Why Positivity Depends on the Contrast
A treatment mechanism can assign some actions very rarely, or never, at particular histories. That does not automatically make every regime contrast unsupported. The relevant question is narrower: does the observed treatment mechanism assign positive probability to the actions prescribed by each regime, at the histories that regime can reach? The answer can differ from one contrast to another.
This distinction matters because a global check—asking whether every treatment is possible at every conceivable history—can flag failures irrelevant to a particular estimand. Conversely, a treatment model can return positive fitted probabilities everywhere while concealing histories for which the data provide little or no support. Support must be evaluated against the histories and actions relevant to the regimes being compared.
“Almost every” is important: the condition concerns histories having positive probability under the regime, not every value a history might take in theory. For continuous histories, the condition is understood relative to the distribution of \(H_t^d\), so a set of histories with probability zero under that regime does not itself cause failure. The condition is also distinct from sequential exchangeability and consistency. Positivity concerns available treatment support; it does not establish that confounding has been controlled or that a regime is well defined.
For a realized history path, the treatment-assignment probability product for regime \(d\) is \(G_d=\prod_{t=0}^{K}g_t(d_t(H_t^d)\mid H_t^d)\). It includes the treatment-assignment factors along that path. It is not the probability of the entire covariate-and-treatment path, because it does not include probabilities for covariate transitions.
Products Along Coherent Regime Histories
The history in a later treatment probability must be compatible with the earlier actions prescribed by the same regime. One cannot combine the first action from one pathway with a later history reached under a different first action and call the result a regime-specific product. The following examples keep that correspondence explicit.
Worked Example: Two Regimes Reaching Different Histories
Consider two treatment times, \(t=0,1\), and one baseline group. At time 0, \(P(A_0=1)=0.40\), so \(P(A_0=0)=0.60\). Let the measured history at time 1 be \(H_1=\text{high}\) if \(A_0=1\), and \(H_1=\text{low}\) if \(A_0=0\). At time 1, suppose \(P(A_1=1\mid H_1=\text{high})=0.02\) and \(P(A_1=1\mid H_1=\text{low})=0.70\). Compare \(d=(1,1)\) with \(d'=(0,1)\).
Under \(d\), the first action is 1, so the regime reaches the high history and prescribes 1 again. Its treatment-assignment probability product is \(G_d=0.40(0.02)=0.008\). The inverse product is \(1/0.008=125\).
Under \(d'\), the first action is 0, so the regime reaches the low history and prescribes 1 there. Its product is \(G_{d'}=0.60(0.70)=0.42\), and the inverse product is \(1/0.42\approx2.38\).
Both products are positive, so both regimes meet the pointwise support condition in this example. But their weights differ sharply: in a sample of \(1{,}000\), the corresponding expected numbers following these prescribed treatment sequences are 8 and 420. These are expectations, not guaranteed sample counts. The calculation shows why positivity alone does not ensure stable weighting: a positive product can still be very small.
Worked Example: One Supported Contrast and One Unsupported Contrast
At a single treatment time, let the baseline covariate \(X\) be binary, with \(P(X=1)=0.30\) and \(P(X=0)=0.70\). Suppose \(P(A=1\mid X=0)=0.80\) and \(P(A=1\mid X=1)=0\). Thus \(P(A=0\mid X=0)=0.20\) and \(P(A=0\mid X=1)=1\). Compare three regimes: \(d_0\), which always assigns 0; \(d_1\), which always assigns 1; and \(d_*\), which assigns 1 when \(X=0\) and 0 when \(X=1\).
The products, here just the single-time treatment probabilities, are:
| Regime | Product at \(X=0\) | Product at \(X=1\) | Support over the full target population? |
|---|---|---|---|
| \(d_0\): always 0 | 0.20 | 1.00 | Yes |
| \(d_1\): always 1 | 0.80 | 0 | No |
| \(d_*\): 1 if \(X=0\), otherwise 0 | 0.80 | 1.00 | Yes |
The contrast \(d_*\) versus \(d_0\) is supported: each regime’s prescribed action has positive probability in both baseline groups. The contrast \(d_1\) versus \(d_0\) is not supported for the full population, because \(d_1\) prescribes action 1 for \(X=1\), where its probability is zero and that group has positive population probability. Restricting attention to \(X=0\) would remove that support failure, but would change the target population and hence the estimand.
Worked Example: A Global Failure Irrelevant to a Particular Contrast
Suppose \(P(A_0=1)=P(A_0=0)=0.50\). At time 1, the history is \(H_1=0\) if \(A_0=0\) and \(H_1=1\) if \(A_0=1\). Let \(P(A_1=0\mid H_1=0)=1\), and let \(P(A_1=1\mid H_1=1)=0.05\). Compare \(d=(0,0)\) and \(d'=(1,1)\).
The first regime reaches \(H_1=0\), where it prescribes the only action with positive probability. Its product is \(G_d=0.50(1)=0.50\). The second reaches \(H_1=1\), where its prescribed action has probability 0.05. Its product is \(G_{d'}=0.50(0.05)=0.025\). Both regimes are supported. Yet global positivity fails: at the history \(H_1=0\), action 1 has probability zero. That failure does not affect this pair, because neither regime prescribes action 1 at that history. It would matter for a regime that reaches \(H_1=0\) and prescribes action 1 there.
The second product is small despite being positive; its inverse is 40. Thus the contrast passes the support check but may still be practically unstable.
