Aggregate-Constrained Component Intervention Distribution

• The ACID framework defines how aggregate-level interventions are distributed among components while satisfying linear aggregation constraints.
• It employs methodologies such as maximum-entropy optimization and instrument-tuned intervention to bridge observed data with policy-relevant causal inference.
• The approach highlights limitations in standard IV estimation, emphasizing the need for explicit distribution laws to identify genuine causal effects.

An aggregate-constrained component intervention distribution (ACID) specifies the law dictating how a value imposed on an aggregate variable is realized at the level of the underlying components, subject to linear aggregation and intervention constraints. In causal inference and policy analysis, “aggregate” treatments such as education years, GDP, or caloric intake often represent sums over finer-grained components. The ACID formalism is essential because the causal effect of changing an aggregate variable is generally not defined unambiguously without specifying exactly how the aggregate change is distributed across its constituent parts. This concept is rigorously defined and explored for the causal interpretation of instrumental variable (IV) estimands in multi-component settings, illustrating fundamental limitations and formal requirements for identification of policy-relevant causal effects (Tsao et al., 17 Jan 2026).

1. Formal Definition and Mathematical Structure

Consider the SCM with observed variables $V = \{I, U, A_1,\ldots,A_k, Y\}$, where $A = \sum_{j=1}^k \alpha_j A_j$ is a linear aggregate treatment, $I$ is an instrument, $U$ an unobserved confounder, and $$Y = \sum_{j=1}^k \beta_j A_j + \gamma_y U + \epsilon_y$$ the outcome. An aggregate intervention $\operatorname{do}(A=a)$ set at the aggregate level requires instantiating component values $(A_1, \ldots, A_k)$ such that $\sum_j \alpha_j a_j = a$. The full interventional law is

$$P(v)_{\operatorname{do}(A=a)} = P(i, u) \cdot P^*(a_1,\ldots,a_k; a) \cdot P(y | a_1,\ldots,a_k, u)$$

where $P^*(a_1,\ldots,a_k; a)$—the ACID—is a joint component distribution supported exactly on the hyperplane $\sum_j \alpha_j a_j = a$ and reflects any regularity, independence, or other constraints imposed by the intervention. Surgicality ensures components are set independently from their natural causes, the aggregation constraint requires compatibility with the imposed aggregate, and value-independence stipulates invariance of the causal effect across adjusted aggregate values.

The aggregate causal effect (ACE) is defined by

$$ACE(A,Y) = E[Y \mid \operatorname{do}(A=a+1)] - E[Y \mid \operatorname{do}(A=a)]$$

which, under the ACID, reduces to

$$ACE(A,Y) = \sum_{j=1}^k \left[ \int \beta_j a_j dP^*(a_j; a+1) - \int \beta_j a_j dP^*(a_j; a) \right ]$$

(Tsao et al., 17 Jan 2026).

2. Role in Causal Identification and IV Estimation

Instrumental variable analysis with aggregate treatments requires strong assumptions regarding the ACID to ensure that the estimand from standard regressions corresponds to a meaningful policy intervention. In linear SCMs, 2SLS with a single instrument $I$ yields the IV estimand:

$$IV = \frac{\operatorname{Cov}(Y, I)}{\operatorname{Cov}(A, I)} = \frac{\sum_j \beta_j \delta_j}{\sum_j \alpha_j \delta_j}$$

where $\delta_j$ is the effect of $I$ on $A_j$. For $IV$ to equal $ACE(A,Y)$ as computed under the ACID, it is necessary that

$$\frac{\sum_j \beta_j \delta_j}{\sum_j \alpha_j \delta_j} = \sum_j \beta_j d_j$$

with $$d_j = E[A_j \mid \operatorname{do}(A=a+1)] - E[A_j \mid \operatorname{do}(A=a)]$$ (independent of $a$ by value-independence). Only two non-contrived cases ensure this:

• Proportional aggregation: $\beta_j/\alpha_j$ is constant for all $j$, so all component effects scale identically, and $ACE(A,Y)=IV$ for any ACID respecting the aggregation constraint.
• Instrument-tuned intervention: the ACID is chosen such that $d_j = \delta_j/\sum_i \alpha_i \delta_i$, explicitly matching the influence of $I$ on components to the aggregate intervention splits.

In typical empirical settings, aggregation does not naturally satisfy these forms, leading to major limitations in the interpretation of the estimated causal effects (Tsao et al., 17 Jan 2026).

