REVIEW 2 major objections 4 minor 18 references
Constructing targeted minimum loss/maximum likelihood estimators: a simple illustration to build intuition
T0 review · 2 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash
Pith's one-line read The paper shows a TMLE for E(Y0) can be built by splitting the EIF equation into two score equations and fitting one fluctuation of the outcome model.
desk verdict A solid, honest TMLE tutorial; the worked algorithms should either cross-fit or say they're only showing the targeting step. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The central object is the score equation of a generalized linear model with a canonical link and an offset: maximum-likelihood estimation forces the weighted sum of residuals to zero. The paper's key observation is that equation (4) is exactly such a score equation, with the clever covariate equal to the inverse-probability weight $I(A=0)/\Pr(A=0\mid W)$. The machinery of the argument is the decomposition of the EIF estimating equation into (4) and (5), followed by the offset-plus-fluctuation form of the targeting model, $E^*(Y\mid A=0,W)=E(Y\mid A=0,W)+\gamma\,I(A=0)/\Pr(A=0\mid W)$ (or the weighted-intercept version). Fitting this fluctuation by maximum likelihood solves (4) by construction, and the sample-mean step solves (5); the appendix repeats this pattern for a two-time-point estimand by splitting its EIF into three estimating equations and solving two of them with sequential targeting models.
What would settle it
Simulate data from a known mechanism with binary treatment and continuous outcome so the true $E(Y_0)$ is computable; implement the paper's two-step recipe with the correct EIF and sample splitting. If the resulting confidence intervals do not cover $E(Y_0)$ at near-nominal rates across replications, the central claim is wrong. The same simulation with an intentionally incorrect EIF (for example, dropping the inverse-probability weight) should show coverage degrade, confirming that validity depends on starting from the right EIF.
Extended reading notes
Core claim
The paper's central claim is that the EIF estimating equation for $E(Y_0)$, $$0=\sum_{i=1}^n\left[\frac{I(A_i=0)}{\Pr(A=0\mid W_i)}(Y_i-E(Y\mid A=0,W_i))+E(Y\mid A=0,W_i)-\psi\right],$$ can be rearranged into two estimating equations. The first, $$0=\sum_{i=1}^n\frac{I(A_i=0)}{\Pr(A=0\mid W_i)}(Y_i-E^*(Y\mid A=0,W_i)),$$ has exactly the form of a maximum-likelihood score equation for a generalized linear model with an offset, so it is solved by fitting a targeting model: regress $Y$ on the clever covariate $I(A=0)/\Pr(A=0\mid W)$ (or fit an intercept-only weighted version), holding the initial prediction $E(Y\mid A=0,W)$ as an offset. The second equation, $$0=\sum_{i=1}^n\left(E^*(Y\mid A=0,W_i)-\hat{\psi}\right),$$ is solved by taking the sample mean of the targeted predictions. Thus the mean of $E^*(Y\mid A=0,W)$ solves the full EIF equation, which is what makes the estimator a TMLE: the initial nuisance models may be machine-learned, but the parametric targeting step guarantees the score equation is solved, yielding an asymptotically linear estimator with valid inference under the paper's stated conditions.
Load-bearing premise
The construction works only if the estimator starts from the correct efficient influence function and if the named regularity conditions (fast enough learning of the intermediate models, or sample-splitting) actually hold; the paper assumes both rather than deriving them or building them into the steps.
Editorial extensions
If this is right
- A plug-in g-computation estimator $\frac{1}{n}\sum_i \hat{E}(Y\mid A=0,W_i)$ can be upgraded to a TMLE by one maximum-likelihood fluctuation, so any initial outcome regression, including a machine-learned one, becomes part of a valid EIF-solving estimator.
- Since only the targeting model is fit by maximum likelihood, the initial outcome and propensity score models may be estimated with flexible machine-learning methods without invalidating the score equation.
- For bounded outcomes, a logistic targeting model keeps the targeted predictions inside the range of $Y$, so the final estimate respects the parameter space of the outcome; this gives TMLE a finite-sample advantage over the one-step estimator.
- The construction extends to longitudinal settings: for the two-time-point estimand, the EIF equation splits into three estimating equations, and the appendix algorithm solves them sequentially with two targeting models after two machine-learned outcome fits.
- The paper argues TMLEs inherit rate double robustness from the EIF construction: setting the EIF mean to zero removes the drift term, leaving a remainder that is a product of nuisance errors, and TMLE and the one-step estimator become asymptotically equivalent under the stated conditions.
- The paper itself flags that the empirical process term needs careful handling because the targeting model is estimated in the full sample, and the worked algorithms omit cross-fitting; a literal implementation should therefore add sample splitting or verify the regularity conditions before trusting the confidence intervals.
Reading between the lines
- The score-equation decomposition is likely portable: for any estimand whose EIF is a sum of terms that can each be written as a weighted residual, a user could derive a TMLE by solving each term with a maximum-likelihood fluctuation, even though the paper only demonstrates $E(Y_0)$ and the two-time-point extension.
