What the nuisances must satisfy#

The union condition, Q̄ = Q̄_0 or g = g_0, is assumed and not checked. It is not enough. The release claim rests on the rate conditions of the remainder section and on the conditioning terms. With one nuisance wrong, those terms vanish only in the cases that section lists. The rate conditions fall on five estimated functions rather than two for the univariate construction:

  1. the primary outcome regression Q̄_n;

  2. the primary propensity g_n;

  3. the reduced outcome regression Q_r;

  4. the reduced propensity g_{r,1};

  5. the reduced propensity g_{r,2}.

The bivariate construction instead has four: the same two primary nuisances, Q_r, and its single two-column reduced probability g_r.

The reduced regressions are the part it is easy to forget. They estimate conditional means given the estimated nuisance. A univariate rate for them does not control the conditioning term, which conditions on the true nuisance as well.

Their consistency is estimated, unmeasured: a saturated learner recovers them exactly on the exact law, which is consistency at one learner on one law and not a rate. An archived study over 6,000 fits measured 0.844/0.848 coverage against a plain TMLE’s 0.532/0.472 in the cell built for it. It was nominal nowhere, the best reading being 0.880, with the three reductions fitted by glm. A committed probe tested the conditioning term as its cause, under a rule fixed before it ran. The shortfall is not attributed. The conditioning term at the mean fit does not account for it (the validation programme).

Practical consequences:

  • A flexible learner for the consistent nuisance does not buy the one-wrong interval. Its bias must keep the level sets of the truth, or be negligible in the direction the theorem page names. No fit can check either. Declare a known mechanism where the design gives one.

  • Choose the reduction learners deliberately. They default to the primary specification. reduced_outcome_learner= and reduced_treatment_learner= are two keywords rather than one because the tasks differ: g_{r,1} (or bivariate g_r) is a conditional probability and Q_r plus univariate g_{r,2} are conditional means of signed quantities. A learner instance built for classification cannot serve Q_r, whose target is an outcome residual.

  • A flexible primary nuisance does not buy a flexible reduction. The reductions are on one scalar by default and two for bivariate g_r, so their bias is a low-dimensional smoothing question independent of how well the primary fit adjusted for W.

  • A random-forest reduction is outside the entropy condition of section 3, and is not warned about at runtime. The automatic Super Learner includes a forest, so choose explicit reduced learners when claiming that condition.