Prior work relies on weak supervision, in the form of counterfactual pre- and post-intervention views or temporal structure; places restrictive assumptions, such as linearity, on the mixing function or latent causal model; or requires partial knowledge of the generative process, such as the causal graph or intervention targets. We instead consider the general setting in which both the causal model and the mixing function are nonparametric. The learning signal takes the form of multiple datasets, or environments, arising from unknown interventions in the underlying causal model. Our goal is to identify both the ground truth latents and their causal graph up to a set of ambiguities which we show to be irresolvable from interventional data. We study the fundamental setting of two causal variables and prove that the observational distribution and one perfect intervention per node suffice for identifiability, subject to a genericity condition. This condition rules out spurious solutions that involve fine-tuning of the intervened and observational distributions, mirroring similar conditions for nonlinear cause-effect inference. For an arbitrary number of variables, we show that at least one pair of distinct perfect interventional domains per node guarantees identifiability. Further, we demonstrate that the strengths of causal influences among the latent variables are preserved by all equivalent solutions, rendering the inferred representation appropriate for drawing causal conclusions from new data. Our study provides the first identifiability results for the general nonparametric setting with unknown interventions, and elucidates what is possible and impossible for causal representation learning without more direct supervision., Comment: NeurIPS 2023 camera-ready version; 36 pages, 4 figures

Independent Component Analysis (ICA) aims to recover independent latent variables from observed mixtures thereof. Causal Representation Learning (CRL) aims instead to infer causally related (thus often statistically dependent) latent variables, together with the unknown graph encoding their causal relationships. We introduce an intermediate problem termed Causal Component Analysis (CauCA). CauCA can be viewed as a generalization of ICA, modelling the causal dependence among the latent components, and as a special case of CRL. In contrast to CRL, it presupposes knowledge of the causal graph, focusing solely on learning the unmixing function and the causal mechanisms. Any impossibility results regarding the recovery of the ground truth in CauCA also apply for CRL, while possibility results may serve as a stepping stone for extensions to CRL. We characterize CauCA identifiability from multiple datasets generated through different types of interventions on the latent causal variables. As a corollary, this interventional perspective also leads to new identifiability results for nonlinear ICA -- a special case of CauCA with an empty graph -- requiring strictly fewer datasets than previous results. We introduce a likelihood-based approach using normalizing flows to estimate both the unmixing function and the causal mechanisms, and demonstrate its effectiveness through extensive synthetic experiments in the CauCA and ICA setting., Comment: NeurIPS 2023 final camera-ready version

Early on during a pandemic, vaccine availability is limited, requiring prioritisation of different population groups. Evaluating vaccine allocation is therefore a crucial element of pandemics response. In the present work, we develop a model to retrospectively evaluate age-dependent counterfactual vaccine allocation strategies against the COVID-19 pandemic. To estimate the effect of allocation on the expected severe-case incidence, we employ a simulation-assisted causal modelling approach which combines a compartmental infection-dynamics simulation, a coarse-grained, data-driven causal model and literature estimates for immunity waning. We compare Israel's implemented vaccine allocation strategy in 2021 to counterfactual strategies such as no prioritisation, prioritisation of younger age groups or a strict risk-ranked approach; we find that Israel's implemented strategy was indeed highly effective. We also study the marginal impact of increasing vaccine uptake for a given age group and find that increasing vaccinations in the elderly is most effective at preventing severe cases, whereas additional vaccinations for middle-aged groups reduce infections most effectively. Due to its modular structure, our model can easily be adapted to study future pandemics. We demonstrate this flexibility by investigating vaccine allocation strategies for a pandemic with characteristics of the Spanish Flu. Our approach thus helps evaluate vaccination strategies under the complex interplay of core epidemic factors, including age-dependent risk profiles, immunity waning, vaccine availability and spreading rates.

One aim of representation learning is to recover the original latent code that generated the data, a task which requires additional information or inductive biases. A recently proposed approach termed Independent Mechanism Analysis (IMA) postulates that each latent source should influence the observed mixtures independently, complementing standard nonlinear independent component analysis, and taking inspiration from the principle of independent causal mechanisms. While it was shown in theory and experiments that IMA helps recovering the true latents, the method's performance was so far only characterized when the modeling assumptions are exactly satisfied. Here, we test the method's robustness to violations of the underlying assumptions. We find that the benefits of IMA-based regularization for recovering the true sources extend to mixing functions with various degrees of violation of the IMA principle, while standard regularizers do not provide the same merits. Moreover, we show that unregularized maximum likelihood recovers mixing functions which systematically deviate from the IMA principle, and provide an argument elucidating the benefits of IMA-based regularization., Comment: 10 pages, 14 figures, UAI CRL 2022 final camera-ready version

