Identifying a Counterfactual Mean by Backward Recursion

Why Identify the Mean Backward?

A regime-specific mean asks what the average outcome would be if everyone followed a specified sequence of treatment decisions. The difficulty is longitudinal: a treatment can affect later covariates, and those covariates can affect both later treatment and the outcome. Under suitable assumptions, the counterfactual mean can nevertheless be expressed entirely through conditional expectations in the observed data. Backward recursion provides a particularly transparent way to construct that expression.

The target parameter is the regime-specific mean introduced in the earlier material on regime-specific mean parameters. The purpose here is to derive its identification formula and show precisely where consistency, sequential exchangeability, and positivity enter. This is an identification result, not yet an estimation algorithm: it does not specify how to fit the conditional expectations from a finite sample.

Histories, Regimes, and Counterfactuals

Consider decision times $t=0,\ldots,K$. Let $W$ denote baseline covariates, $A_t$ the treatment at time $t$, and $L_{t+1}$ the covariates observed after treatment $A_t$ and before the next decision. The terminal outcome is $Y$, observed after the final treatment $A_K$. Write $\bar A_t=(A_0,\ldots,A_t)$ and $\bar L_t=(L_1,\ldots,L_t)$. The information available immediately before decision $t$ is

H_t=(W,\bar L_t,\bar A_{t-1}), \qquad t=0,\ldots,K,

where $H_0=W$, and $\bar A_{-1}$ is empty. A regime $d=(d_0,\ldots,d_K)$ specifies a treatment value as a function of the history: $A_t=d_t(H_t)$. A regime is static if each $d_t$ is constant and dynamic if its prescribed value can depend on observed history.

Under $d$, treatment is set according to the rule at each time. The resulting covariates and outcome are denoted $L_{t+1}^{d}$ and $Y^{d}$; the history generated under the regime before time $t$ is $H_t^{d}$. The target is $\psi(d)=E[Y^{d}]$. Assume $E[|Y^{d}|]<\infty$, so this mean exists.

Definition: Define $Q_{K+1}^{d}=Y$ as a terminal function of the terminal history. For $t=K,\ldots,0$, define the observed-data regressions recursively by $$ Q_t^{d}(h_t)=E\!\left[Q_{t+1}^{d}(H_{t+1})\mid H_t=h_t,\ A_t=d_t(h_t)\right], $$ where $H_{t+1}=(H_t,A_t,L_{t+1})$ for $t<K$, and $H_{K+1}$ includes $H_K,A_K,Y$, with $Q_{K+1}^{d}(H_{K+1})=Y$. The recursion is understood on histories where the conditioning events are supported.

In the final step, $Q_K^{d}$ is the conditional mean of $Y$ given the history at time $K$ and the prescribed final treatment. At earlier steps, the recursion averages the next regression over the covariate distribution following the prescribed treatment. Thus it works from the outcome back toward baseline.

Identification Assumptions

For clarity, the statement below uses discrete treatments, so the probability of a prescribed treatment value can be positive. The same identification principle applies to continuous treatments when the intervention is supported by the observed conditional treatment distribution, expressed using the appropriate density or absolute-continuity condition.

Assumptions:

Consistency. If a person's observed treatment history agrees with a specified treatment history, their observed covariates and outcome equal the corresponding counterfactual covariates and outcome. In particular, when observed treatment follows $d$ through time $K$, $Y=Y^{d}$.
Sequential exchangeability. At each time $t$, conditional on the observed history $H_t$, treatment assignment is independent of the counterfactual future covariates and outcome under the regime being considered. This condition is required among histories reached when prior treatment has followed that regime.
Positivity. At every history reached under $d$, the prescribed value is available: $P(A_t=d_t(h_t)\mid H_t=h_t)>0$, for histories with positive probability under the regime.
Well-defined histories and integrability. The conditional distributions and expectations in the recursion exist, and $Y^{d}$ is integrable.

Sequential exchangeability is a no-unmeasured-confounding condition for each treatment decision, conditional on the recorded history. It is not a claim that treatment is randomized marginally. Positivity is distinct: even with no confounding, the data cannot directly reveal the outcome distribution under a treatment value never used at a relevant history.

The Backward Identification Result

Theorem: Under consistency, sequential exchangeability, positivity, and the existence conditions above, the regime-specific mean is identified by $$ \psi(d)=E[Q_0^{d}(W)]. $$ The expectation on the right is over the observed baseline distribution. Each regression $Q_t^{d}$ is defined by the backward recursion above, using observed-data conditional distributions.

