Why TMLE Plug-In Estimates Respect Parameter Bounds

Why Respecting Bounds Is a Structural Property

A causal estimand for a bounded outcome must lie within the outcome’s range. For a binary outcome, an intervention-specific risk belongs to \([0,1]\); for an outcome known to lie between \(a\) and \(b\), its mean belongs to \([a,b]\). A longitudinal TMLE plug-in estimate inherits this restriction when its targeted predictions stay in the outcome range and the estimate averages them using valid probability distributions.

This is a property of the estimator’s construction, not a guarantee that the estimate is close to the truth. In particular, a bounded estimate can still be biased by a misspecified or poorly estimated nuisance function, or target the wrong intervention. The guarantee also does not apply indiscriminately to every quantity called a TMLE: it depends on which quantity is averaged and how.

For a longitudinal problem, the outcome regression may be represented by sequential conditional means and then evaluated recursively under the intervention. Write \(\hat Q^*\) for the resulting targeted predictions, including the final prediction for the outcome and the earlier recursive predictions. The relevant requirement is that each prediction being averaged at a step lies in \([a,b]\). For a binary outcome, a logistic-scale fluctuation naturally produces fitted values in \([0,1]\); the previous material on choosing fluctuation submodels for outcome support explains why the choice of fluctuation scale matters.

A deterministic intervention fixes the treatment at each decision point, while a stochastic intervention assigns a probability distribution over treatment choices. In either case, the plug-in calculation averages predictions over probability distributions: the target population distribution and any distributions over histories induced by the intervention. Each such average uses nonnegative weights that sum to one. That normalization is essential to the bound.

The Convex-Averaging Result

Proof. First consider any probability distribution \(G\) and any measurable function \(q\) satisfying \(a \leq q(z) \leq b\) for \(G\)-almost every \(z\). Integrating the two inequalities gives

The integral exists because \(q\) is bounded. This proves that one probability average preserves the interval. Apply this fact at each recursive step: if the conditional predictions being averaged lie in \([a,b]\), their average does too. The finite-sample case is the same argument with a discrete distribution assigning probability equal to each averaging weight. Finally, averaging the resulting predictions over the empirical distribution of the observed baseline covariates is another probability average, so the final plug-in estimate also lies in \([a,b]\). \(\square\)

For example, with baseline observations \(X_1,\ldots,X_n\), a simple plug-in estimate has the form

If the intervention requires averaging over additional histories, replace each summand by the corresponding average under the intervention’s conditional probability distributions. The same proof applies. With nonuniform target-population weights \(w_i\), it applies when \(w_i \geq 0\) and \(\sum_{i=1}^n w_i=1\).

Stochastic Interventions and Recursive Predictions

The bound is not restricted to deterministic treatment regimes. Under a stochastic intervention, treatment choices are averaged according to the intervention’s probability law. These are still probability averages, provided the intervention probabilities are nonnegative and sum to one at each history.

The same reasoning applies to bounded continuous outcomes, not only binary risks. For instance, if a survival-time summary is restricted to the interval from zero to a fixed horizon \(\tau\), a plug-in average of predictions in \([0,\tau]\) remains in that interval. The fact that individual predictions are bounded is substantive: it can follow from the outcome definition, or from a model and fluctuation that enforce the outcome support. If an implementation produces a prediction outside the claimed range, the premise of the theorem has failed.

Why an Influence-Function Update Can Leave the Bounds

An influence-function-based one-step estimate adds a correction to an initial estimate. Unlike a probability average, that addition need not have nonnegative weights or stay inside the parameter space. The empirical correction is not constrained to be small enough to keep the result in range.

Derivation by example. Let \(Y\) be binary, let the target be the mean outcome under treatment \(A=1\), and suppose there are two observations. The fitted outcome regression predicts \(\hat Q(1,X_i)=0.10\) for both people, so the initial plug-in estimate is \(\hat\psi_0=0.10\). Suppose the fitted treatment probability is \(g(1\mid X_i)=0.25\) for both people, and the observed data are \(A_1=1,Y_1=0\) and \(A_2=0\). The empirical residual correction for this target is

Thus the one-step estimate is \(0.10-0.20=-0.10\), outside the possible risk range \([0,1]\). This example shows that the correction can move an estimate outside the parameter space; it does not say that every one-step estimate will do so. In this construction, the inverse-probability factor magnifies a negative residual. The resulting number is an algebraic estimate, not a valid probability.

Bounds, Diagnostics, and Limitations

The plug-in guarantee relies on the actual computation matching the mathematical form. Useful checks include:

• Check the prediction range. Confirm that predictions entering each average are within the outcome bounds, not merely that the final number looks plausible.
• Check the averaging weights. For each averaging operation, verify that weights are nonnegative and sum to one. Signed weights or unnormalized weights do not define a probability average.
• Check what is being reported. A single intervention-specific mean inherits the outcome’s bounds. A difference of means need not: for two risks it lies in \([-1,1]\), and it can be negative.
• Check the target and its support. Extrapolation to histories poorly represented in the data may produce misleading predictions even when a link keeps those predictions inside the outcome range.

A common mistake is to reason, “The estimate uses an influence-function correction, so it must be a mean of bounded predictions.” That does not follow. An additive correction can be negative or positive and can be too large in magnitude, as the calculation above demonstrates. The correction may be useful for other statistical properties, but it is not itself a convex averaging step. Clipping an out-of-range one-step estimate to the nearest bound restores the range numerically, but it changes the estimator and does not establish the properties of the original update.

The reverse mistake is to treat a bounded plug-in estimate as proof that the analysis is valid. The bound alone does not establish identification, adequate positivity, consistency of nuisance estimates, or accurate finite-sample estimation. Nor does it ensure that a fluctuation fit converged successfully. A software pipeline can also accidentally average with incorrect or unnormalized weights, or return predictions outside the outcome range because of a coding or model-scale error. Bound checks are therefore useful diagnostics, but they do not replace checks of the intervention, data support, and estimation procedure.

The key distinction is structural: a plug-in mean is bounded when it is built from in-range predictions using probability averages. An influence-function update adds a signed quantity and has no automatic range constraint. Neither fact, by itself, settles which estimator will perform better in a particular sample.