Counterfactual Models in Causal Inference

• Counterfactual Models are defined as frameworks that link observed evidence with hypothetical changes via structural equations and latent variable updates.
• They extend classical causal models by incorporating transport-based couplings, set-valued outcomes, and generative techniques to handle high-dimensional and dynamic data.
• Applications span policy analysis, fairness evaluation, decision support, and epidemiology, demonstrating practical insights from diverse methodological innovations.

Counterfactual models formalize contrary-to-fact questions such as “If $A$ were true, would $C$ have been true?” by linking an observed world to a hypothetical world in which actions, treatments, policies, or environments are changed while relevant background conditions are held fixed. In the classical structural-equation tradition, counterfactuals are evaluated by updating beliefs about latent circumstances from actual evidence, modifying the structural equations that encode the hypothetical action, and predicting in the altered system. More recent work broadens this notion to couplings between observable distributions, canonical cross-world process laws, set-valued distributions in incomplete models, and conditional generative models for high-dimensional outcomes (Balke et al., 2013, Lara, 22 Jul 2025, Lara et al., 2021, Kline et al., 2024, Wu et al., 2023).

1. Structural foundations

In structural causal models, observable variables $V=\{V_1,\dots,V_n\}$ are generated by structural equations

$V_i = f_i(V_1,\dots,V_n,U_1,\dots,U_m),$

where $U=\{U_1,\dots,U_m\}$ are disturbances or exogenous variables. Balke and Pearl formulate a counterfactual conditional as

$a > c \mid o,$

to be read as “Given that we have observed $o$, if $a$ were true, then $c$ would have been true.” Their procedure has three steps: use the observations $o$ to update beliefs about the root causes, replace the structural equations of the intervened variables by fixed assignments, and solve the altered system with the updated beliefs carried over from the actual world (Balke et al., 2013).

For Gaussian linear structural equation models, this counterfactual logic can be rewritten in terms of total effects and observed covariance structure rather than only disturbance parameters. Under an intervention $C$0, the counterfactual mean and variance of a response $C$1 can be written as

$C$2

and, conditional on actual evidence $C$3,

$C$4

This reformulation makes counterfactual computation depend on identified total effects such as back-door or conditional-IV estimands together with observed covariances (Cai et al., 2012).

A more recent reformulation of the structural tradition appears in the Markovian setting, where “counterfactual models,” or canonical representations of structural causal models, are defined as pairs $C$5. Here $C$6 is a causal graphical model that fixes the observational and interventional layers, while $C$7 is a collection of one-step-ahead counterfactual process distributions with prescribed marginals. The central claim is that, once the graphical/interventional content is fixed, the remaining counterfactual layer is a choice of cross-world joint law. The paper characterizes counterfactual equivalence of Markovian SCMs exactly through equality of their canonical representations (Lara, 22 Jul 2025).

2. Representations beyond point-identified structural models

When a full structural causal model is unavailable, one response is to replace unobserved counterfactual laws by observable distributions plus an explicit coupling. Transport-based counterfactual models do this by defining a collection of couplings

$C$8

where each $C$9 couples $V=\{V_1,\dots,V_n\}$0 and $V=\{V_1,\dots,V_n\}$1, satisfies an identity condition for $V=\{V_1,\dots,V_n\}$2, and obeys reciprocity under swapping coordinates. In the deterministic case these become transport maps $V=\{V_1,\dots,V_n\}$3, often selected by an optimal-transport criterion such as quadratic cost. Under relative exogeneity and invertibility assumptions, the paper shows that these transport operators can coincide with structural causal counterfactual operators, especially in linear additive SCMs (Lara et al., 2021).

A different extension arises when the structural model is itself incomplete or only partially identified. In empirical games with partially identified parameters, multiple equilibria, and randomized strategies, Kline and Tamer define the counterfactual predictive distribution set (CPDS). For each parameter value $V=\{V_1,\dots,V_n\}$4, the relevant random-set-valued object is

$V=\{V_1,\dots,V_n\}$5

and the population CPDS is

$V=\{V_1,\dots,V_n\}$6

The point is that, in incomplete and partially identified models, a counterfactual is not a single predictive distribution but a set of distributions of random sets. The paper proves that the population CPDS is sharp and develops posterior consistency results for mappings of estimated sets, so counterfactual inference can be layered on top of an existing identification-and-estimation step for the underlying model parameter (Kline et al., 2024).

These two strands emphasize different failures of point identification. Transport-based models respond to missing causal structure by choosing a coupling between observable marginals. CPDS responds to structural incompleteness by treating the counterfactual object itself as set-valued. This suggests that “counterfactual model” is not a single mathematical form but a representation chosen to match the source of non-uniqueness.