Two Consequences of the Definition
The product calculation gives a direct test for support along a regime path. The contrast definition also clarifies why support should be assessed separately for both components.
Proof. Each factor \(g_t(d_t(H_t^d)\mid H_t^d)\) is a probability, hence is in \([0,1]\). A finite product of nonnegative numbers is positive exactly when none of its factors is zero. Therefore \(G_d>0\) along a history if and only if the prescribed action has positive conditional probability at every time on that history. Applying this argument at every time and \(\mu_{d,t}\)-almost every history gives the stated sequential positivity criterion. \(\square\)
Justification. If the two regime means are identified under the assumptions used for the analysis, their difference is identified by subtracting those means. This requires support for each mean’s regime, but does not require the regimes to prescribe the same actions or to have the same histories. For the other direction, a support failure can prevent identification of a component mean and therefore of the contrast. For example, in the single-time setting above, suppose \(Y^0=0\) for everyone. Consider two causal models with the same treatment and baseline distributions and the same observed outcomes: in both, \(Y^1=0\) when \(X=0\); when \(X=1\), let \(Y^1=0\) in one model and \(Y^1=1\) in the other. Since \(P(A=1\mid X=1)=0\), \(Y^1\) is never observed in that group. The observed data are identical under the two models, but the mean under the always-1 regime differs by \(P(X=1)=0.30\), while the mean under the always-0 regime is zero in both. The contrast therefore differs as well. This demonstrates why support failure gives no general nonparametric identification guarantee; special structural restrictions could supply information, but positivity alone does not. \(\square\)
Diagnostics, Alternatives, and Limitations
A useful diagnostic starts with the target contrast rather than with a list of all possible treatment-history combinations. For each regime, trace the histories it can reach, identify its prescribed action at each such history, and examine the corresponding conditional treatment probabilities. In data, estimated probabilities and observed frequencies are diagnostics, not proofs of population positivity: a zero count in a finite sample might reflect rarity rather than a true structural zero.
Record the prescribed action at each time as a function of the history, including how earlier actions affect later histories.
Ask which histories have positive probability under each regime and in the target population. Do not test actions at histories irrelevant to that regime as though they were required.
Inspect treatment frequencies and estimated conditional probabilities for the action the regime requires at each relevant history. Calculate or summarize the cumulative products along coherent paths.
A zero probability is structural failure; probabilities close to zero indicate practical positivity problems and potential instability. Check whether a conclusion depends on a few small-probability histories.
With continuous histories, exact history strata may not occur repeatedly. Diagnostics then rely on the support of the history distribution and on how well the data cover regions where a regime prescribes a given action. A fitted model can produce positive probabilities in regions with little empirical information. That is not evidence of strong support; it may be model-based extrapolation.
The principal alternative to an estimand-specific check is a global positivity requirement, which demands treatment availability more broadly across actions and histories. It can be a useful conservative standard, but may be stronger than needed for a specific contrast. Conversely, an estimand-specific check must not omit histories that the relevant regime actually reaches. Outcome-model extrapolation can produce estimates beyond observed support only by adding modeling assumptions; it does not remove the support limitation. Weight truncation may reduce the influence of very large weights, but it does not make a zero-probability treatment observable and can introduce bias. Changing the target population can sometimes avoid unsupported strata, but must be reported as a change in the estimand.
A support diagnosis is therefore not a single threshold decision. Structural zeros preclude the usual data-based identification of the affected regime without extra assumptions. Very small positive probabilities preserve formal positivity but may make estimation highly variable. Estimated probabilities, product distributions, and influential observations can reveal practical problems, but no finite-sample diagnostic can prove that the true conditional probability is strictly positive at every relevant history.
Common Pitfall
Wrong reasoning: “Treatment 1 is impossible at one history, so every contrast involving treatment 1 fails positivity.” This ignores which histories each regime reaches and which actions it prescribes there. In the third example, treatment 1 is impossible at \(H_1=0\), yet the regime prescribing treatment 1 reaches \(H_1=1\), where the probability is positive.
Correction: Check the action prescribed by each regime at histories relevant to that regime. Also check path coherence: if two regimes prescribe different actions at an earlier time, their later histories may differ. A calculation using a later history that is incompatible with the regime’s earlier action is not a valid regime-specific probability product.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- For a two-time regime, the prescribed treatment probabilities along its histories are 0.30 and 0.04. What is the product, and does the regime satisfy pointwise positivity along this path?
- A treatment has probability zero at a history that neither regime in a contrast can reach. Does that fact alone violate positivity for the contrast?
- In the second worked example, why does restricting the target population to \(X=0\) alter the estimand?
- Why does truncating an inverse-probability weight not repair structural positivity failure?
Solutions. (1) The product is \(0.30(0.04)=0.012\). Both factors are positive, so the path satisfies pointwise positivity, although the inverse product is about 83.3 and may signal practical instability. (2) No. Positivity is assessed at histories relevant to the regimes and for their prescribed actions; an unreachable history alone does not violate the contrast-specific condition. (3) The original target averages over both \(X=0\) and \(X=1\), whereas the restricted target averages only over \(X=0\). The population whose regime outcomes are being contrasted has changed. (4) Truncation modifies large finite weights. At a structural zero, the prescribed treatment is never observed at the relevant history, so no finite modification of observed weights supplies the missing outcome information.