Pulling back the curtain: the road from statistical estimand to machine-learning based estimator for epidemiologists (no wizard required)

Audrey Renson; Dana E. Goin; Iv\'an D\'iaz; Lina Montoya; Rachael K. Ross

REVIEW 2 major objections 5 minor 3 cited by

Pulling back the curtain: the road from statistical estimand to machine-learning based estimator for epidemiologists (no wizard required)

T0 review · 2 major / 5 minor · reviewed 2026-08-08 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This paper shows epidemiologists how to build machine-learning-compatible estimators by conjecturing an efficient influence function and proving the estimator stays root-n consistent, with a worked example for the mean outcome under no…

desk verdict A genuinely useful tutorial for epidemiologists, with one real error in the formal rate condition that must be fixed before the proof is authoritative. read the letter →

arxiv 2502.05363 v1 pith:Z4KO7HTK submitted 2025-02-07 stat.ME stat.AP

Audrey Renson , Lina Montoya , Dana E. Goin , Iván Díaz , Rachael K. Ross This is my paper · ORCID

classification stat.MEstat.AP MSC 62D2062G0562G2062F12

keywords causalinferencemachinelearningefficientinfluencefunctionratedoublerobustnessone-stepestimatorcross-fittingsemiparametrictheoryvonMisesexpansion

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper is a tutorial aimed at epidemiologists who want to build their own machine-learning-compatible estimators for causal parameters that are not off the shelf. It walks through the full route: translate a causal question into a statistical estimand, conjecture its efficient influence function using only sum and product derivative rules, turn that influence function into a one-step estimator, and then prove that the estimator remains root-n consistent when the nuisance functions are fitted by machine learning. The running example is the estimand $\psi = E(E(Y \mid W, A = 0))$, the average outcome if everyone were untreated; its one-step estimator is the augmented inverse probability weighted estimator. The central proof shows that the estimator's remainder term equals $-E\{ (g-\hat g)(q-\hat q)/\hat g\}$, a product of the two nuisance errors, so by Cauchy-Schwarz the estimator converges at the product of the nuisance convergence rates and can reach root-n even when each machine learning fit converges at, say, $n^{1/4}$. If the route works, epidemiologists need not wait for statisticians to publish an estimator for every new research question; they can derive one and verify that it permits valid confidence intervals.

What carries the argument

The efficient influence function (EIF) is the object that carries the argument: it acts as a derivative of the estimand with respect to the data distribution, and the paper conjectures it by pretending covariates are discrete and applying ordinary sum and product derivative rules to known influence functions of marginal and conditional means. The one-step estimator is formed by adding the sample mean of the estimated EIF to the plug-in estimator, which cancels the first-order error in the von Mises expansion. The remainder term of that expansion is then massaged into products of nuisance errors, and sample splitting, in the form of cross-fitting, is used to keep the empirical-process term negligible. The load-bearing identity is $R = -E\{ \hat{g}^{-1}(g-\hat g)(q-\hat q)\}$, which turns rate double robustness into a Cauchy-Schwarz application: under a positivity bound on $g$ and $\hat g$, the factor $1/\hat g$ is bounded in probability, so the remainder is controlled by the product of the two nuisance $L^2$ errors.

What would settle it

Run a simulation study with cross-fitted machine learning for both nuisance functions, choose a data-generating process where the true propensity score has small but nonzero values (for example, $g(W)$ between 0.01 and 0.99), fit nuisance models with $n^{1/4}$-rate estimators, and record empirical coverage of 95% confidence intervals for $\psi = E(E(Y \mid W, A = 0))$ at $n = 500$, $2000$, and $10000$. If coverage falls materially below 0.95 even at the largest sample size while the product of estimated nuisance rates is at least $\sqrt{n}$, the paper's practical claim that rate double robustness suffices for valid inference is falsified. Alternatively, directly compare the one-step estimator's bias with the computed Monte Carlo version of $-E\{ \hat{g}^{-1}(g-\hat g)(q-\hat q)\}$; a systematic mismatch would indicate the remainder identity is not the right diagnostic.

Watch

Extended reading notes

Core claim

The paper's central claim is that for an estimand of the form $\psi = E(E(Y \mid W, A = 0))$, the one-step estimator constructed from the conjectured efficient influence function, namely the AIPW estimator, is rate doubly robust: its scaled error $\sqrt{n}(\hat{\psi}_{\mathrm{os}} - \psi)$ converges to a mean-zero normal distribution as long as the true and estimated propensity scores are bounded away from 0 and 1 and the product of the $L^2$ convergence rates of the outcome regression and propensity score estimators reaches $\sqrt{n}$. This follows because the von Mises expansion separates the plug-in error into a first-order term, which the one-step correction removes, and a remainder that reduces by algebra to $R = -E\{ \hat{g}^{-1}(g-\hat g)(q-\hat q)\}$, which is bounded by the product of the two nuisance $L^2$ errors via Cauchy-Schwarz. The paper is careful to call the influence function conjectured rather than proven, since the derivative-rule derivation does not by itself establish pathwise differentiability; what is proven is that the resulting estimator has the stated asymptotic properties. The appendix carries out the same rate double robustness proof for the estimand $E[E(Y \mid W, A = 0) \mid A = 1]$, one piece of the average treatment effect in the treated.

