Conditional Generative Models are Sufficient to Sample from Any Causal Effect Estimand

Causal inference from observational data has recently found many applications in machine learning. While sound and complete algorithms exist to compute causal effects, many of these algorithms require explicit access to conditional likelihoods over the observational distribution, which is difficult to estimate in the high-dimensional regime, such as with images. To alleviate this issue, researchers have approached the problem by simulating causal relations with neural models and obtained impressive results. However, none of these existing approaches can be applied to generic scenarios such as causal graphs on image data with latent confounders, or obtain conditional interventional samples. In this paper, we show that any identifiable causal effect given an arbitrary causal graph can be computed through push-forward computations of conditional generative models. Based on this result, we devise a diffusion-based approach to sample from any (conditional) interventional distribution on image data. To showcase our algorithm's performance, we conduct experiments on a Colored MNIST dataset having both the treatment ($X$) and the target variables ($Y$) as images and obtain interventional samples from $P(y|do(x))$. As an application of our algorithm, we evaluate two large conditional generative models that are pre-trained on the CelebA dataset by analyzing the strength of spurious correlations and the level of disentanglement they achieve.

Modular Learning of Deep Causal Generative Models for High-dimensional Causal Inference

Pearl's causal hierarchy establishes a clear separation between observational, interventional, and counterfactual questions. Researchers proposed sound and complete algorithms to compute identifiable causal queries at a given level of the hierarchy using the causal structure and data from the lower levels of the hierarchy. However, most of these algorithms assume that we can accurately estimate the probability distribution of the data, which is an impractical assumption for high-dimensional variables such as images. On the other hand, modern generative deep learning architectures can be trained to learn how to accurately sample from such high-dimensional distributions. Especially with the recent rise of foundation models for images, it is desirable to leverage pre-trained models to answer causal queries with such high-dimensional data. To address this, we propose a sequential training algorithm that, given the causal structure and a pre-trained conditional generative model, can train a deep causal generative model, which utilizes the pre-trained model and can provably sample from identifiable interventional and counterfactual distributions. Our algorithm, called Modular-DCM, uses adversarial training to learn the network weights, and to the best of our knowledge, is the first algorithm that can make use of pre-trained models and provably sample from any identifiable causal query in the presence of latent confounders with high-dimensional data. We demonstrate the utility of our algorithm using semi-synthetic and real-world datasets containing images as variables in the causal structure.

Approximate Causal Effect Identification under Weak Confounding

Causal effect estimation has been studied by many researchers when only observational data is available. Sound and complete algorithms have been developed for pointwise estimation of identifiable causal queries. For non-identifiable causal queries, researchers developed polynomial programs to estimate tight bounds on causal effect. However, these are computationally difficult to optimize for variables with large support sizes. In this paper, we analyze the effect of "weak confounding" on causal estimands. More specifically, under the assumption that the unobserved confounders that render a query non-identifiable have small entropy, we propose an efficient linear program to derive the upper and lower bounds of the causal effect. We show that our bounds are consistent in the sense that as the entropy of unobserved confounders goes to zero, the gap between the upper and lower bound vanishes. Finally, we conduct synthetic and real data simulations to compare our bounds with the bounds obtained by the existing work that cannot incorporate such entropy constraints and show that our bounds are tighter for the setting with weak confounders.

Towards Characterizing Domain Counterfactuals For Invertible Latent Causal Models

Learning latent causal models from data has many important applications such as robustness, model extrapolation, and counterfactuals. Most prior theoretic work has focused on full causal discovery (i.e., recovering the true latent variables) but requires strong assumptions such as linearity or fails to have any analysis of the equivalence class of solutions (e.g., IRM). Instead of full causal discovery, we focus on a specific type of causal query called the domain counterfactual, which hypothesizes what a sample would have looked like if it had been generated in a different domain (or environment). Concretely, we assume domain-specific invertible latent structural causal models and a shared invertible observation function, both of which are less restrictive assumptions than prior theoretic works. Under these assumptions, we define domain counterfactually equivalent models and prove that any model can be transformed into an equivalent model via two invertible functions. This constructive property provides a tight characterization of the domain counterfactual equivalence classes. Building upon this result, we prove that every equivalence class contains a model where all intervened variables are at the end when topologically sorted by the causal DAG, i.e., all non-intervened variables have non-intervened ancestors. This surprising result suggests that an algorithm that only allows intervention in the last $k$ latent variables may improve model estimation for counterfactuals. In experiments, we enforce the sparse intervention hypothesis via this theoretic result by constraining that the latent SCMs can only differ in the last few causal mechanisms and demonstrate the feasibility of this algorithm in simulated and image-based experiments.

Front-door Adjustment Beyond Markov Equivalence with Limited Graph Knowledge

Causal effect estimation from data typically requires assumptions about the cause-effect relations either explicitly in the form of a causal graph structure within the Pearlian framework, or implicitly in terms of (conditional) independence statements between counterfactual variables within the potential outcomes framework. When the treatment variable and the outcome variable are confounded, front-door adjustment is an important special case where, given the graph, causal effect of the treatment on the target can be estimated using post-treatment variables. However, the exact formula for front-door adjustment depends on the structure of the graph, which is difficult to learn in practice. In this work, we provide testable conditional independence statements to compute the causal effect using front-door-like adjustment without knowing the graph under limited structural side information. We show that our method is applicable in scenarios where knowing the Markov equivalence class is not sufficient for causal effect estimation. We demonstrate the effectiveness of our method on a class of random graphs as well as real causal fairness benchmarks.