Variational autoencoders (VAEs) are a popular framework for modeling complex data distributions; they can be efficiently trained via variational inference by maximizing the evidence lower bound (ELBO), at the expense of a gap to the exact (log-)marginal likelihood. While VAEs are commonly used for representation learning, it is unclear why ELBO maximization would yield useful representations, since unregularized maximum likelihood estimation cannot invert the data-generating process. Yet, VAEs often succeed at this task. We seek to elucidate this apparent paradox by studying nonlinear VAEs in the limit of near-deterministic decoders. We first prove that, in this regime, the optimal encoder approximately inverts the decoder -- a commonly used but unproven conjecture -- which we refer to as {\em self-consistency}. Leveraging self-consistency, we show that the ELBO converges to a regularized log-likelihood. This allows VAEs to perform what has recently been termed independent mechanism analysis (IMA): it adds an inductive bias towards decoders with column-orthogonal Jacobians, which helps recovering the true latent factors. The gap between ELBO and log-likelihood is therefore welcome, since it bears unanticipated benefits for nonlinear representation learning. In experiments on synthetic and image data, we show that VAEs uncover the true latent factors when the data generating process satisfies the IMA assumption., Comment: NeurIPS2022 final version

Model identifiability is a desirable property in the context of unsupervised representation learning. In absence thereof, different models may be observationally indistinguishable while yielding representations that are nontrivially related to one another, thus making the recovery of a ground truth generative model fundamentally impossible, as often shown through suitably constructed counterexamples. In this note, we discuss one such construction, illustrating a potential failure case of an identifiability result presented in "Desiderata for Representation Learning: A Causal Perspective" by Wang & Jordan (2021). The construction is based on the theory of nonlinear independent component analysis. We comment on implications of this and other counterexamples for identifiable representation learning., Comment: 5 pages, 1 figure

We introduce an approach to counterfactual inference based on merging information from multiple datasets. We consider a causal reformulation of the statistical marginal problem: given a collection of marginal structural causal models (SCMs) over distinct but overlapping sets of variables, determine the set of joint SCMs that are counterfactually consistent with the marginal ones. We formalise this approach for categorical SCMs using the response function formulation and show that it reduces the space of allowed marginal and joint SCMs. Our work thus highlights a new mode of falsifiability through additional variables, in contrast to the statistical one via additional data., Comment: 32 pages (9 pages main paper + bibliography and appendix), 6 figures

Independent component analysis provides a principled framework for unsupervised representation learning, with solid theory on the identifiability of the latent code that generated the data, given only observations of mixtures thereof. Unfortunately, when the mixing is nonlinear, the model is provably nonidentifiable, since statistical independence alone does not sufficiently constrain the problem. Identifiability can be recovered in settings where additional, typically observed variables are included in the generative process. We investigate an alternative path and consider instead including assumptions reflecting the principle of independent causal mechanisms exploited in the field of causality. Specifically, our approach is motivated by thinking of each source as independently influencing the mixing process. This gives rise to a framework which we term independent mechanism analysis. We provide theoretical and empirical evidence that our approach circumvents a number of nonidentifiability issues arising in nonlinear blind source separation., Comment: NeurIPS 2021 final camera-ready version

Self-supervised representation learning has shown remarkable success in a number of domains. A common practice is to perform data augmentation via hand-crafted transformations intended to leave the semantics of the data invariant. We seek to understand the empirical success of this approach from a theoretical perspective. We formulate the augmentation process as a latent variable model by postulating a partition of the latent representation into a content component, which is assumed invariant to augmentation, and a style component, which is allowed to change. Unlike prior work on disentanglement and independent component analysis, we allow for both nontrivial statistical and causal dependencies in the latent space. We study the identifiability of the latent representation based on pairs of views of the observations and prove sufficient conditions that allow us to identify the invariant content partition up to an invertible mapping in both generative and discriminative settings. We find numerical simulations with dependent latent variables are consistent with our theory. Lastly, we introduce Causal3DIdent, a dataset of high-dimensional, visually complex images with rich causal dependencies, which we use to study the effect of data augmentations performed in practice., Comment: NeurIPS 2021 final camera-ready revision (with minor corrections)

In this paper, we investigate the principle that `good explanations are hard to vary' in the context of deep learning. We show that averaging gradients across examples -- akin to a logical OR of patterns -- can favor memorization and `patchwork' solutions that sew together different strategies, instead of identifying invariances. To inspect this, we first formalize a notion of consistency for minima of the loss surface, which measures to what extent a minimum appears only when examples are pooled. We then propose and experimentally validate a simple alternative algorithm based on a logical AND, that focuses on invariances and prevents memorization in a set of real-world tasks. Finally, using a synthetic dataset with a clear distinction between invariant and spurious mechanisms, we dissect learning signals and compare this approach to well-established regularizers., Comment: From v1: extended 2.2 and 2.3, added details for reproducibility and link to codebase

Learning expressive probabilistic models correctly describing the data is a ubiquitous problem in machine learning. A popular approach for solving it is mapping the observations into a representation space with a simple joint distribution, which can typically be written as a product of its marginals -- thus drawing a connection with the field of nonlinear independent component analysis. Deep density models have been widely used for this task, but their maximum likelihood based training requires estimating the log-determinant of the Jacobian and is computationally expensive, thus imposing a trade-off between computation and expressive power. In this work, we propose a new approach for exact training of such neural networks. Based on relative gradients, we exploit the matrix structure of neural network parameters to compute updates efficiently even in high-dimensional spaces; the computational cost of the training is quadratic in the input size, in contrast with the cubic scaling of naive approaches. This allows fast training with objective functions involving the log-determinant of the Jacobian, without imposing constraints on its structure, in stark contrast to autoregressive normalizing flows.

