Positivity Along Counterfactual Histories

Why the Histories Under a Regime Matter

In longitudinal studies, treatment can change covariates that affect later treatment and outcome. Consequently, the histories encountered under a specified treatment regime need not resemble the histories most often seen under usual care. A regime might make a particular covariate value common, even when that value is rare in the observed data. Positivity must therefore be assessed along the histories the regime induces.

The key distinction is between a history being uncommon under observed care and the required treatment being unavailable at that history. Uncommon histories can create unstable estimates; histories at which the required treatment has probability zero create a failure of support. If the regime makes such a history possible, the observed data do not provide the comparison needed to identify its effect.

Definition: Let $L_0$ be baseline covariates, and, for decision times $t=0,\ldots,K$, let $A_t$ be treatment and $L_t$ the covariate history measured before that decision. Define $H_t=(L_0,A_0,L_1,\ldots,A_{t-1},L_t)$. A deterministic regime is $\bar a=(a_0,\ldots,a_K)$. Its counterfactual history at time $t$, $H_t^{\bar a_{t-1}}$, is the history that would arise under the regime’s actions through time $t-1$. Positivity along $\bar a$ requires $P(A_t=a_t\mid H_t=h)>0$ for $Q_t$-almost every $h$, where $Q_t$ is the distribution of $H_t^{\bar a_{t-1}}$. For $t=0$, $Q_0$ is the baseline covariate distribution.

This condition is about the population distribution of regime-induced histories, not just about a list of histories that happen to appear in a finite sample. For continuous covariates, “$Q_t$-almost every” allows exceptions on sets that have probability zero under the regime. The treatment actions in these examples are discrete; continuous treatments require the corresponding density or local-support formulation.

From Sequential Support to Regime-History Support

The following derivation makes the support requirement precise. Assume consistency and sequential exchangeability, and suppose the conditional distributions used below are well-defined. In particular, exchangeability must suffice to transport the observed covariate transition laws to the counterfactual regime. Treatment is discrete, and $p_t(a_t\mid h)=P(A_t=a_t\mid H_t=h)$. For ease of display, suppose the covariates have densities; the same argument applies to probability mass functions or general conditional probability measures.

Result: Under consistency and sequential exchangeability, the density of the covariate history generated by $\bar a$ through time $t$ factors as $$ q_t(l_0,\ldots,l_t)=f(l_0)\prod_{s=0}^{t-1} f(l_{s+1}\mid h_s,A_s=a_s). $$ The observed joint density of that covariate history and the matching treatment sequence is $$ m_t(l_0,\ldots,l_t) =f(l_0)\prod_{s=0}^{t-1} p_s(a_s\mid h_s)f(l_{s+1}\mid h_s,A_s=a_s). $$ If $p_s(a_s\mid h_s)>0$ for $Q_s$-almost every regime-induced history at every $s<t$, then $Q_t$ is absolutely continuous with respect to the observed covariate-history distribution among those with $\bar A_{t-1}=\bar a_{t-1}$.

Proof. Under the intervention, the baseline distribution is $f(l_0)$. At each time $s$, the regime sets $A_s=a_s$, and the conditional law of the next covariate is $f(l_{s+1}\mid h_s,A_s=a_s)$, by consistency and the exchangeability condition. Multiplying these successive conditional densities gives the displayed factorization for $q_t$.

Under observed care, the joint law additionally requires each action in the sequence to be assigned. At history $h_s$, that assignment has conditional probability $p_s(a_s\mid h_s)$. The chain rule therefore gives the displayed factorization for $m_t$. Wherever $q_t>0$ and the required treatment probabilities are positive, division yields $$ \frac{q_t(l_0,\ldots,l_t)}{m_t(l_0,\ldots,l_t)} =\prod_{s=0}^{t-1}\frac{1}{p_s(a_s\mid h_s)}. $$ Thus a set with zero observed joint density $m_t$ cannot have positive $q_t$ density under the stated condition. Equivalently, the regime-induced history distribution is absolutely continuous with respect to the observed distribution on the matching treatment path. This establishes the result. $\square$

The ratio also explains why an uncommon path can be difficult even when it is supported: small treatment probabilities produce large ratios. The derivation does not say that every regime-induced history must be common in the observed data. It says that the observed treatment path must have positive probability wherever the regime assigns positive probability to a history.

Worked Example: A Common Regime History From a Rare Treatment Path

Restrict attention to people with baseline $L_0=0$, and consider the regime $A_0=1,A_1=1$. Suppose $P(A_0=1\mid L_0=0)=0.02$; under $A_0=1$, $P(L_1=1\mid L_0=0,A_0=1)=0.80$; and at that history $P(A_1=1\mid L_0=0,A_0=1,L_1=1)=0.25$.

Under the regime, $A_0$ is set to 1, so $P(L_1^{A_0=1}=1\mid L_0=0)=0.80$. Thus $L_1=1$ is a common regime-induced history. The probability under observed care of the matching path $A_0=1,L_1=1,A_1=1$, conditional on $L_0=0$, is $$ 0.02\times 0.80\times 0.25=0.004. $$ It is only 0.4%, but it is positive. At this history the regime’s required second treatment is supported, because its observed conditional probability is $0.25>0$. The density ratio for this path is $1/(0.02\times0.25)=200$. Therefore the path is available in principle but can contribute very unstable estimates: a few observations may carry large weights.

What Zero Support Implies

A structural zero along a regime-induced history is different from a small probability. If, on a set of histories with positive probability under the regime, the required treatment never occurs, the data contain no observations of the needed treatment at those histories. The following construction shows why this can be a failure of identification, rather than merely a computational inconvenience.

Result: If a target regime assigns positive probability to a stratum of histories on which its required treatment has probability zero, then, without additional assumptions, the observed data need not identify the regime’s mean outcome.

