Counterfactual inference aims to answer retrospective ''what if'' questions and thus belongs to the most fine-grained type of inference in Pearl's causality ladder. Existing methods for counterfactual inference with continuous outcomes aim at point identification and thus make strong and unnatural assumptions about the underlying structural causal model. In this paper, we relax these assumptions and aim at partial counterfactual identification of continuous outcomes, i.e., when the counterfactual query resides in an ignorance interval with informative bounds. We prove that, in general, the ignorance interval of the counterfactual queries has non-informative bounds, already when functions of structural causal models are continuously differentiable. As a remedy, we propose a novel sensitivity model called Curvature Sensitivity Model. This allows us to obtain informative bounds by bounding the curvature of level sets of the functions. We further show that existing point counterfactual identification methods are special cases of our Curvature Sensitivity Model when the bound of the curvature is set to zero. We then propose an implementation of our Curvature Sensitivity Model in the form of a novel deep generative model, which we call Augmented Pseudo-Invertible Decoder. Our implementation employs (i) residual normalizing flows with (ii) variational augmentations. We empirically demonstrate the effectiveness of our Augmented Pseudo-Invertible Decoder. To the best of our knowledge, ours is the first partial identification model for Markovian structural causal models with continuous outcomes.

Partial Counterfactual Identification of Continuous Outcomes with a Curvature Sensitivity Model

The geometric structure of data is an important inductive bias in machine learning. In this work, we characterize the data manifolds entailed by structural causal models. The strengths of the proposed framework are twofold: firstly, the geometric structure of the data manifolds is causally informed, and secondly, it enables causal reasoning about the data manifolds in an interventional and a counterfactual sense. We showcase the versatility of the proposed framework by applying it to the generation of causally-grounded counterfactual explanations for machine learning classifiers, measuring distances along the data manifold in a differential geometric-principled manner.

On Data Manifolds Entailed by Structural Causal Models

For a given causal question, it is important to efficiently decide which causal inference method to use for a given dataset. This is challenging because causal methods typically rely on complex and difficult-to-verify assumptions, and cross-validation is not applicable since ground truth causal quantities are unobserved. In this work, we propose CAusal Method Predictor (CAMP), a framework for predicting the best method for a given dataset. To this end, we generate datasets from a diverse set of synthetic causal models, score the candidate methods, and train a model to directly predict the highest-scoring method for that dataset. Next, by formulating a self-supervised pre-training objective centered on dataset assumptions relevant for causal inference, we significantly reduce the need for costly labeled data and enhance training efficiency. Our strategy learns to map implicit dataset properties to the best method in a data-driven manner. In our experiments, we focus on method prediction for causal discovery. CAMP outperforms selecting any individual candidate method and demonstrates promising generalization to unseen semi-synthetic and real-world benchmarks.

Learned Causal Method Prediction

Capturing the underlying structural causal relations represented by Directed Acyclic Graphs (DAGs) has been a fundamental task in various AI disciplines. Causal DAG learning via the continuous optimization framework has recently achieved promising performance in terms of both accuracy and efficiency. However, most methods make strong assumptions of homoscedastic noise, i.e., exogenous noises have equal variances across variables, observations, or even both. The noises in real data usually violate both assumptions due to the biases introduced by different data collection processes. To address the issue of heteroscedastic noise, we introduce relaxed and implementable sufficient conditions, proving the identifiability of a general class of SEM subject to these conditions. Based on the identifiable general SEM, we propose a novel formulation for DAG learning that accounts for the variation in noise variance across variables and observations. We then propose an effective two-phase iterative DAG learning algorithm to address the increasing optimization difficulties and to learn a causal DAG from data with heteroscedastic variable noise under varying variance. We show significant empirical gains of the proposed approaches over state-of-the-art methods on both synthetic data and real data.

pdf/causal-discovery-under-identifiable-heteroscedastic-noise-1vjj33i5dg.pdf

Causal Discovery under Identifiable Heteroscedastic Noise Model

Estimating mutual correlations between random variables or data streams is essential for intelligent behavior and decision-making. As a fundamental quantity for measuring statistical relationships, mutual information has been extensively studied and utilized for its generality and equitability. However, existing methods often lack the efficiency needed for real-time applications, such as test-time optimization of a neural network, or the differentiability required for end-to-end learning, like histograms. We introduce a neural network called InfoNet, which directly outputs mutual information estimations of data streams by leveraging the attention mechanism and the computational efficiency of deep learning infrastructures. By maximizing a dual formulation of mutual information through large-scale simulated training, our approach circumvents time-consuming test-time optimization and offers generalization ability. We evaluate the effectiveness and generalization of our proposed mutual information estimation scheme on various families of distributions and applications. Our results demonstrate that InfoNet and its training process provide a graceful efficiency-accuracy trade-off and order-preserving properties. We will make the code and models available as a comprehensive toolbox to facilitate studies in different fields requiring real-time mutual information estimation.