The central step is to show that each observed-data regression equals the conditional mean of the counterfactual outcome among people whose histories have been generated under the regime. At time $t$, let $Y^{d}$ denote the outcome that would follow the regime from the observed history at $t$ onward, with the regime applied at subsequent decisions. On histories consistent with $d$ up to $t$, the induction claim is

Q_t^{d}(H_t)=E[Y^{d}\mid H_t,\ \bar A_{t-1}=\bar d_{t-1}]

where $\bar d_{t-1}$ means that each earlier treatment was the value prescribed at that person's history. At $t=0$, this earlier-treatment condition is empty.

Start at the final decision.
By consistency, among people with the regime-consistent history and $A_K=d_K(H_K)$, the observed outcome equals the outcome under the prescribed final treatment. Sequential exchangeability allows its conditional mean to represent the corresponding counterfactual mean. Therefore $Q_K^{d}(H_K)=E[Y^{d}\mid H_K,\bar A_{K-1}=\bar d_{K-1}]$.

Move one time point earlier.
Suppose the claim holds at $t+1$. Given a history consistent with the regime through $t-1$, average $Q_{t+1}^{d}(H_{t+1})$ over the observed distribution of the next covariates among those with $A_t=d_t(H_t)$. Consistency links that covariate distribution to the counterfactual distribution under the prescribed treatment, and sequential exchangeability permits this conditional distribution to identify the counterfactual one.

Reach baseline.
Repeating the step yields $Q_0^{d}(W)=E[Y^{d}\mid W]$. Taking expectation over the baseline distribution gives the marginal counterfactual mean.

For completeness, the induction step can be written directly. Suppose at $t+1$ the conditional mean representation holds. Conditional on a regime-consistent history at time $t$, sequential exchangeability and consistency identify the distribution of $H_{t+1}^{d}$ with the observed distribution of $H_{t+1}$ conditional on $H_t$ and $A_t=d_t(H_t)$. Hence

\begin{aligned} E[Y^{d}\mid H_t,\bar A_{t-1}=\bar d_{t-1}] &=E\!\left[E[Y^{d}\mid H_{t+1}^{d},\bar A_t=\bar d_t]\mid H_t,\bar A_{t-1}=\bar d_{t-1}\right]\\ &=E\!\left[Q_{t+1}^{d}(H_{t+1})\mid H_t,A_t=d_t(H_t)\right]\\ &=Q_t^{d}(H_t). \end{aligned}

The first equality is the tower property of conditional expectation, with the next counterfactual history as the conditioning information. The second uses the induction claim and the identified conditional distribution of the next history. Positivity ensures that the observed conditional expectation in the last two lines is defined on the histories needed by the regime. This proves the claim backwards to $t=0$. Finally, the law of total expectation gives $E[Y^{d}]=E\{E[Y^{d}\mid W]\}=E[Q_0^{d}(W)]$. $\square$

Worked Calculations

Worked Example: One Decision and Baseline Standardization

Let $W$ be binary, with $P(W=0)=0.6$ and $P(W=1)=0.4$. There is one treatment decision, and the target regime sets $A_0=1$. Suppose the observed conditional outcome means are $E[Y\mid W=0,A_0=1]=2$ and $E[Y\mid W=1,A_0=1]=5$. Assume consistency, sequential exchangeability, and positivity in both strata.

With $K=0$, the recursion has just $Q_0^{d}(w)=E[Y\mid W=w,A_0=1]$. Thus $Q_0^{d}(0)=2$ and $Q_0^{d}(1)=5$. Average over the baseline distribution:

\psi(d)=E[Q_0^{d}(W)]=(0.6)(2)+(0.4)(5)=1.2+2.0=3.2.

The calculation does not average the treated outcomes using the proportion treated in the sample. It averages the stratum-specific conditional means using the target population's baseline distribution. The result is the mean identified for assigning treatment to everyone, under the stated assumptions.

Worked Example: Two Decisions With an Intermediate Covariate

Suppose $W$ is constant, $A_0,A_1\in\{0,1\}$, and the regime sets both treatments to 1. After $A_0$, the binary covariate $L_1$ is observed. In the observed data, $P(L_1=1\mid A_0=1)=0.3$, and the final conditional means are $E[Y\mid A_0=1,L_1=0,A_1=1]=2$ and $E[Y\mid A_0=1,L_1=1,A_1=1]=8$. Assume the identification conditions and positive probability for the prescribed treatments at relevant histories.

At the final decision, the backward recursion gives $Q_1^{d}(0)=2$ and $Q_1^{d}(1)=8$. The time-zero regression averages these values over the distribution of $L_1$ following $A_0=1$:

Q_0^{d}=E[Q_1^{d}(L_1)\mid A_0=1] =(0.7)(2)+(0.3)(8)=1.4+2.4=3.8.