Proof by construction. Let baseline $X$ be binary, with $P(X=1)=0.4$. Suppose $A=1$ is never observed when $X=1$, while $P(A=1\mid X=0)>0$. Consider the target regime that sets $A=1$ for everyone. For individuals with $X=0$, let $Y^1=0.2$ in both of two causal models. For individuals with $X=1$, let $Y^0=0$ in both models, but set $Y^1=0$ in the first model and $Y^1=1$ in the second.

The observed data have the same distribution in both models: when $X=1$, only $A=0$ is observed, so the different values of $Y^1$ are never revealed. Yet the target regime means differ. In the first model the mean is $0.6(0.2)+0.4(0)=0.12$; in the second it is $0.6(0.2)+0.4(1)=0.52$. Since identical observed data correspond to different target means, the mean is not identified from those data alone. $\square$

Worked Example: Marginal Treatment Frequency Hides a Zero

Suppose $P(X=0)=0.60$, $P(X=1)=0.40$, and the target is $A=1$ for everyone. Observed treatment probabilities are $P(A=1\mid X=0)=0.10$ and $P(A=1\mid X=1)=0$.

The marginal treatment probability is $$ P(A=1)=0.60(0.10)+0.40(0)=0.06. $$ A check using only this marginal value might conclude that $A=1$ occurs in the data. But the regime includes the $X=1$ stratum, which has probability $0.40$, and in that stratum the required treatment has probability zero. Positivity fails for the target regime despite the positive marginal frequency. Restricting attention to $X=0$ would answer a different target question; it would not identify the original population mean without further assumptions.

Worked Example: A Rare History Is Not Automatically a Structural Failure

Suppose a history $H_t=h$ occurs in 1% of the observed population, but among people at that history the required treatment has probability $0.30$. If the regime-induced distribution assigns positive probability to $h$, the treatment is supported there: $0.30>0$. The expected observed fraction following the required treatment from this history is $0.01\times0.30=0.003$, or 0.3% of the population.

This small fraction warns that the relevant conditional distribution may be estimated imprecisely. It does not establish a population-level zero. Conversely, if the true conditional probability is zero, observing no treated individuals is consistent with a structural failure, regardless of how large the overall sample is. The distinction requires subject-matter knowledge and careful assessment of the treatment mechanism; a sample count alone cannot prove a population probability is exactly zero.

Assessment, Alternatives, and Limitations

A useful assessment follows the target regime forward. At baseline, identify the population histories included by the target. At each decision, ask whether the regime-induced histories up to that point can occur under observed care, and whether the required treatment has positive probability at those histories. Then consider whether the supported paths occur often enough for reliable estimation.

Specify the target regime.
Write down the treatment action required at each decision, including how actions depend on history if the regime is dynamic.

Trace its induced histories.
For each decision, determine which covariate histories can arise after imposing the earlier actions. Do not substitute the marginal distribution of histories under usual care.

Check conditional support there.
Assess the probability of the required action at histories that have positive probability under the regime. For continuous covariates, assess support in the relevant conditional distribution, not probability at an exact point.

Separate zero from rarity.
Look for structural restrictions, sparse strata, and unstable estimated probabilities. Sparse data make support difficult to assess, but are not by themselves proof of a population zero.

This history-by-history assessment is more informative than a marginal treatment-frequency check. A joint positivity statement about the entire treatment sequence can also express support, but it can hide which decision and which history cause the problem. Sequential assessment makes the source of a failure visible. For a stochastic intervention, the corresponding question is whether the intervention’s treatment distribution places mass only where the observed conditional treatment distribution has support; a stochastic rule cannot create observed data at an unsupported history.

Diagnostics include tabulating treatment probabilities within clinically meaningful histories, examining fitted probabilities over regime-relevant histories, and comparing how often those histories occur under observed care and under the target regime. These tools are limited: fitted probabilities can conceal separation or sparse strata, and a model’s prediction of a positive probability is not proof of empirical support. Truncating weights can control extreme estimates, but it does not restore identification at an exact zero. Restricting the target population or changing the intervention can address the scientific question differently; extrapolating outcomes into a region with no support requires additional, explicit modeling assumptions.

Common Mistake

Pitfall: “The required treatment occurs in the sample, so positivity holds.” This checks only whether treatment occurs somewhere. The correction is to check its conditional probability at histories that the target regime can generate. A treatment may be common overall and entirely absent in one regime-relevant stratum.

A related error is to treat histories that are rare under usual care as irrelevant. The regime can shift the history distribution, so a rare observed history may account for a substantial share of the target population. The support question concerns the regime’s distribution of histories; the observed frequency matters for precision and diagnostics, but does not replace that target-specific condition.

Check Your Understanding

Answer each question by following the target regime’s histories, not just overall treatment frequencies.

A regime induces a history with probability 0.20, and the required treatment probability there is 0. What part of the positivity condition fails?
At a regime-relevant history, the required treatment probability is 0.05. Is this necessarily a structural positivity failure? What concern remains?
In the marginal-frequency example, why does $P(A=1)=0.06$ not establish positivity for the regime $A=1$ for everyone?
Under what condition can a structural zero produce nonidentification of the regime mean?

Solutions. (1) The regime gives positive probability to the history, but the required treatment has conditional probability zero there, violating positivity along the regime. (2) No. The probability is positive, so this is not a structural zero; however, the relevant observations may be sparse and estimates unstable. (3) The marginal probability averages over $X$. It is positive because treatment occurs when $X=0$, but the required treatment is absent when $X=1$, a stratum included by the target. (4) If the regime assigns positive probability to a history where its required treatment is never observed, the observed data may leave the corresponding counterfactual outcome unrestricted. Distinct values for that outcome can yield the same observed distribution but different regime means.