Load-bearing premise

The proof depends on both the true propensity score $g(W)$ and the estimated propensity score $\hat{g}(W)$ staying bounded away from 0 and 1 with probability 1; if the estimated propensity approaches 0 or 1, the factor $1/\hat{g}$ in the remainder term is no longer bounded and the product-rate argument collapses.

Editorial extensions

If this is right

The AIPW one-step estimator retains root-n consistency and nominal confidence intervals when the outcome regression and propensity score are fit by machine learning, provided the product of their convergence rates is at least $\sqrt{n}$.
G-computation and inverse-probability-weighted estimators generally inherit the slower of the two nuisance rates, so they do not offer the same guarantee without additional conditions.
A researcher who has derived a conjectured EIF and verified a product-form remainder can reuse the same template for novel estimands, with the sample variance of the estimated EIF divided by $n$ as the variance estimator.
The one-step estimator is doubly robust in the usual consistency sense as a byproduct of the remainder calculation: if either nuisance function is consistent, the remainder vanishes.
Sample splitting or cross-fitting is recommended rather than optional when machine learning is used, because it removes the first-order error term and controls the empirical-process term.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Beyond the paper, the same remainder-product template suggests a practical diagnostic for a newly derived EIF-based estimator: simulate from a known distribution, compute the empirical analogue of $-E\{ \hat{g}^{-1}(g-\hat g)(q-\hat q)\}$, and check whether the full estimator's bias tracks it; a mismatch would reveal a derivation error before asymptotic theory is invoked.
A reader should expect the most fragile part of the argument to be the positivity bound on the estimated propensity score; in finite samples with near-violations, the $1/\hat g$ factor can inflate the remainder even when the asymptotic rates are formally satisfied.
The paper's conjectured-EIF route is explicitly not available for pathwise non-differentiable estimands such as conditional effects with continuous covariates, so the boundary of where the template stops is as informative as the template itself and could be probed in future work.
The sample-splitting requirement suggests that implementation choices for cross-fitting, such as the number of folds and the stability of machine learning fits within folds, will matter for finite-sample coverage even though they drop out asymptotically.

Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

This is a tutorial paper, not a methods paper. That is fine: the target reader is an epidemiologist who wants to derive an estimator for a novel causal estimand without reading semiparametric theory. The paper does that well. It walks through the derivation of the EIF for the ATE-type estimand, builds the one-step (AIPW) estimator, and shows the rate double robustness property via a von Mises expansion. The algebra in the main text and appendices C–E is essentially correct, and the paper is honest about outsourcing rigorous proofs (e.g., the von Mises expansion, the EIF conjecture) to the cited literature. The worked example in Appendix E (expected untreated outcome among the treated) is a nice addition.

The core technical claim—that the remainder is a product of nuisance errors and therefore the estimator can tolerate slower-converging machine learning—is right in spirit. But the formal statement in Appendix C has a sign error that matters. The paper defines op(a) as 'asymptotically equal to 0 with convergence rate a,' then concludes R = op(n^{1/2}) when g−ĝ = op(n^a) and q−q̂ = op(n^b) with a+b=1/2. With that definition, op(n^{1/2}) is a diverging object, and a+b=1/2 would give a remainder whose product rate is n^{1/2}, not n^{-1/2}. The correct condition is a+b > 1/2 (or a+b ≥ 1/2 with log factors) if you write rates as n^{-a} and n^{-b}. The heuristic in Section 5.1 (n^{1/4} × n^{1/4} = √n) is fine; the formal line is wrong. An applied reader following the algebra literally will be misled about the required nuisance rates.

The positivity assumption (Assumption 1) is standard and the paper states it explicitly. It is the main external condition on which the proof's Cauchy–Schwarz step relies, so it could deserve more discussion for users whose data have near violations, but that is not a flaw in the proof.

Is it new? No. The estimator and the double robustness property are in [9], [11], [20]. But the pedagogical contribution is real: it shows the derivation steps in a way that many applied readers will follow. The paper deserves a serious referee, mainly to fix the rate condition and clean up the op notation. I would not cite it for the methodology, but I would point students to it as a readable entry point.