InfoNet: Neural Estimation of Mutual Information without Test-Time Optimization

We study the problem of identifying cause and effect over two univariate continuous variables X and Y from a sample of their joint distribution. Our focus lies on the setting where the variance of the noise may be dependent on the cause. We propose to partition the domain of the cause into multiple segments when the noise indeed is dependent. To this end, we minimize a scale-invariant, penalized regression score, finding the optimal partitioning using dynamic programming. We show under which conditions this allows us to identify the causal direction for the linear setting with heteroscedastic noise, for the non-linear setting with homoscedastic noise, as well as empirically confirm that these results generalize to the non-linear and heteroscedastic case. Altogether, the ability to model heteroscedasticity translates into an improved performance in telling cause from effect on a wide range of synthetic and real-world datasets.

Inferring Cause and Effect in the Presence of Heteroscedastic Noise

Mutual information quantifies the dependence between two random variables and remains invariant under diffeomorphisms. In this paper, we explore the pointwise mutual information profile, an extension of mutual information that maintains this invariance. We analytically describe the profiles of multivariate normal distributions and introduce the family of fine distributions, for which the profile can be accurately approximated using Monte Carlo methods. We then show how fine distributions can be used to study the limitations of existing mutual information estimators, investigate the behavior of neural critics used in variational estimators, and understand the effect of experimental outliers on mutual information estimation. Finally, we show how fine distributions can be used to obtain model-based Bayesian estimates of mutual information, suitable for problems with available domain expertise in which uncertainty quantification is necessary.

pdf/the-mixtures-and-the-neural-critics-on-the-pointwise-mutual-2zgag73rdw.pdf

The Mixtures and the Neural Critics: On the Pointwise Mutual Information Profiles of Fine Distributions

We study the class of location-scale or heteroscedastic noise models (LSNMs), in which the effect $Y$ can be written as a function of the cause $X$ and a noise source $N$ independent of $X$, which may be scaled by a positive function $g$ over the cause, i.e., $Y = f(X) + g(X)N$. Despite the generality of the model class, we show the causal direction is identifiable up to some pathological cases. To empirically validate these theoretical findings, we propose two estimators for LSNMs: an estimator based on (non-linear) feature maps, and one based on neural networks. Both model the conditional distribution of $Y$ given $X$ as a Gaussian parameterized by its natural parameters. When the feature maps are correctly specified, we prove that our estimator is jointly concave, and a consistent estimator for the cause-effect identification task. Although the the neural network does not inherit those guarantees, it can fit functions of arbitrary complexity, and reaches state-of-the-art performance across benchmarks.

On the Identifiability and Estimation of Causal Location-Scale Noise Models

Mutual information is a general statistical dependency measure which has found applications in representation learning, causality, domain generalization and computational biology. However, mutual information estimators are typically evaluated on simple families of probability distributions, namely multivariate normal distribution and selected distributions with one-dimensional random variables. In this paper, we show how to construct a diverse family of distributions with known ground-truth mutual information and propose a language-independent benchmarking platform for mutual information estimators. We discuss the general applicability and limitations of classical and neural estimators in settings involving high dimensions, sparse interactions, long-tailed distributions, and high mutual information. Finally, we provide guidelines for practitioners on how to select appropriate estimator adapted to the difficulty of problem considered and issues one needs to consider when applying an estimator to a new data set.

pdf/beyond-normal-on-the-evaluation-of-mutual-information-j7e65m1q.pdf

Beyond Normal: On the Evaluation of Mutual Information Estimators

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Papers

Inferring Cause and Effect in the Presence of Heteroscedastic Noise

The Mixtures and the Neural Critics: On the Pointwise Mutual Information Profiles of Fine Distributions

On the Identifiability and Estimation of Causal Location-Scale Noise Models

Beyond Normal: On the Evaluation of Mutual Information Estimators