Since $W$ is constant, $E[Q_0^{d}(W)]=3.8$. The intermediate covariate is not held fixed at one value: the regime-specific mean averages over the covariate distribution that would follow the prescribed first treatment.

Worked Example: A Positivity Failure

Let $W\in\{0,1\}$, with both values occurring, and suppose the regime prescribes $A_0=1$ for everyone. If $P(A_0=1\mid W=0)=0$, the conditional mean $E[Y\mid W=0,A_0=1]$ is not identified from the observed data. The recursion therefore cannot evaluate $Q_0^{d}(0)$.

This is not repaired by knowing the baseline probability $P(W=0)$ or by having many observations with $W=0$: none of those observations received the prescribed treatment. For example, two causal models can agree on all observed data at $W=0$ (where everyone has $A_0=0$) but assign different values, say 1 and 9, to $E[Y^{A_0=1}\mid W=0]$. They produce the same observed distribution but different values of $\psi(d)$. The counterfactual mean is thus not identified without additional assumptions or extrapolation.

Why the Recursion Matters: Diagnostics and Alternatives

Backward recursion is a factorization of the identifying calculation into conditional means. A forward g-formula instead writes the mean as a sum or integral over the full sequence of covariate histories under the regime. The two are equivalent: repeated application of the tower property collapses the forward average into the backward conditional expectations. The backward form emphasizes the outcome regression at each stage; the forward form emphasizes averaging over the regime-induced history distribution.

Inverse-probability weighting is another g-method. Rather than estimating a sequence of outcome regressions, it reweights observed outcomes according to treatment probabilities. These approaches have different statistical behavior, but neither makes the identification assumptions optional. In particular, weighting cannot create support for a treatment value absent from a history stratum, and regression-based recursion also cannot identify an unsupported conditional mean without extrapolation.

Useful checks focus on whether the recursion is being evaluated where the data provide information:

For each time and regime-relevant history, inspect the frequency or estimated probability of the prescribed treatment. Near-zero probabilities indicate practical positivity problems; zeros indicate a failure of the discrete-treatment condition.
Check that the covariates required by the regime are measured before each treatment decision. Omitting a time-varying confounder can invalidate sequential exchangeability even if the recursion is computed correctly.
Check that each conditional mean is averaged over the covariate distribution following the prescribed treatment, not an unrelated distribution such as the pooled distribution of $L_{t+1}$.
Check that the time ordering is coherent. A covariate measured after $A_t$ cannot be included in $H_t$ as if it were known before that treatment.

The identification result is not a guarantee of accurate estimation. With many time points, histories can become sparse, conditional mean models can be misspecified, and small treatment probabilities can make the data weakly informative. The theorem also depends on no unmeasured confounding and correct consistency; those are substantive assumptions, not features that can be verified from the observed distribution alone.

Common Pitfall

A frequent error is to replace the regime-induced covariate distribution with the observed marginal distribution. In the two-decision example, using a pooled $P(L_1=1)$ in place of $P(L_1=1\mid A_0=1)$ would average the final means over the wrong distribution. The correction is to use the conditional distribution after the regime-prescribed treatment at each step. This distinction matters because earlier treatment can change later covariates, and those covariates can change the outcome.

Key takeaway: Under consistency, sequential exchangeability, and positivity, identify the regime-specific mean by starting with the terminal outcome regression, conditioning at each time on the prescribed treatment, and averaging backward until only baseline covariates remain. Then average over the observed baseline distribution.

Check Your Understanding

Answer each question, then compare with the brief solutions.

In a two-decision study, what information belongs in $H_1$, and why must it be measured before $A_1$?
Suppose $Q_1^{d}(0)=4$, $Q_1^{d}(1)=10$, and under the prescribed first treatment $P(L_1=1)=0.25$. Calculate $Q_0^{d}$.
Which assumption is directly violated if an unmeasured prognostic factor affects both a treatment decision and the counterfactual outcome, even after conditioning on the recorded history?
Why does a zero probability of the prescribed treatment at a regime-relevant history prevent identification by the backward recursion?

Solutions. (1) $H_1$ contains $W,L_1,A_0$: baseline information, the covariate observed after the first decision, and the earlier treatment. It must be available before $A_1$ so it can define the history used to assign or condition on that decision. (2) $Q_0^{d}=(0.75)(4)+(0.25)(10)=3+2.5=5.5$. (3) Sequential exchangeability fails because the recorded history does not suffice to remove treatment-outcome confounding. (4) The observed conditional expectation given the prescribed treatment is unsupported at that history, so the relevant conditional mean cannot be recovered from the observed data without extra assumptions.