Positivity Failures and Unstable Sequential Updates

Why Small Probabilities Matter in Sequential Targeting

Longitudinal TMLE targets a parameter by updating sequential outcome regressions. For a specified treatment regime, the clever covariate in an update includes inverse probabilities for following that regime; when censoring is present, it also includes inverse probabilities of remaining observed. These factors connect the observed data to the histories relevant to the target. If a required treatment or observation probability is close to zero, the corresponding clever covariate can be very large.

The issue is not merely that a number in a fitted model is small. A large clever covariate gives a small number of observations substantial influence on a targeting equation. The resulting update can be sensitive to those observations, to small changes in nuisance predictions, and to the details of the fitting procedure. This tutorial focuses on that instability: how it arises, how to quantify it, and what diagnostics and responses are defensible.

Identification requires more than a well-fitting regression. In particular, consistency and sequential exchangeability are paired with positivity: histories with positive probability under the target regime must have positive probability of the required treatment and of remaining observed. A structural violation means that some required probability is zero. A practical positivity problem occurs when these probabilities are positive but so small that the available sample contains little information about the relevant histories. No fitted model can create information about a treatment path that the data essentially never contain.

How Inverse Probabilities Amplify Variation

The amplification can be seen at a single time point. Conditional on history \(H_t\), write \(p=g_t(d_t(H_t)\mid H_t)\) and \(X=I(A_t=d_t(H_t))/p\). Provided \(p>0\),

The mean remains one, but the variance diverges as \(p\) approaches zero. The derivation is direct: \(X\) equals \(1/p\) with conditional probability \(p\) and zero otherwise, so \(E[X^2\mid H_t]=p(1/p^2)=1/p\), and subtracting the squared mean gives \(1/p-1\). Censoring contributes the same kind of amplification: \(R_t/c_t\) has conditional mean one and variance \((1-c_t)/c_t\) among those at risk of censoring.

Proof. Under the stated fixed-probability setup, the probability of matching both treatment decisions is \(p_1p_2\). Thus \(W\) equals \(1/(p_1p_2)\) with probability \(p_1p_2\), and is zero otherwise. Consequently \(E[W]=1\) and \(E[W^2]=1/(p_1p_2)\), which yields the stated variance. With independent observation probability \(c\), the event defining \(W_R\) has probability \(p_1p_2c\), and the same calculation gives \(E[W_R]=1\), \(E[W_R^2]=1/(p_1p_2c)\), and the claimed variance. \(\square\)

A key feature of a sequential update is that the large factor multiplies a residual in the targeting score. For a binary outcome regression with fitted value \(Q\), a logistic fluctuation commonly uses a linear predictor of the form \(\operatorname{logit}(Q)+\epsilon H_t^*\). Its score contribution for an observation is proportional to \(H_t^*(Y-Q^\epsilon)\). A large \(H_t^*\) can therefore make that observation influential in determining \(\epsilon\). The update is sequential: changes to a regression at one time can also affect the pseudo-outcomes or fitted values used at earlier times. Near-positivity can consequently show up as unstable changes across several targeting steps, not just as one conspicuous weight.

Diagnosing Instability in Practice

Diagnostics should examine the probabilities and the resulting cumulative covariates, not only the final estimate. In particular, inspect treatment probabilities for the regime decision at each time, observation probabilities among those still at risk, and the cumulative clever covariate at every targeted regression. Summaries such as minimum and lower-tail probabilities, upper quantiles of the cumulative covariate, and the largest individual score contributions can reveal whether a small set of records dominates the update.

A useful supplementary summary is the weight-based effective sample size, \(n_{\mathrm{eff}}=(\sum_i H_i^*)^2/\sum_i(H_i^*)^2\), when the covariates being summarized are nonnegative. It is a diagnostic of concentration, not a replacement for the sampling variance of the estimator. Also compare fitted predictions and the target estimate before and after targeting, and check whether small changes in nuisance fits or probability bounds produce large changes in the result. A large maximum alone is a warning, not a universal cutoff; its importance depends on how often large values occur and on their associated residuals.

Responses, Alternatives, and Limits

Several responses address different problems and should not be treated as interchangeable. First, check coding, time indexing, and fitted probabilities: an apparent extreme value can come from a history encoded incorrectly or a probability computed for the wrong treatment. If the model is correct but estimates are noisy, improved nuisance estimation or additional data may help, although neither removes a genuine lack of support. Probability truncation can limit leverage but changes the estimating equation, as the example showed.

Restricting analysis to a population with better overlap can improve stability, but generally changes the target population and therefore the estimand. Changing a deterministic regime to a stochastic intervention can sometimes reduce demands for unsupported treatment decisions; it changes the intervention being evaluated, so the result answers a different question. The appropriate response depends on the scientific target, not only on which choice produces a more stable estimate. Unlike an analysis that simply omits the targeting step, a targeted estimate still faces the same weak-support limitation: targeting does not manufacture observations in rare histories.

A structural zero is especially consequential. If the intervention requires a treatment that never occurs for a history with positive probability, the observed data do not identify the regime-specific outcome there without additional assumptions. A numerical algorithm may still return a value, but that does not establish identification. With near-zero probabilities, the target may be identified in principle yet estimated imprecisely in the available sample. In either case, report the support diagnostics, the treatment and censoring models used, and sensitivity to any stabilization choice.

A common mistake is to see a finite targeted estimate and conclude that positivity must have been adequate. The reasoning is wrong because finite software output only shows that the fitted calculations completed; it does not show that the target histories were represented well. The correction is to inspect the probabilities, cumulative clever covariates, and score contributions, and then assess whether the scientific target is supported by the observed histories.