Rubin–Neyman Causal framework for Probability Measures

Many excellent resources introduce causal inference to economists and applied statisticians. Yet, at its mathematical core, causal inference does not start with regression. It starts with a question of identification:

when can a functional/estimand of an unknown probability measure be identifiable from an observed probability distribution?

Causal Inference: the Rubin-Neyman framework

For each unit, there are two potential outcomes:

\[Y^1, Y^0\]

where $Y^1$ is the outcome under treatment and $Y^0$ is the outcome under control. Randomization reveals only the potential outcome corresponding to the assigned treatment, leaving the other counterfactual outcome unobserved. ¹²

The full counterfactual random variables $Z$ is

\[Z=(X,Y^1,Y^0),\]

with full-data law

\[P_Z = \mathcal L(X,Y^1,Y^0).\]

But we never observe $Z$ completely. The observed triplet is baseline covariates, treatment received, and factual outcome

\(D=(X,T,Y),\) where observe the factual outcome $Y = TY^1 + (1-T)Y^0$ and treatment assignment $T \in {0,1}$.

The observed random element $D=(X,T,Y)$ is not the full causal object $Z$; the observed object is a projection/function of it.

This distinction is the entire mathematical foundation of causal inference.

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1. Observation as a Projection of the Counterfactual Triplet

Define the projection map $\Gamma$

\[\Gamma(X,Y^1,Y^0,T) = (X,T,TY^1+(1-T)Y^0).\]

Then the observed law is

\[P_{D}(\cdot) = P_{Z} \circ \Gamma^{-1}(\cdot).\]

In other words, $P_Z \longrightarrow P_{D}$ is a many-to-one map and $Gamma$ may not have .

Many different full counterfactual laws may produce the same observed law. This is why causal inference is not ordinary supervised learning.

A supervised learning problem begins with

\[P_{D} = \mathcal L(X,T,Y)\]

and learns associations inside the observed world.

A causal problem asks about a functional of

\[P_Z = \mathcal L(X,Y^1,Y^0),\]

which is partly unobserved.

Thus the core question is:

When can a functional of $P_Z$ be represented as a functional of $P_{D}$?

This is the mathematical meaning of identification.³⁴

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2. Causal Estimands Are Functionals of the Full-Data Law

A causal estimand is a functional

\[\psi(P_Z).\]

Examples include the average treatment effect:

\[\tau = E_F[Y^1-Y^0],\]

the conditional average treatment effect:

\[\tau(x) = E_F[Y^1-Y^0 \mid X=x],\]

and the optimal treatment rule:

\[d^*(x) = 1\{E_F[Y^1-Y^0 \mid X=x]>0\}.\]

These are population-level causal functionals.

The individual treatment effect,

\[\Delta_i = Y_i^1-Y_i^0,\]

is conceptually meaningful but generally not identifiable from observed data because only one of $Y_i^1$ and $Y_i^0$ is observed.

The probability of benefit is

\[P_Z(Y^1>Y^0 \mid X=x).\]

This is also generally not identifiable without additional assumptions because it depends on the joint conditional law of $(Y^1,Y^0)$, not only their two marginal distributions.

This distinction is important:

\[\text{CATE depends on conditional means;}\] \[\text{PoB depends on the joint counterfactual coupling.}\]

Observed data usually identify the former under standard assumptions, but not the latter.

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3. Identification as Constancy on Observational Equivalence Classes

Let $\mathcal M_F$ be a model for possible full-data laws. Each full-data law $P_Z$ induces an observed-data law $P_{D}$.

Two full-data laws $P_Z$ and $P_Z’$ are observationally equivalent if they induce the same observed law:

\[P_{D}(P_Z)=P_{D}(P_Z').\]

A causal functional $\psi(P_Z)$ is identifiable if

\[P_{D}(P_Z)=P_{D}(P_Z') \quad \Rightarrow \quad \psi(P_Z)=\psi(P_Z')\]

for all $P_Z,P_Z’ \in \mathcal M_F$.

Equivalently, $\psi$ must be constant on every fiber of the projection map

\[P_Z \mapsto P_{D}.\]

This is the most precise way to say:

If two causal worlds look identical in the observed data but imply different causal effects, the causal effect is not identifiable.

Machine learning cannot solve this problem. ML can estimate functionals of $P_{D}$. It cannot distinguish two full counterfactual worlds that generate the same $P_{D}$ but have different $\psi(P_Z)$.

So causal inference has two layers:

\[\text{Identification: } \psi(P_Z)=\phi(P_{D})\]

and

\[\text{Estimation: } \hat\phi(P_n) \approx \phi(P_{D}).\]

Only the second layer is a standard statistical learning problem.