Recommendation: engage, but only after the Appendix C error is corrected.

Referee Report

2 major / 5 minor

Summary. The paper is a tutorial aimed at epidemiologists. It walks through the translation of a causal question into the statistical estimand ψ = E[E(Y|W,A=0)], conjectures its efficient influence function using derivative rules, constructs the one-step (AIPW) estimator, and then attempts to demonstrate rate double robustness via the von Mises expansion, so that the estimator can validly incorporate machine-learning nuisance estimates. The main text gives a heuristic proof, Appendix C gives a formal remainder calculation, and Appendix E repeats the derivation for the treated estimand. The paper is explicitly framed as an accessible guide, with honest caveats that the EIF is conjectured and that the von Mises expansion is quoted rather than derived.

Significance. If the rate-double-robustness proof is corrected, this would be a genuinely useful pedagogical bridge: the algebraic derivations are transparent and checkable, the paper is honest about which external results are assumed, and the alternative estimand in Appendix E gives readers a template for adapting the approach. The manuscript's main weakness is not the positivity assumption, which is standard and explicitly stated, but the formal rate condition in Appendix C, which currently gives readers the wrong boundary for admissible nuisance convergence rates. Because that condition is the central justification for the tutorial's practical message, the paper needs a revision before its claims can be relied upon.

major comments (2)

[Appendix C, final paragraph] The proof concludes 'R = o_p(n^{1/2}), so long as g − ĝ = o_p(n^a) and q − q̂ = o_p(n^b) where a + b = 1/2.' This is not the condition needed for equation (5). The asymptotic normality claim requires √n R = o_p(1), i.e. R = o_p(n^{−1/2}). Under the usual reading of o_p(n^{1/2}), the displayed statement does not even assert that R vanishes; at best it is on the wrong side of the required rate. Concretely, if the nuisance errors satisfy ||g−ĝ||_2 = O_p(n^{−a}) and ||q−q̂||_2 = O_p(n^{−b}), then |R| = O_p(n^{−(a+b)}), so √n R = o_p(1) requires a+b > 1/2, not a+b = 1/2. At a+b = 1/2 the remainder is the same order as the CLT term and the Gaussian approximation in equation (5) is not established. This is load-bearing because rate double robustness is the paper's central claim; it must be corrected in both the formal statement and the accompanying prose, with rates stated for L2(P) norms rather than as ambiguous pointwise o_p statements.
[Section 5.1, Section 5.2, and Discussion] The informal statements encode the same boundary error as Appendix C. Section 5.1 says 'slower rates in one nuisance estimator can be traded for faster rates in the other, as long as the product is √n,' and the Discussion allows 'one may be n^{1/8} if the other is at least n^{3/8}.' These examples describe nuisance-error products that are O_p(n^{−1/2}), which do not guarantee √n R = o_p(1); the remainder must converge strictly faster than n^{−1/2} (or at n^{−1/2} with additional logarithmic factors such that n^{1/2}R = o_p(1)). The tutorial should state this explicitly, since the practical message about which machine-learning convergence rates are admissible depends on it.

minor comments (5)

[Appendix C] The definition of o_p(a) as 'a random variable asymptotically equal to 0 with convergence rate a' is nonstandard and contributes to the sign confusion in the final paragraph; the manuscript should define nuisance rates explicitly as O_p(n^{−a}) or o_p(n^{−a}).
[Appendix C] In the Cauchy-Schwarz step, the inequality should be |R| ≤ C ||g−ĝ||_2 ||q−q̂||_2 with C a bound on 1/ĝ; the displayed 'R ≤ √||g−ĝ||√||q−q̂||' omits the absolute value and the bounded factor.
[Section 5.2] The 'mild condition' that g and ĝ are bounded away from 0 and 1 is a substantive positivity assumption; it deserves to be listed alongside the other assumptions rather than described as mild, especially because near-violations are common in epidemiologic applications.
[Appendix D.2] The text says the von Mises expansion is used 'with both sides divided by √n,' but the displayed equation is the unscaled expansion; this inconsistency should be repaired.
[Throughout] There are several typographical slips (e.g., 'estiman d' in the title line, 'Rerranging' in Appendix C, and inconsistent capitalization in 'we adopt'); a careful proofread is needed.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivations are self-contained algebra with explicit, independent external inputs.