Characterization and Learning of Causal Graphs with Small Conditioning Sets

Constraint-based causal discovery algorithms learn part of the causal graph structure by systematically testing conditional independences observed in the data. These algorithms, such as the PC algorithm and its variants, rely on graphical characterizations of the so-called equivalence class of causal graphs proposed by Pearl. However, constraint-based causal discovery algorithms struggle when data is limited since conditional independence tests quickly lose their statistical power, especially when the conditioning set is large. To address this, we propose using conditional independence tests where the size of the conditioning set is upper bounded by some integer $k$ for robust causal discovery. The existing graphical characterizations of the equivalence classes of causal graphs are not applicable when we cannot leverage all the conditional independence statements. We first define the notion of $k$-Markov equivalence: Two causal graphs are $k$-Markov equivalent if they entail the same conditional independence constraints where the conditioning set size is upper bounded by $k$. We propose a novel representation that allows us to graphically characterize $k$-Markov equivalence between two causal graphs. We propose a sound constraint-based algorithm called the $k$-PC algorithm for learning this equivalence class. Finally, we conduct synthetic, and semi-synthetic experiments to demonstrate that the $k$-PC algorithm enables more robust causal discovery in the small sample regime compared to the baseline PC algorithm.

Entropic Causal Inference: Identifiability and Finite Sample Results

Entropic causal inference is a framework for inferring the causal direction between two categorical variables from observational data. The central assumption is that the amount of unobserved randomness in the system is not too large. This unobserved randomness is measured by the entropy of the exogenous variable in the underlying structural causal model, which governs the causal relation between the observed variables. Kocaoglu et al. conjectured that the causal direction is identifiable when the entropy of the exogenous variable is not too large. In this paper, we prove a variant of their conjecture. Namely, we show that for almost all causal models where the exogenous variable has entropy that does not scale with the number of states of the observed variables, the causal direction is identifiable from observational data. We also consider the minimum entropy coupling-based algorithmic approach presented by Kocaoglu et al., and for the first time demonstrate algorithmic identifiability guarantees using a finite number of samples. We conduct extensive experiments to evaluate the robustness of the method to relaxing some of the assumptions in our theory and demonstrate that both the constant-entropy exogenous variable and the no latent confounder assumptions can be relaxed in practice. We also empirically characterize the number of observational samples needed for causal identification. Finally, we apply the algorithm on Tuebingen cause-effect pairs dataset.

Active Structure Learning of Causal DAGs via Directed Clique Tree

A growing body of work has begun to study intervention design for efficient structure learning of causal directed acyclic graphs (DAGs). A typical setting is a causally sufficient setting, i.e. a system with no latent confounders, selection bias, or feedback, when the essential graph of the observational equivalence class (EC) is given as an input and interventions are assumed to be noiseless. Most existing works focus on worst-case or average-case lower bounds for the number of interventions required to orient a DAG. These worst-case lower bounds only establish that the largest clique in the essential graph could make it difficult to learn the true DAG. In this work, we develop a universal lower bound for single-node interventions that establishes that the largest clique is always a fundamental impediment to structure learning. Specifically, we present a decomposition of a DAG into independently orientable components through directed clique trees and use it to prove that the number of single-node interventions necessary to orient any DAG in an EC is at least the sum of half the size of the largest cliques in each chain component of the essential graph. Moreover, we present a two-phase intervention design algorithm that, under certain conditions on the chordal skeleton, matches the optimal number of interventions up to a multiplicative logarithmic factor in the number of maximal cliques. We show via synthetic experiments that our algorithm can scale to much larger graphs than most of the related work and achieves better worst-case performance than other scalable approaches. A code base to recreate these results can be found at https://github.com/csquires/dct-policy

Experimental Design for Cost-Aware Learning of Causal Graphs

We consider the minimum cost intervention design problem: Given the essential graph of a causal graph and a cost to intervene on a variable, identify the set of interventions with minimum total cost that can learn any causal graph with the given essential graph. We first show that this problem is NP-hard. We then prove that we can achieve a constant factor approximation to this problem with a greedy algorithm. We then constrain the sparsity of each intervention. We develop an algorithm that returns an intervention design that is nearly optimal in terms of size for sparse graphs with sparse interventions and we discuss how to use it when there are costs on the vertices.

Entropic Latent Variable Discovery

We consider the problem of discovering the simplest latent variable that can make two observed discrete variables conditionally independent. This problem has appeared in the literature as probabilistic latent semantic analysis (pLSA), and has connections to non-negative matrix factorization. When the simplicity of the variable is measured through its cardinality, we show that a solution to this latent variable discovery problem can be used to distinguish direct causal relations from spurious correlations among almost all joint distributions on simple causal graphs with two observed variables. Conjecturing a similar identifiability result holds with Shannon entropy, we study a loss function that trades-off between entropy of the latent variable and the conditional mutual information of the observed variables. We then propose a latent variable discovery algorithm -- LatentSearch -- and show that its stationary points are the stationary points of our loss function. We experimentally show that LatentSearch can indeed be used to distinguish direct causal relations from spurious correlations.