Potential outcomes and identification

Fix a target population with probability law $P$, a real outcome, and a nonempty countable discrete set $\mathcal{A}$ of treatment versions. A treatment version $a\in \mathcal{A}$ is a specified intervention protocol, not just a treatment label. The potential outcome $Y^a$ is the real random variable giving the outcome for a unit in the target population under version $a$. Write $A$ for the observed treatment assignment or receipt, with values in $\mathcal{A}$, and $Y$ for the observed outcome. The observed potential outcome is the measurable random variable

$Y^A={\sum }_{a\in \mathcal{A}}{\mathbf{1}}_{\{A=a\}}{Y}^a,$

where the countable sum has at most one nonzero summand at each sample point. Consistency is the assumption $Y=Y^A$ $P$-almost surely. It links the observed outcome to the potential outcome for the received version; it does not assert that treatment assignment is unconfounded.

The full-data law is the joint law of the potential-outcome family, $A$, any baseline covariate random element $L$, and $Y$. The observed-data law is its marginal law of $(A,L,Y)$. A target population, outcome, intervention protocol, and contrast must be fixed before the associated causal target is specified. If interference between units matters, the version or treatment regime must include the relevant assignment or exposure of other units; the notation $Y^a$ by itself does not assert no interference.

Let $A$ take values in a countable discrete set on a target probability space, and let $L$ be the baseline covariate random element. For a treatment value $a$ of interest, the conditional treatment probability is the almost-sure class

$e_a=E[{\mathbf{1}}_{\{A=a\}}\mid \sigma (L)],$

using the conditional-expectation convention of Conditional expectation as an ae class under the The Axiom of Choice. Positivity for $a$ on the target population means both $P(A=a)>0$ and $e_a>0$ almost surely under the target law of $L$. Positivity for a contrast is required at every treatment value in that contrast.

The almost-sure condition is relative to the target population: values of $L$ with zero target probability impose no condition. Strict positivity does not mean that $e_a$ is bounded away from zero. The marginal condition makes the observed treatment-stratum law $P(\cdot \mid A=a)$ available.

Assume the The Axiom of Choice for the conditional-expectation classes. Let $U$ be a real integrable random variable, let $Z$ be a random element, and let $\mathcal{G}$ be a sub-sigma-algebra. If $U$ and $Z$ are conditionally independent given $\mathcal{G}$ in the bounded-function sense of Conditional independence given a sigma-algebra, then for every bounded measurable real function $h$ on the state space of $Z$,

$E[Uh(Z)\mid \mathcal{G}]=E[U\mid \mathcal{G}]E[h(Z)\mid \mathcal{G}]\text{almost surely}\mathrm{.}$

Let $A$ be the observed treatment assignment, $L$ a baseline covariate random element, and $(Y^a{)}_{a\in \mathcal{A}}$ the potential-outcome family of Potential outcomes and consistency. Conditional exchangeability given $L$ means

$\sigma (Y^a:a\in \mathcal{A})\perp \text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\perp \sigma (A)\mid \sigma (L),$

where conditional independence is the bounded-measurable-function factorization of Conditional independence given a sigma-algebra under the The Axiom of Choice. Each fixed $Y^a$ is then conditionally independent of $A$ given $L$ by measurable-map preservation in Conditional-independence equivalences and preservation. This is a full-data assumption; the observed distribution alone does not establish it.

In a fixed target population with two specified treatment versions $0$ and $1$ and integrable potential outcomes $Y^0,Y^1$ from Potential outcomes and consistency, the average treatment effect (ATE) is the target-population mean contrast $E[Y^1-Y^0]$. For a baseline covariate random element $L$, the conditional average treatment effect (CATE) given $L$ is the conditional-expectation almost-sure class

$E[Y^1-Y^0\mid \sigma (L)],$

using the conditional-expectation convention of Conditional expectation as an ae class under the The Axiom of Choice. Thus a CATE is determined only up to equality almost surely under the target law of $L$; a pointwise value at a null covariate value is not specified by this definition. The ATE is the mean of the individual contrast, not an assertion that both potential outcomes are observed for any unit.

Assume the Axiom of Choice. Let $\mathcal{A}$ be a countable discrete set containing treatment versions $0$ and $1$. On a full-data probability space $(\mathrm{\Omega },\mathcal{F},P)$, let $A$ take values in $\mathcal{A}$, let $L$ take values in a standard-Borel space $\mathcal{L}$ (Standard Borel spaces), let $Y$ be the observed real outcome, and let $(Y^b{)}_{b\in \mathcal{A}}$ be the potential outcomes of Potential outcomes and consistency. For each treatment value $a$ under consideration, assume $Y^a\in L^1(P)$, consistency $Y=Y^A$ almost surely, conditional exchangeability $\sigma (Y^b:b\in \mathcal{A})\perp \text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\text{\hspace{0.17em}⁣}\perp \sigma (A)\mid \sigma (L)$, and positivity $P(A=a)>0$ and $E[{\mathbf{1}}_{\{A=a\}}\mid \sigma (L)]>0$ almost surely. These are assumptions on the full-data law; Choice is required for the stated conditional-expectation classes and regular conditional distributions.

Write $P_L(B)=P(L\in B)$ for the target-population law of $L$. For such an $a$, define the treatment-stratum probability $P_a(C)=P(C\cap \{A=a\})\mathrm{/}P(A=a)$ for $C\in \mathcal{F}$. Under $P_a$, let $K_a(l,dy)$ be a regular conditional law of $Y$ given the standard-Borel random element $L$, and put $m_a(l)=\int y{K}_a(l,dy)$ wherever the absolute integral is finite, with value $0$ on the exceptional set. The suppliers Conditional law given a random element, Existence of regular conditional distributions for standard borel targets, Regular conditional kernels factor through a standard borel conditioning variable, and Conditional integration through a regular conditional law give this construction. Then $m_a$ is $P_L$-integrable and is uniquely determined $P_L$-almost everywhere by the observed-data law, and

$E_P[Y^a]={\int }_{\mathcal{L}}{m}_a(l)P_L(dl)\mathrm{.}$

If $L$ is countably discrete, this is

$E_P[Y^a]={\sum }_{l:P(L=l)>0}{E}_P[Y\mid A=a,L=l]P(L=l)\mathrm{.}$

In this sum, $E[Y\mid A=a,L=l]$ denotes $E[Y{\mathbf{1}}_{\{A=a,L=l\}}]\mathrm{/}P(A=a,L=l)$, whose denominator is positive for each displayed atom.

For the binary contrast, both treatment levels must satisfy positivity. The CATE is then represented $P_L$-almost everywhere by $m_1-m_0$, and the ATE is $\int (m_1-m_0)d{P}_L$. In the model whose parameters are all full-data laws satisfying these assumptions and whose observation map sends a full-data law to its $(A,L,Y)$ marginal, each $E[Y^a]$ is identifiable in the sense of Identifiability of an estimand: it is constant over full-data laws with the same observed-data law.

This theorem is for discrete treatment values. For continuous $A$, conditional-law versions are uniquely determined only $P_A$-almost everywhere: any two regular conditional laws agree outside a single conditioning-law null set by Simultaneous ae uniqueness of regular conditional distributions, and factoring through $A$ transfers that qualification to $P_A$. The theorem asserts no pointwise causal effect at a $P_A$-null value; a specified version and any continuity or smoothness assumptions need a separate proof. Conditional exchangeability is generally untestable from the observed law. If positivity fails on a target stratum, that stratum's counterfactual mean is unsupported by the observed treatment data; the theorem does not identify the full target by this formula.