3. Dynamic, temporal, and stochastic-process counterfactuals

In continuous-time decision support, counterfactual models often target future trajectories rather than static outcomes. The Counterfactual Gaussian Process (CGP) is designed for irregularly sampled longitudinal traces with interleaved actions and measurements, and its target is

$V=\{V_1,\dots,V_n\}$7

The model embeds a Gaussian-process outcome model inside a marked point process and estimates it by an adjusted maximum-likelihood objective that accounts for outcome dynamics, event timing, and treatment assignment. Its identification relies on consistency, continuous-time no unmeasured confounders, and non-informative measurement times (Schulam et al., 2017).

For discrete longitudinal settings with time-varying treatment, recent work replaces explicit density estimation of the counterfactual law by weighted conditional generative modeling. The target is the full counterfactual distribution $V=\{V_1,\dots,V_n\}$8 of $V=\{V_1,\dots,V_n\}$9, and the key identity is that $V_i = f_i(V_1,\dots,V_n,U_1,\dots,U_m),$0 can be written as a weighted version of the observed-data law using inverse probability of treatment weights. This yields a generic objective of the form

$V_i = f_i(V_1,\dots,V_n,U_1,\dots,U_m),$1

which can be instantiated with a conditional VAE or a diffusion model. The method is explicitly aimed at high-dimensional and multimodal outcomes under time-varying treatment histories (Wu et al., 2023).

In linear multivariate time series, exact counterfactual reasoning becomes possible. For a VAR($V_i = f_i(V_1,\dots,V_n,U_1,\dots,U_m),$2) structural model

$V_i = f_i(V_1,\dots,V_n,U_1,\dots,U_m),$3

a hypothetical intervention encoded by perturbations $V_i = f_i(V_1,\dots,V_n,U_1,\dots,U_m),$4 propagates through total causal effect matrices $V_i = f_i(V_1,\dots,V_n,U_1,\dots,U_m),$5 satisfying

$V_i = f_i(V_1,\dots,V_n,U_1,\dots,U_m),$6

and the counterfactual difference process is

$V_i = f_i(V_1,\dots,V_n,U_1,\dots,U_m),$7

Because the same exogenous noise realization is shared across factual and counterfactual worlds, the noise cancels in the difference process, so explicit noise abduction is unnecessary in this linear additive setting (Butler et al., 2024).

Event streams require yet another construction. Counterfactual Temporal Point Processes reinterpret thinning as a structural causal model based on the Gumbel-Max trick. Given a proposal process and a factual intensity $V_i = f_i(V_1,\dots,V_n,U_1,\dots,U_m),$8, each candidate event is accepted or rejected through a latent Gumbel competition; replacing $V_i = f_i(V_1,\dots,V_n,U_1,\dots,U_m),$9 by an alternative intensity $U=\{U_1,\dots,U_m\}$0 while keeping the same latent variables yields a counterfactual event trajectory. Proposition 1 establishes a monotonicity condition for thinning that is sufficient to identify counterfactual acceptance dynamics, and the paper combines this with superposition and branching-process arguments to handle inhomogeneous Poisson and linear Hawkes processes (Noorbakhsh et al., 2021).

When hidden states are present, the counterfactual problem can cease to be point identified even if the observed transition and emission model is known. In dynamic latent-state models, many distinct SCMs induce the same observed stochastic state-space model. The proposed solution is to sample posterior hidden-state trajectories in an abduction step and then optimize over all SCMs consistent with the observed dynamic model, producing sharp feasible upper and lower bounds on the counterfactual quantity of interest rather than a single answer (Haugh et al., 2022).

4. High-dimensional generative and explanatory counterfactual models

In high-dimensional settings, especially images, one line of work preserves Pearl’s abduction–action–prediction semantics but implements each step with deep generative machinery. Diff-SCM assumes a known causal structure, treats forward diffusion as a route from observed endogenous variables to exogenous noise, performs abduction by deterministic DDIM inversion, and generates the counterfactual by guided reverse diffusion from the same latent state. The intervention is implemented through gradients of an anti-causal predictor with respect to the image, so the same inferred latent information is preserved while the target ancestor is changed (Sanchez et al., 2022).

Other image frameworks use “counterfactual” in a more practical robustness or explanation sense rather than as a full structural causal model. “Reinforcing Pre-trained Models Using Counterfactual Images” explicitly describes its framework as “not formally causal”: BLIP-2 generates captions, a LoRA-fine-tuned LLaMA-7B perturbs those captions, Stable Diffusion with prompt-to-prompt editing and null-text inversion produces edited images, and the resulting images are used both to diagnose spurious classifier dependencies and to fine-tune the classification head. The central object is a label-preserving, semantically controlled image variant rather than a structurally identified causal counterfactual (Li et al., 2024).

In source-code models, counterfactual explanations are local, actionable perturbations. The definition requires a perturbation $U=\{U_1,\dots,U_m\}$1 of a program $U=\{U_1,\dots,U_m\}$2 such that

$U=\{U_1,\dots,U_m\}$3

The method does not rely on gradients or internal access to the task model; instead it uses a masked LLM to propose plausible token replacements and searches over increasingly large token subsets until the model “changes its mind.” Minimality is therefore defined operationally through sparse token replacement, not by a continuous optimization objective (Cito et al., 2021).

