Counterfactual Upper-Bound Analysis

Updated 1 February 2026

Counterfactual upper-bound analysis is a method to determine the maximum effect of interventions in partially identified models by characterizing sharp bounds under uncertainty.
It utilizes exact inference via credal networks and approximate causal EM algorithms to efficiently optimize counterfactual probabilities.
This approach is pivotal in applications like palliative care, offering actionable insights when data or model uncertainty precludes precise counterfactual estimation.

Counterfactual upper-bound analysis refers to a rigorous methodology for computing the maximal values that counterfactual probabilities, expectations, or functional effects can attain, given uncertainty or partial identification in the underlying structural model, data, or estimation procedure. This paradigm is central in causal inference—whether in structural causal models (SCMs), econometric identification, bandit optimization, or algorithmic recourse—whenever the exact value of a counterfactual is not point-identified by the available data and/or assumptions, but well-defined bounds can be sharply characterized. The formulation, computation, and interpretation of such upper bounds depend crucially on the structure of the model (e.g., graph topology, unobserved variables), the type of information available (e.g., observational, interventional, or mixed datasets), and the application context, as detailed below.

1. Foundations: Partially Identified Counterfactuals in Structural Causal Models

Let $M$ be a structural causal model over discrete random variables partitioned as exogenous ( $U$ ) and endogenous ( $V$ ), with deterministic structural equations $F$ and observational data $D$ on $V$ . A counterfactual quantity is typically of the form

$P\big(Y_x = y \mid e \big)$

where $X = x$ is an intervention, $Y = y$ is the event of interest, and $e$ is evidence. In general, this counterfactual is partially identifiable: it depends on the unobserved distribution $P(U)$ , and thus, only the set of all SCMs $(F, D)$ -compatible—denoted $\mathcal{M}_{M,P}$ —is known. The counterfactual upper bound is then

$\overline{P} = \max_{M' \in \mathcal{M}_{M,P}} P_{M'}(Y_x = y \mid e)$

and the lower bound is the analogous minimum. These bounds summarize the identifiability of the counterfactual in the absence of full knowledge of $P(U)$ (Zaffalon et al., 2023).

2. Computational Methodologies: From SCMs to Credal Networks to Causal EM

Exact Inference via Credal Networks

For SCMs with known structural equations and observed sufficient marginals (Markovian or quasi-Markovian), there exists a mapping to an equivalent credal network. The distributional constraints induced by $(F, D)$ are translated into (possibly degenerate) credal sets for exogenous variables. In the Markovian case, the mapping is as follows: for each exogenous $U$ , let $V$ be its unique endogenous child, and let $W$ be the remaining parents of $V$ . One enforces

$\sum_{u: f_V(u, W) = v} P(u) = P_{\mathrm{data}}(v \mid W) \qquad \forall v, W$

This results in a feasible set $K$ for $P(U)$ : all distributions matching the observed marginals through the deterministic mechanisms.

Given a counterfactual functional $q(P(U)) = P(Y_x = y \mid e)$ , the upper bound is

$\overline{q} = \max_{P(U) \in K} q(P(U))$

Complexity and Inference Hardness

Causal inference—computing such bounds—is NP-hard even in polytree SCMs. This follows from the known NP-hardness of inference in credal networks with polytree structure, as every such network corresponds to an SCM of the same structure (Zaffalon et al., 2023).

Approximate Inference via Causal EM

When exact computation is infeasible, a causal EM algorithm (EMCC) is applied:

E-step: For each $u \in Q(U)$ and $v$ in data, compute posterior $P^{(t)}(u \mid v)$ under current $P^{(t)}(U)$ .
M-step: Update $P^{(t+1)}(u)$ by averaging posteriors over $D$ . This scheme is iterated to stationarity; multiple random restarts sample extreme points of the feasible set $K$ , and for each solution, the counterfactual is evaluated. The reported upper bound is the maximum value across EM solutions. The accuracy of the approximation is quantified via credible intervals explicitly derived as functions of the empirical range and number of restarts (Zaffalon et al., 2023).

3. Theoretical Guarantees and Properties

Feasibility and Likelihood (M-compatibility): The feasible set $K$ is non-empty if and only if the log-likelihood of the data is maximized at some $P(U)$ . This check is essential: if the constraints from $F$ and $D$ are incompatible, the counterfactual bound is ill-posed.
Sharpness: The computed bounds are tight; no value outside $[\underline{P}, \overline{P}]$ is consistent with $(F, D)$ . For credal network representations, the set of compatible Bayesian networks is precisely the set of fully specified SCMs matched to the data (Zaffalon et al., 2023).
Approximation Accuracy: Relative RMSE decays rapidly as the number of EM restarts increases (RRMSE $<0.1$ with $\approx 200$ runs on synthetic benchmarks), and the method scales to models with up to 19 nodes and 256 exogenous cardinality.

4. Limitations and Model Requirements

Necessity of Structural Equations: Knowledge of the explicit structural equations $F$ is required. Attempts to compute bounds solely from graphical structure or observed marginals may yield invalid intervals if the data are incompatible with plausible $F$ (Zaffalon et al., 2023).
Dependence on Data Compatibility: M-compatibility must be checked; otherwise, the computational procedure can yield spurious “bounds” that do not correspond to any SCM.
Canonical $F$ and Bound Tightness: Using "canonical" specification of $F$ (encoding all deterministic mechanisms) guarantees M-compatibility at large data, but the resulting bounds may be conservative, i.e., wider than necessary.

5. Empirical Applications: Case Study in Palliative Care

A practical application is presented for modeling place of death of terminal cancer patients:

Graph includes patient/family awareness, home-assistance (“Triangolo”), symptoms, age, performance, and more.
Interventions on home-assistance and awareness are considered, with home death as the effect variable.
EMCC is run with $r=160$ restarts and about $500$ iterations per run.
Resulting probability of necessity and sufficiency (PNS) intervals:
- Triangolo: $[0.30, 0.31]$
- Patient awareness: $[0.03, 0.10]$
- Family awareness: $[0.06, 0.10]$
- These indicate that expanding home-assistance has a much stronger counterfactual effect than communication-based interventions (Zaffalon et al., 2023).

6. Algorithmic and Practical Considerations

Step	Approach	Remarks
Feasibility	Map $(F,D)$ to credal network / linear constraints	Markovian/quasi-Markovian distinction matters
Upper bound	Optimize counterfactual over feasible $P(U)$	Linear/nonlinear programming
Approximation	Multiple causal EM runs sampling $K$	Range approximates bound
Quality control	RRMSE, credible intervals on [a, b]	Theoretical interval based on Beta model

While exact algorithms are infeasible for large discrete models due to NP-hardness, the EM approach provides scalable and practical approximations with quantifiable error. The range of counterfactual estimates over multiple EM runs closely matches true bounds in synthetic and real datasets, such as the palliative care study.

7. Context and Broader Significance

Counterfactual upper-bound analysis generalizes naturally across distinct inferential paradigms: from linear programming for deterministic SCMs (Balke et al., 2013), to convex optimization in moment or $\phi$ -divergence neighborhoods (Christensen et al., 2019), to robust causal inference in empirical games with partially identified equilibria (Kline et al., 2024). The paradigm is essential wherever partial identifiability or model uncertainty renders point estimation of counterfactuals ill-posed. Within the core SCM literature, the sharpness, computational feasibility, and data requirements for such bounds remain crucial theoretical and methodological concerns (Zaffalon et al., 2023).