3. ACIDs in Causal Inference: Maximum Entropy and Mixtures

The ACID is inherently underdetermined except for trivial cases. To resolve joint component distributions, maximum-entropy (MaxEnt) methods are deployed when only marginal interventional distributions are available. Given marginals $P_i(v) = P(X_i = v \mid \operatorname{do}(I_i))$ for each component, the MaxEnt ACID solves:

$\max_{P(x_1,\ldots,x_n)} -\sum_x P(x) \log P(x)$

subject to all marginal constraints and normalization. The unique solution is an exponential-family distribution over supported aggregations (Mejia et al., 2024):

$$P(x) = \exp\left( \sum_{i} \lambda_{i,x_i} - \psi(\lambda) \right)$$

with the $\lambda$ chosen so that model marginals match experimental constraints. This “uninformative” joint is maximally entropic except for imposed marginals; higher-order dependencies, if present, remain unmodeled.

In the context of mixtures, dis-entangling the proportions of component-level interventions from aggregate-observed data requires identifiability conditions and algorithmic constructions. Only when exclusion restrictions (some intervention levels are absent per node) hold does the system of mixture equations admit a unique solution for the mixing proportions (Sinha et al., 2019).

4. ACIDs and Forecasting: Entropic Tilting

In the context of constrained Bayesian forecasting, imposing aggregate constraints $S(y)=A$ on a predictive joint $p(y)$ yields an ACID via entropic tilting. The optimal $q^*(y)$ minimizes $\mathrm{KL}(q \| p)$ subject to $\mathbb{E}_q[S(y)] = A$:

$$q(y; \lambda) = \frac{p(y) \exp(\lambda S(y))}{Z(\lambda)}, \quad Z(\lambda) = \mathbb{E}_p[\exp(\lambda S(y))]$$

with $\lambda$ chosen so that $\mathbb{E}_q[S(y)] = A$. For multivariate normal $p(y)$, the tilt shifts only the mean, preserving covariance, with explicit formulas available. This approach provides “decision-analytic interventions” rather than direct inferential conditionals (West, 2020).

5. Examples, Generalizations, and Limitations

Several illustrative ACID constructions appear in the literature:

• Gaussian ACID: $$(A_1,\ldots,A_k)^\top \sim \mathcal{N}(c + a\cdot d, \Sigma)$$ with aggregation and support constraints. The $ACE$ depends entirely on $d_j$ and only matches $IV$ under proportional or instrument-tuned splits (Tsao et al., 17 Jan 2026).
• Discrete interventions and mixtures: Sinha et al. show exact recovery of mixing proportions with aggregate marginals and a constructive algorithm for a known DAG under exclusion assumptions (Sinha et al., 2019).
• Energy systems flexibility: The aggregate-flexibility region for energy-constrained resource fleets is graphically represented, with aggregate demands decomposed optimally via greedy allocation, subject to total resource constraints (Evans et al., 2018, Mukhi et al., 19 Sep 2025).

The general lesson is that without precise specification or “contrived tuning” of the ACID, causal or operational claims about aggregate interventions are ambiguous and fundamentally under-identified. Standard IV methods only identify policy-relevant effects when aggregation structure supports proportionality or explicit instrument-compatible splitting.

6. Implications for Causal Claims and Exclusion Restrictions

The aggregation structure in treatment variables introduces a latent violation of the exclusion restriction in IV analysis, except in proportional aggregation. Proposition 5.1 from (Tsao et al., 17 Jan 2026) shows that, unless $\beta_j/\alpha_j$ is constant, the system behaves observationally like a standard IV model with a direct instrument-outcome link (an exclusion violation). This introduces the need for a much stronger justificatory basis for exclusion restrictions in aggregate treatment settings. The notion of aggregate-constrained component intervention distributions forces a formal distinction between estimable statistical effects and genuine policy or intervention targets, emphasizing that definitive policy conclusions require explicit specification of the underlying ACID.

7. Connections to Network Optimization and Distributed Systems

Aggregate-constrained intervention frameworks extend to optimization in networked systems, such as electric vehicle charging or energy storage fleets. Aggregate flexibility polytopes, intersection with network constraints, and optimal decomposition of aggregate profiles are precisely determined by component-level constraints, with all aggregate operational decisions requiring component-level distribution laws analogous to the ACID (Mukhi et al., 19 Sep 2025). These formulations reveal systematic parallels in how aggregate policy constraints propagate into feasible intervention scheduling and optimization.

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ACID Aspect	Formal Requirement	Identification Conditions
Aggregation constraint	$\sum_j \alpha_j a_j = a$	Proportional/Instrument-tuned split
Surgicality	$A_j$ set independent of parents	Physically implementable assignment
Value-independence	ACE invariant to $a$ choice	Gaussian or deterministic split
IV-ACE matching	Constraint (6) above	Special forms only

The above table summarizes the core ACID ingredients for identification of aggregate causal effects under intervention, capturing the need for explicit intervention distributions and codified splitting laws.

This body of research denotes a major conceptual advance in the formalization of aggregate-level interventions, instrumental variable inference, and multicomponent causal identification (Tsao et al., 17 Jan 2026, Mejia et al., 2024, Sinha et al., 2019, Mukhi et al., 19 Sep 2025, Evans et al., 2018).