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Causal Imputation for Counterfactual SCMs: Bridging Graphs and Latent Factor Models

Alvaro Ribot, Chandler Squires, Caroline Uhler

TL;DR

This work addresses causal imputation across a wide set of actions and contexts by framing outcomes as a partially observed matrix and deriving a structural causal model (SCM) based model class that induces a latent factor representation with fixed action effects. Under a linear SCM, the model yields a latent factor model with a fixed-row effect, justifying a fixed-effect–aware extension of Synthetic Interventions (SI) and enabling tensor-to-matrix extrapolation. The authors prove theoretical connections, propose FE-directed enhancements, and demonstrate that SI variants outperform other matrix completion methods on the PRISM drug–cell line dataset, especially in low-data regimes. The framework provides a principled bridge between SCMs and latent-factor models, highlighting the non-symmetric nature of causal matrix completion and offering practical guidance for leveraging structure in contexts and actions. Overall, the approach advances causal prediction in biological settings where only index-based information is available and data are incomplete, with potential applicability to other domains and richer data modalities.

Abstract

We consider the task of causal imputation, where we aim to predict the outcomes of some set of actions across a wide range of possible contexts. As a running example, we consider predicting how different drugs affect cells from different cell types. We study the index-only setting, where the actions and contexts are categorical variables with a finite number of possible values. Even in this simple setting, a practical challenge arises, since often only a small subset of possible action-context pairs have been studied. Thus, models must extrapolate to novel action-context pairs, which can be framed as a form of matrix completion with rows indexed by actions, columns indexed by contexts, and matrix entries corresponding to outcomes. We introduce a novel SCM-based model class, where the outcome is expressed as a counterfactual, actions are expressed as interventions on an instrumental variable, and contexts are defined based on the initial state of the system. We show that, under a linearity assumption, this setup induces a latent factor model over the matrix of outcomes, with an additional fixed effect term. To perform causal prediction based on this model class, we introduce simple extension to the Synthetic Interventions estimator (Agarwal et al., 2020). We evaluate several matrix completion approaches on the PRISM drug repurposing dataset, showing that our method outperforms all other considered matrix completion approaches.

Causal Imputation for Counterfactual SCMs: Bridging Graphs and Latent Factor Models

TL;DR

This work addresses causal imputation across a wide set of actions and contexts by framing outcomes as a partially observed matrix and deriving a structural causal model (SCM) based model class that induces a latent factor representation with fixed action effects. Under a linear SCM, the model yields a latent factor model with a fixed-row effect, justifying a fixed-effect–aware extension of Synthetic Interventions (SI) and enabling tensor-to-matrix extrapolation. The authors prove theoretical connections, propose FE-directed enhancements, and demonstrate that SI variants outperform other matrix completion methods on the PRISM drug–cell line dataset, especially in low-data regimes. The framework provides a principled bridge between SCMs and latent-factor models, highlighting the non-symmetric nature of causal matrix completion and offering practical guidance for leveraging structure in contexts and actions. Overall, the approach advances causal prediction in biological settings where only index-based information is available and data are incomplete, with potential applicability to other domains and richer data modalities.

Abstract

We consider the task of causal imputation, where we aim to predict the outcomes of some set of actions across a wide range of possible contexts. As a running example, we consider predicting how different drugs affect cells from different cell types. We study the index-only setting, where the actions and contexts are categorical variables with a finite number of possible values. Even in this simple setting, a practical challenge arises, since often only a small subset of possible action-context pairs have been studied. Thus, models must extrapolate to novel action-context pairs, which can be framed as a form of matrix completion with rows indexed by actions, columns indexed by contexts, and matrix entries corresponding to outcomes. We introduce a novel SCM-based model class, where the outcome is expressed as a counterfactual, actions are expressed as interventions on an instrumental variable, and contexts are defined based on the initial state of the system. We show that, under a linearity assumption, this setup induces a latent factor model over the matrix of outcomes, with an additional fixed effect term. To perform causal prediction based on this model class, we introduce simple extension to the Synthetic Interventions estimator (Agarwal et al., 2020). We evaluate several matrix completion approaches on the PRISM drug repurposing dataset, showing that our method outperforms all other considered matrix completion approaches.
Paper Structure (39 sections, 3 theorems, 52 equations, 17 figures, 2 tables)

This paper contains 39 sections, 3 theorems, 52 equations, 17 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Organization of the paper and contributions.
  3. Related Work
  4. Local Approaches to Matrix Completion.
  5. Global Approaches to Matrix Completion.
  6. Causal Prediction in Biological Applications.
  7. Background and Setup
  8. Background
  9. An SCM-based Model Class for Causal Prediction
  10. Synthetic Interventions
  11. Theoretical Results
  12. Theoretically-motivated extensions of Synthetic Interventions
  13. Tensor case.
  14. Exploiting Additional Structure in the Model Class
  15. Empirical results
  16. ...and 24 more sections

Key Result

Theorem 1

Under assumption:linear, we have for some $U_i, U'_i \in \mathbb{R}^{|\mathcal{X}| \times |Z|}, {\mathbf{v}}_i \in \mathbb{R}^{|Z|}$ depending on the action index $i$, and some ${\mathbf{w}}_j \in \mathbb{R}^{|Z|}$ depending on the context index $j$.

Figures (17)

  • Figure 1: The causal matrix completion problem. Each row of ${\mathbf{Y}}$ corresponds to an action, and each column corresponds to a context. $Y_{ij}$ denotes the outcome after performing action $i$ in context $j$. We represent missing entries with "$?$".
  • Figure 2: The latent factor model (LFM) written as a simple structural causal model (SCM). LFMs assume that $Y_{ij} = \langle {\mathbf{u}}_i, {\mathbf{v}}_i \rangle + \varepsilon_{ij}$ for ${\mathbf{u}}_i,{\mathbf{v}}_i \in \mathbb{R}^r$. This generative process can be viewed as an SCM over 3 observed (shaded) and 2 latent (unshaded) variables. Here, $I_A$ and $I_C$ represent action and context indices, respectively (note that they are independent).
  • Figure 3: DAG defining our model class. Shaded nodes are observed, while unshaded nodes are unobserved. Data is generated by conditioning on the context index $I_C = j$ (indicated by the double-edge for the node $I_C$) followed by intervening to set the action index $I_A = i$ (indicated by the square node $I_A$). The context may potentially depend on any subset of ${\mathbf{Z}}$ and the action may potentially affect any subset of ${\mathbf{Z}}$. The exogenous noise terms $\varepsilon_1, \varepsilon_2, \ldots, \varepsilon_q$ are shared between the pre-interventional and post-interventional SCMs. Here, the observed outcome is ${\mathbf{Y}}_{ij} \sim \mathbb{P} ( {\mathbf{Z}}_\mathcal{X}(i) \mid I_C = j)$ for $\mathcal{X} = \{ 2, 4 \}$.
  • Figure 4: (Left) Translating the PRISM data to our model class. Letting $\mathcal{C} = \{ k, l \}$ indicates that cell state is defined solely in terms of $Z_k$ and $Z_l$. (Right) Matrix of viability scores. Each entry represents the viability score for a drug-cell line pair. Negative viability indicates cell death. Viability scores are normalized for visualization purposes.
  • Figure 5: Missing data patterns used in our experiments. Observed entries are denoted with black. For each missing entry we have the same number of observations along rows and columns.
  • ...and 12 more figures

Theorems & Definitions (8)

  • Definition 1: Structural Causal Model (SCM)
  • Theorem 1
  • proof : Sketch
  • Proposition 2
  • Corollary 3
  • proof
  • proof
  • remark 1