- The two targeting forms (clever covariate versus inverse-probability weights) solve the same estimating equation but may have different finite-sample behavior in small or sparse samples; the paper presents both but does not compare them, so a simulation or empirical comparison would be a natural next step.
- A quick way to verify the recipe is to implement it in standard GLM software with an offset and weights and compare the resulting estimate and coverage against the one-step estimator on simulated data with known $E(Y_0)$; the paper's claims imply they should agree asymptotically but differ in finite samples.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. This letter is a pedagogical supplement to Renson et al. (2025) that illustrates how to construct a TMLE starting from the EIF estimating equation. It reviews score equations for GLMs, defines the EIF for the mean outcome under no treatment E(Y0), and shows that the EIF estimating equation (3) can be split into two estimating equations: (4), solved by fitting a fluctuation model for the outcome regression with I(A=0)/Pr(A=0|W) as a clever covariate or as weights, and (5), solved by taking the sample mean of the targeted predictions. The appendix extends the same logic to a two-time-point sustained treatment estimand and gives a six-step algorithm. The core algebraic claim—that the displayed targeting models solve the EIF score equation by the GLM score property—is correct. The main weakness is that the worked algorithms do not implement the cross-fitting that the paper itself lists as a condition for valid asymptotic inference, and the Discussion leaves the resulting empirical-process issue unresolved.
Significance. The letter addresses a real pedagogical gap: TMLE construction is often presented as an opaque sequence of steps, and the score-equation viewpoint is genuinely clarifying. If the manuscript is revised to make the implementation consistent with its own regularity conditions, it could be a useful teaching resource. Strengths include the explicit derivation of two targeting models from the EIF, the clear use of the GLM score property, and the worked longitudinal example. The manuscript does not claim new statistical theory, and its reliance on the companion paper for the EIF is appropriate for a supplement. However, the asymptotic-validity guarantee is only as good as the conditions attached to Eq. (3), and those conditions are not reflected in the provided algorithms; this is the main issue blocking acceptance.
major comments (2)
- [TMLE; Appendix] The paper's stated goal is valid inference when ML is used for nuisance functions, and the paragraph around Eq. (3) explicitly lists 'use of cross-fitting' as a condition for asymptotic linearity. Yet neither the main-text construction nor the six-step algorithm in the Appendix performs sample splitting: the propensity score, initial outcome model, targeting model, and final mean are all computed on the full sample. The Discussion acknowledges that the empirical process term 'needs careful handling since the targeting model is estimated in the entire sample,' but no such handling is described. A literal implementation with random forests or lasso will solve the in-sample EIF equation while possibly failing to be asymptotically linear, so the promised confidence intervals are not guaranteed. Please add an explicit cross-fitting step to the algorithms, or clearly state that the displayed recipe is an asymptotic illustration and point readers to a cross-fitted implementation.
- [Discussion; TMLE] The Discussion claims that 'TMLEs produce estimates that respect the parameter space of Y' because the targeting outcome model ensures bounding of E*(Y|A=0,W). This is true for the logistic fluctuation described in the main text, but the fully worked examples use linear targeting models, which do not bound the predictions and can therefore produce a final mean outside the outcome space (e.g., a probability outside [0,1] for binary Y). Moreover, for a bounded continuous outcome the recommended logistic model requires rescaling Y to [0,1] before taking logit, and the letter does not mention that step. Please qualify the Discussion claim to refer to appropriate (e.g., logistic) fluctuations and give the scaling/back-transformation details.
minor comments (4)
- [Appendix, step 2] Step 2 says to fit a flexible model for E(Y|W0,A0=0,W1,A1=0) but does not state whether the regression is fit on the subset with A0=A1=0 or on the full data; making this explicit would help applied readers avoid an avoidable ambiguity.
- [Score equations section] The symbol f(X) is used for a generic transformation and then effectively reused for the clever covariate I(A=0)/Pr(A=0|W); a sentence explicitly defining H_i = I(A_i=0)/\hat{g}(W_i) would improve readability.
- [Abstract] The abstract's phrase 'validly incorporate machine learning' should carry the caveats stated near Eq. (3) (cross-fitting, fast-enough nuisance convergence); consider adding a short qualifier.
- [Discussion] The sentence 'The only difference with the analysis in Renson et al. is that the empirical process term needs careful handling' is imprecise, because the absence of sample splitting also changes how nuisance estimators enter the remainder analysis. Please rephrase or cite a reference where the analysis without sample splitting is spelled out.