Group studies involving large cohorts of subjects are important to draw general conclusions about brain functional organization. However, the aggregation of data coming from multiple subjects is challenging, since it requires accounting for large variability in anatomy, functional topography and stimulus response across individuals. Data modeling is especially hard for ecologically relevant conditions such as movie watching, where the experimental setup does not imply well-defined cognitive operations. We propose a novel MultiView Independent Component Analysis (ICA) model for group studies, where data from each subject are modeled as a linear combination of shared independent sources plus noise. Contrary to most group-ICA procedures, the likelihood of the model is available in closed form. We develop an alternate quasi-Newton method for maximizing the likelihood, which is robust and converges quickly. We demonstrate the usefulness of our approach first on fMRI data, where our model demonstrates improved sensitivity in identifying common sources among subjects. Moreover, the sources recovered by our model exhibit lower between-session variability than other methods.On magnetoencephalography (MEG) data, our method yields more accurate source localization on phantom data. Applied on 200 subjects from the Cam-CAN dataset it reveals a clear sequence of evoked activity in sensor and source space. The code is freely available at https://github.com/hugorichard/multiviewica., Comment: Accepted to NeurIPS 2020

We point out an instantiation of Simpson's paradox in Covid-19 case fatality rates (CFRs): comparing a large-scale study from China (17 Feb) with early reports from Italy (9 Mar), we find that CFRs are lower in Italy for every age group, but higher overall. This phenomenon is explained by a stark difference in case demographic between the two countries. Using this as a motivating example, we introduce basic concepts from mediation analysis and show how these can be used to quantify different direct and indirect effects when assuming a coarse-grained causal graph involving country, age, and case fatality. We curate an age-stratified CFR dataset with >750k cases and conduct a case study, investigating total, direct, and indirect (age-mediated) causal effects between different countries and at different points in time. This allows us to separate age-related effects from others unrelated to age and facilitates a more transparent comparison of CFRs across countries at different stages of the Covid-19 pandemic. Using longitudinal data from Italy, we discover a sign reversal of the direct causal effect in mid-March which temporally aligns with the reported collapse of the healthcare system in parts of the country. Moreover, we find that direct and indirect effects across 132 pairs of countries are only weakly correlated, suggesting that a country's policy and case demographic may be largely unrelated. We point out limitations and extensions for future work, and, finally, discuss the role of causal reasoning in the broader context of using AI to combat the Covid-19 pandemic., Comment: Journal version with full Appendix. The first two authors contributed equally to this work

The use of inverse probability weighting (IPW) methods to estimate the causal effect of treatments from observational studies is widespread in econometrics, medicine and social sciences. Although these studies often involve sensitive information, thus far there has been no work on privacy-preserving IPW methods. We address this by providing a novel framework for privacy-preserving IPW (PP-IPW) methods. We include a theoretical analysis of the effects of our proposed privatisation procedure on the estimated average treatment effect, and evaluate our PP-IPW framework on synthetic, semi-synthetic and real datasets. The empirical results are consistent with our theoretical findings.

We consider the problem of recovering a common latent source with independent components from multiple views. This applies to settings in which a variable is measured with multiple experimental modalities, and where the goal is to synthesize the disparate measurements into a single unified representation. We consider the case that the observed views are a nonlinear mixing of component-wise corruptions of the sources. When the views are considered separately, this reduces to nonlinear Independent Component Analysis (ICA) for which it is provably impossible to undo the mixing. We present novel identifiability proofs that this is possible when the multiple views are considered jointly, showing that the mixing can theoretically be undone using function approximators such as deep neural networks. In contrast to known identifiability results for nonlinear ICA, we prove that independent latent sources with arbitrary mixing can be recovered as long as multiple, sufficiently different noisy views are available.

The problem of inferring the direct causal parents of a response variable among a large set of explanatory variables is of high practical importance in many disciplines. Recent work exploits stability of regression coefficients or invariance properties of models across different experimental conditions for reconstructing the full causal graph. These approaches generally do not scale well with the number of the explanatory variables and are difficult to extend to nonlinear relationships. Contrary to existing work, we propose an approach which even works for observational data alone, while still offering theoretical guarantees including the case of partially nonlinear relationships. Our algorithm requires only one estimation for each variable and in our experiments we apply our causal discovery algorithm even to large graphs, demonstrating significant improvements compared to well established approaches., Comment: first author uploaded a new version as "Causal Feature Selection via Orthogonal Search"