For mixed tabular classification, ProCE integrates feature-level causal structure directly into counterfactual explanation. It combines a prediction-validity loss, a prototype loss in latent space, and a causality-preserving loss derived from a structural causal model over features, then optimizes the resulting vector objective with NSGA-II rather than a weighted scalarization. The intended effect is to generate counterfactuals that are not only prediction-valid and close to the original sample but also causally consistent with relations among input features (Duong et al., 2021).

5. Inference, evaluation, and soundness

Counterfactual models require their own estimands and evaluation criteria. In prediction problems, the relevant target is not the factual outcome law but an intervention-specific quantity such as

$U=\{U_1,\dots,U_m\}$4

and model performance under intervention is correspondingly defined by counterfactual loss, AUC, or calibration. A central performance estimand is

$U=\{U_1,\dots,U_m\}$5

The paper shows that such quantities can be identified by outcome-model, inverse-probability-weighted, or doubly robust estimators, and emphasizes that performance evaluation remains valid even if the candidate prediction model is misspecified (Boyer et al., 2023).

For approximate image counterfactual engines, one cannot usually compare generated images to ground-truth counterfactual images. An alternative is to evaluate axiomatic soundness. The counterfactual image-model paper rewrites the counterfactual mechanism as

$U=\{U_1,\dots,U_m\}$6

and uses three axioms as necessary constraints: composition,

$U=\{U_1,\dots,U_m\}$7

effectiveness,

$U=\{U_1,\dots,U_m\}$8

and, for invertible mechanisms, reversibility,

$U=\{U_1,\dots,U_m\}$9

The paper then defines metricized violations of these equalities to compare approximate models and diagnose trade-offs between intervention strength and instance preservation (Monteiro et al., 2023).

Counterfactual competence can also be evaluated at the level of reasoning systems rather than generative models. A probing study of LLMs uses psycholinguistic materials and large synthetic datasets to test whether models truly track counterfactual conditionals. The main finding is that models often can override real-world knowledge in counterfactual contexts, but for most models this behavior appears largely driven by lexical cues; when lexical cues and world knowledge are controlled away, only GPT-3 shows sensitivity to linguistic nuances of counterfactuals, and that sensitivity remains non-trivially affected by lexical associative factors (Li et al., 2023).

In set-valued econometric counterfactual models, the inferential problem is harder because the target is itself a mapping of sets. The CPDS paper addresses this by developing a general Bayesian consistency theory for posterior distributions of mappings of estimated sets, including equivalences between intersection-based consistency and Hausdorff-consistency notions in Euclidean settings (Kline et al., 2024).

6. Applications, interpretive scope, and open issues

Counterfactual models are now used across policy analysis, fairness, decision support, software engineering, empirical industrial organization, and epidemiology. Transport-based counterfactual models were developed partly to make counterfactual fairness computationally feasible when a full SCM is unavailable, while CGP and counterfactual prediction methods are used for untreated risk prediction, individualized treatment planning, and statin-naive cardiovascular risk modeling (Lara et al., 2021, Schulam et al., 2017, Boyer et al., 2023). In empirical games, Kline and Tamer’s airline-entry application uses one solution concept for estimation and another for counterfactuals, and the framework recovers intervals for behavioral and welfare outcomes rather than point predictions even in the “real world” benchmark (Kline et al., 2024). Counterfactual temporal point processes are used to study alternative epidemic-intervention thresholds and contact-reduction policies on Ebola-like outbreak trajectories (Noorbakhsh et al., 2021).

A recurring controversy concerns what exactly the word “counterfactual” denotes. Some frameworks are formally embedded in structural causal models, while others are explicitly presented as practical robustness or explanation tools. The image-reinforcement paper states that its approach is “not formally causal,” and the transport-based paper presents optimal transport as a principled surrogate that can coincide with causal counterfactuals only under specific assumptions rather than as a general identification result (Li et al., 2024, Lara et al., 2021). The canonical-representation paper sharpens this point by arguing that, once a Markovian causal graphical model fixes the observational and interventional layers, the remaining counterfactual layer is a choice of cross-world dependence structure rather than an empirically estimable object (Lara, 22 Jul 2025).

Limitations also recur across otherwise different literatures. Exact point-valued counterfactuals typically require strong structure: linearity and shared exogenous noise in VARs, Gaussian linear SEM assumptions, or a known SCM with identifiable interventions (Butler et al., 2024, Cai et al., 2012, Sanchez et al., 2022). Once hidden confounding, partial identification, multiple equilibria, high-dimensional generation, or latent dynamic states enter, the literature shifts toward couplings, bounds, set-valued targets, or axiomatic evaluation rather than exact recovery (Kline et al., 2024, Haugh et al., 2022, Monteiro et al., 2023). This suggests that the theory of counterfactual models is fundamentally a theory of how to represent and constrain cross-world quantities when direct empirical access to those quantities is limited or absent.