Circularity Check
No material circularity: the TMLE construction directly solves the EIF score equation, with the EIF taken as a standard input from the companion paper.
full rationale
The paper's central claim is constructive: it rewrites the EIF estimating equation (3) as two estimating equations (4) and (5), and then shows that a maximum-likelihood fluctuation of the initial outcome regression with offset and the clever covariate produces a targeted predictor E* whose score equation is exactly (4). This is verified algebraically in the text ('Plugging this into (4), we get the score equation ...'), and psi is then set to the sample mean of E*, which solves (5). The result is a solution to the EIF equation by construction, not by circular assumption: the EIF itself is an input, and the paper says it follows Renson et al. That self-citation is minor and non-load-bearing because the EIF formula is standard and independently available (e.g., Hines et al. and Kennedy are cited), and the paper does not invoke any uniqueness theorem or forbid alternative targeting models. The appendix repeats the same valid pattern for the longitudinal estimand: the displayed three-score decomposition matches the displayed EIF, and each targeting model is fit by maximum likelihood to solve its corresponding weighted score equation. The Discussion's caveat about the empirical process term and full-sample targeting is an implementation/regularity limitation, not a circularity. No fitted parameter is renamed as a prediction: the fluctuation parameter is explicitly solved for from the score equation and is not an independent prediction.
Assumptions & free parameters
assumptions (4)
- domain assumption Causal identification: conditional exchangeability, positivity, and causal consistency hold, so E(Y0)=E(E(Y|A=0,W)).
- standard math The EIF formulas for the target estimands are correct under a nonparametric statistical model.
- domain assumption Asymptotic linearity conditions hold, including cross-fitting or fast enough nuisance consistency, so solving the EIF estimating equation yields valid inference.
- standard math A maximum-likelihood fit of the targeting GLM solves the targeted score equation exactly.
Cite this review
Pith. "Pith review of Constructing targeted minimum loss/maximum likelihood estimators: a simple illustration to build intuition." pith.science (2026). https://pith.science/paper/UBLOUWS5
@misc{pith2026250711680,
author = {Pith},
title = {Pith review of: Constructing targeted minimum loss/maximum likelihood estimators: a simple illustration to build intuition},
year = {2026},
howpublished = {\url{https://pith.science/paper/UBLOUWS5}},
note = {Machine review of arXiv:2507.11680}
}
read the original abstract
Use of machine learning to estimate nuisance functions (e.g. outcomes models, propensity score models) in estimators used in causal inference is increasingly common, as it can mitigate bias due to model misspecification. However, it can be challenging to achieve valid inference (e.g., estimate valid confidence intervals). The efficient influence function (EIF) provides a recipe to go from a statistical estimand relevant to our causal question, to an estimator that can validly incorporate machine learning. Our companion paper, Renson et al. 2025 (arXiv:2502.05363), provides a thorough but approachable description of the EIF, along with a guide through the steps to go from a unique statistical estimand to development of one type of EIF-based estimator, the so-called one-step estimator. Another commonly used estimator based on the EIF is the targeted maximum likelihood/minimum loss estimator (TMLE). Construction of TMLEs is well-discussed in the statistical literature, but there remains a gap in translation to a more applied audience. In this letter, which supplements Renson et al., we provide a more accessible illustration of how to construct a TMLE.
Reference graph
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[13]
Propensity score models: Fit flexible models for the treatment at each time point, possibly using machine learning, to estimate cPr(A0 = 0|W0) and cPr(A1 = 0|W0, A0 = 0, W1)
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[14]
Predictions from this model arebµ
Outcome model at time point 1: Fit a flexible model for E(Y|W 0, A0 = 0, W1, A1 = 0), possibly using machine learning. Predictions from this model arebµ
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[15]
Predictions from this model arebµ ∗
Targeting model at time point 1: Fit an intercept-only linear model ofYwithbµas an offset, weighted by I(A0=A1=0) cPr(A0=0|W0) cPr(A1=0|W0,A0=0,W1) , using maximum likelihood. Predictions from this model arebµ ∗
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[16]
Predictions from this model are bE(bµ∗|W0, A0 = 0)
Outcome model at time point 0: Fit a flexible model for E(bµ ∗|W0, A0 = 0), possibly using machine learning. Predictions from this model are bE(bµ∗|W0, A0 = 0)
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[17]
Predictions from this model are bE ∗ (bµ∗|W0, A0 = 0)
Targeting model at time point 0: Fit an intercept-only linear model ofbµ ∗ with bE(bµ∗|W0, A0 = 0) as an offset, weighted by I(A0=0) cPr(A0=0|W0) , using maximum likelihood. Predictions from this model are bE ∗ (bµ∗|W0, A0 = 0)
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[18]
Further, as discussed in the main text, although we have shown linear targeting models to simplify notation, logistic models are recommended for better finite sample properties
Estimate estimand as the sample mean of bE ∗ (bµ∗|W0, A0 = 0) 10 As in the example in the main text, we could have alternatively considered targeting models where I(A0=A1=0) cPr(A0=0|W0) cPr(A1=0|W0,A0=0,W1) and I(A0=0) cPr(A0=0|W0) are included as covariates. Further, as disc...
Reviewed August 6, 2026 · model on record in the stance chip above.
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