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4. The Standard Identification Assumptions

For binary treatment, a common route from $P_Z$ to $P_{D}$ uses three assumptions.⁵

Consistency

Let $T=t$, then

\[Y = Y^t.\]

Consistency is what makes the observed outcome a factual realization of one potential outcome.

Conditional exchangeability

Treatment assignment is independent of potential outcomes given covariates:

\[(Y^1,Y^0) \perp T \mid X.\]

This says that after conditioning on $X$, treatment assignment contains no additional information about the potential outcomes.

In randomized trials, this can be justified by design. In observational studies, it is an untestable no-unmeasured-confounding assumption.

Positivity

For all covariate values in the target population,

\[0 < P(T=1 \mid X=x) < 1.\]

If positivity fails, then some counterfactual comparisons are unsupported by observed data.

These assumptions imply that causal estimands are determined by the observed-data law. In the slide language, this is the step where causal quantities become machine-learnable from observed data.

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5. The G-Formula: Identification Through Outcome Regression

Define the observed conditional outcome functions

\[m_t(x)=E[Y \mid T=t,X=x], \qquad t \in \{0,1\}.\]

Under consistency and exchangeability,

\[E_F[Y^t \mid X=x] = E_{D}[Y \mid T=t,X=x] = m_t(x).\]

Therefore,

\[\tau(x)=m_1(x)-m_0(x),\]

and

\[\tau = E_{D}[m_1(X)-m_0(X)].\]

This is the g-formula.⁶

It is the bridge between counterfactual mathematics and machine learning:

\[\text{full-data causal estimand} \quad \Rightarrow \quad \text{observed regression functional}.\]

Once this bridge is established, outcome-modeling methods can be used:

\[\hat m_1(x), \quad \hat m_0(x).\]

This includes S-learners, T-learners, X-learners, Bayesian additive regression trees, random forests, boosting, neural networks, and other flexible regressors.⁷⁸

But the causal interpretation comes from the identification assumptions, not from the modeling algorithm.

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6. IPW: Identification Through the Treatment Mechanism

Define the propensity score

\[e(x)=P(T=1 \mid X=x).\]

Then the ATE can also be identified as

\[\tau = E_{D}\left[ \frac{TY}{e(X)} - \frac{(1-T)Y}{1-e(X)} \right].\]

The derivation is a change-of-measure argument:

\[E_{D}\left[\frac{TY}{e(X)} \mid X\right] = E_{D}[Y \mid T=1,X] = E_F[Y^1 \mid X],\]

and similarly,

\[E_{D}\left[\frac{(1-T)Y}{1-e(X)} \mid X\right] = E_F[Y^0 \mid X].\]

This is why inverse-probability weighting treats treatment assignment as a sampling design.⁹

The mathematical strength is unbiasedness under a correct treatment model. The mathematical weakness is variance. If $e(X)$ is close to $0$ or $1$, the weights explode:

\[\frac{1}{e(X)} \quad \text{or} \quad \frac{1}{1-e(X)}.\]

Thus positivity is not merely a philosophical assumption. It is a numerical stability condition.

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7. Doubly Robust Estimation: Combining Two Identification Routes

Outcome regression uses $m_t(x)$. IPW uses $e(x)$. Doubly robust estimation combines both.

The efficient influence-function representation for the ATE is

\[\varphi(D;\eta) = m_1(X)-m_0(X) + \frac{T}{e(X)}[Y-m_1(X)] - \frac{1-T}{1-e(X)}[Y-m_0(X)] - \tau,\]

where

\[\eta=(m_0,m_1,e).\]

The corresponding estimator is

\[\hat\tau_{DR} = \frac{1}{n} \sum_{i=1}^n \left[ \hat m_1(X_i)-\hat m_0(X_i) + \frac{T_i}{\hat e(X_i)}[Y_i-\hat m_1(X_i)] - \frac{1-T_i}{1-\hat e(X_i)}[Y_i-\hat m_0(X_i)] \right].\]

It is doubly robust because consistency holds if either:

\[\hat m_t \rightarrow m_t\]

or

\[\hat e \rightarrow e.\]

The deeper idea is Neyman orthogonality. At the truth $\eta_0$,

\[\partial_\eta E[\varphi(D;\eta)]|_{\eta=\eta_0}=0.\]

This means first-order nuisance estimation errors do not create first-order bias in the target parameter.¹⁰¹¹

This is the modern reason causal inference and machine learning fit together:

ML estimates high-dimensional nuisance functions; orthogonal estimating equations protect the low-dimensional causal target.

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8. CATE, Policy Learning, and the Decision View

The conditional average treatment effect is

\[\tau(x)=m_1(x)-m_0(x).\]

A treatment rule is a map

\[d: \mathcal X \to \{0,1\}.\]

Its value is

\[V(d)=E_F[Y^{d(X)}].\]

Under identification,

\[V(d) = E_{D}[d(X)m_1(X)+(1-d(X))m_0(X)].\]

If larger outcomes are better and treatment has no additional cost, the optimal rule is

\[d^*(x)=1\{\tau(x)>0\}.\]

This is a population-level rule, not an individual-level revelation of $Y_i^1-Y_i^0$.

Policy learning methods optimize $V(d)$ directly rather than first estimating the full CATE surface.¹²¹³ Outcome weighted learning, for example, can be interpreted as an IPW-weighted classification problem. The classifier has causal meaning only through the value functional it optimizes.

This gives three distinct causal-learning cultures:

1. Outcome modeling: learn $m_1(x)$ and $m_0(x)$, then subtract.
2. Moment weighting: use $e(x)$ and estimating equations to identify causal means.
3. Policy learning: learn a decision boundary that maximizes value.

They are not the same task. They target different functionals.

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9. From CATE to Subgroups

A subgroup selection rule is usually an interpretable set

\[G \subseteq \mathcal X.\]

The subgroup average treatment effect is

\[\tau_G = E[Y^1-Y^0 \mid X \in G].\]

Under identification,

\[\tau_G = E[m_1(X)-m_0(X) \mid X \in G].\]

Equivalently,

\[\tau_G = \frac{E[1\{X \in G\}\tau(X)]} {P(X \in G)}.\]

This shows the mathematical translation:

\[\text{individual covariate-level CATE} \quad \Rightarrow \quad \text{subgroup-level causal contrast}.\]

A subgroup discovery method is therefore not merely finding a region where outcomes are high. It is searching for a region where the treatment contrast is clinically meaningful.

This creates two competing objectives:

\[\text{large subgroup size} \quad \text{and} \quad \text{large subgroup treatment effect}.\]

The hard part is that searching over many possible regions creates multiplicity. Without statistical control, the best-looking subgroup may simply be a selected noise artifact.

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10. Main Takeaway

The causal-inference pipeline is:

\[P_Z \longrightarrow P_{D} \longrightarrow \hat\psi.\]

The first arrow is projection: we lose one potential outcome per unit.

The second arrow is statistical estimation: we estimate an identified functional from observed data.

The key mathematical statement is:

\[\psi(P_Z)=\phi(P_{D}).\]

If this equality is not justified, the problem is not learnable from observed data. If it is justified, then machine learning can help estimate nuisance functions, treatment effects, policy values, and subgroup rules.

In one sentence:

Causal inference is the discipline of turning counterfactual functionals into observed-data functionals before applying statistical learning.

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References

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2. Rubin, D. B. (1974). “Estimating Causal Effects of Treatments in Randomized and Nonrandomized Studies.” Journal of Educational Psychology, 66(5), 688–701. ↩

3. Holland, P. W. (1986). “Statistics and Causal Inference.” Journal of the American Statistical Association, 81(396), 945–960. ↩

4. Pearl, J. (2009). Causality: Models, Reasoning, and Inference. 2nd ed. Cambridge University Press. ↩

5. Hernán, M. A., & Robins, J. M. (2020). Causal Inference: What If. Chapman & Hall/CRC. ↩

6. Robins, J. M. (1986). “A New Approach to Causal Inference in Mortality Studies with a Sustained Exposure Period — Application to Control of the Healthy Worker Survivor Effect.” Mathematical Modelling, 7(9–12), 1393–1512. ↩

7. Künzel, S. R., Sekhon, J. S., Bickel, P. J., & Yu, B. (2019). “Metalearners for Estimating Heterogeneous Treatment Effects Using Machine Learning.” Proceedings of the National Academy of Sciences, 116(10), 4156–4165. ↩

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9. Rosenbaum, P. R., & Rubin, D. B. (1983). “The Central Role of the Propensity Score in Observational Studies for Causal Effects.” Biometrika, 70(1), 41–55. ↩

10. Robins, J. M., Rotnitzky, A., & Zhao, L. P. (1994). “Estimation of Regression Coefficients When Some Regressors Are Not Always Observed.” Journal of the American Statistical Association, 89(427), 846–866. ↩

11. Chernozhukov, V., Chetverikov, D., Demirer, M., Duflo, E., Hansen, C., Newey, W., & Robins, J. (2018). “Double/Debiased Machine Learning for Treatment and Structural Parameters.” The Econometrics Journal, 21(1), C1–C68. ↩

12. Zhao, Y., Zeng, D., Rush, A. J., & Kosorok, M. R. (2012). “Estimating Individualized Treatment Rules Using Outcome Weighted Learning.” Journal of the American Statistical Association, 107(499), 1106–1118. ↩

13. Athey, S., & Wager, S. (2021). “Policy Learning with Observational Data.” Econometrica, 89(1), 133–161. ↩