full rationale

The paper's central chain is to define a statistical estimand, conjecture an efficient influence function using derivative rules and known EIFs, construct a one-step estimator, and then verify its asymptotic properties via the von Mises expansion. The only imported results are the von Mises expansion and the formal definition of pathwise differentiability, which are explicitly attributed to external references (Hines et al. 2022; Kennedy 2022), not to the authors' own prior work. The EIF derivation is explicitly labeled as only conjecturing the EIF, and the estimator's properties are then verified by direct algebraic calculation of the remainder term, so the proof is not defined in terms of the conclusion it seeks to establish. The remainder derivation in Appendix C is algebra from the definition of R, and the rate-double-robustness argument follows from Cauchy-Schwarz under the stated positivity Assumption 1. No fitted parameter is renamed as a prediction, and no uniqueness theorem is invoked to force a choice. The self-citations to the authors' other papers are illustrative examples of uncommon estimands and do not bear the weight of the derivation. The Appendix C rate condition a+b=1/2 is arguably a correctness or presentational concern about the formal rate statement, but it does not create circularity: the proof structure remains explicit algebra against external theory. Therefore, no circular step is present, and the appropriate score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are fitted; the paper is purely analytical. The axioms listed are the background model and the assumptions explicitly stated for the proofs. No new entities are introduced.

assumptions (5)

domain assumption The observed data are iid draws from a nonparametric distribution P.
Stated in Step 1 and used throughout for EIF and von Mises expansions.
domain assumption Positivity: g(W) and ĝ(W) are bounded in (ε, 1-ε) for some ε>0.
Assumption 1 in Appendix C; bounds 1/ĝ so Cauchy-Schwarz applies to the remainder term.
domain assumption Consistency of nuisance estimators: ĝ→g and q̂→q in probability.
Assumption 2 in Appendix C; needed for the empirical process term and for the variance estimator.
standard math Von Mises expansion (2) holds: ψ(P̂)-ψ(P) = -E{φ(O,P̂)} - R.
Invoked in Section 5.2 and used to define the remainder; derivation is deferred to [9].
domain assumption Causal identification: conditional exchangeability, positivity, and consistency of potential outcomes.
Used in Step 1 to map the causal parameter E[Y0] to the statistical estimand ψ.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Pulling back the curtain: the road from statistical estimand to machine-learning based estimator for epidemiologists (no wizard required)." pith.science (2026). https://pith.science/paper/Z4KO7HTK

@misc{pith2026250205363,
  author       = {Pith},
  title        = {Pith review of: Pulling back the curtain: the road from statistical estimand to machine-learning based estimator for epidemiologists (no wizard required)},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Z4KO7HTK}},
  note         = {Machine review of arXiv:2502.05363}
}

read the original abstract

Epidemiologists increasingly use causal inference methods that rely on machine learning, as these approaches can relax unnecessary model specification assumptions. While deriving and studying asymptotic properties of such estimators is a task usually associated with statisticians, it is useful for epidemiologists to understand the steps involved, as epidemiologists are often at the forefront of defining important new research questions and translating them into new parameters to be estimated. In this paper, our goal was to provide a relatively accessible guide through the process of (i) deriving an estimator based on the so-called efficient influence function (which we define and explain), and (ii) showing such an estimator's ability to validly incorporate machine learning, by demonstrating the so-called rate double robustness property. The derivations in this paper rely mainly on algebra and some foundational results from statistical inference, which are explained.

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Constructing g-computation estimators: two case studies in selection bias
stat.ME 2025-06 conditional novelty 6.0 of 10

New g-computation estimators, expressed as stacked estimating equations, recover average causal effects under treatment-induced selection and under confounding plus selection bias when no single adjustment set exists.
Transporting results from a trial to an external target population when trial participation impacts adherence
stat.ME 2025-05 conditional novelty 5.0 of 10

A method to transport trial treatment effects to a target population under user-specified assumptions about adherence differences, with double-robust estimators and an opioid use disorder application.
Constructing targeted minimum loss/maximum likelihood estimators: a simple illustration to build intuition
stat.ME 2025-07 conditional novelty 2.0 of 10

This letter shows, with a simple causal example and a longitudinal appendix, how to construct a TMLE by solving the efficient influence function's estimating equation with a sequence of weighted regression updates.

Reference graph

Works this paper leans on

21 extracted references · 19 canonical work pages · cited by 3 Pith papers

[9]

Demystifying statistical learning based on eﬃcie nt inﬂuence functions

Oliver Hines et al. “Demystifying statistical learning based on eﬃcie nt inﬂuence functions”. In: The American Statistician 76.3 (2022), pp. 292–304

work page 2022
[11]

Semiparametric doubly robust targeted d ouble machine learning: a review

Edward H Kennedy. “Semiparametric doubly robust targeted d ouble machine learning: a review”. In: arXiv preprint arXiv:2203.06469 (2022)

arXiv 2022
[20]

Targeted learning: causal inference for observational and experimental data

Mark J Van der Laan, Sherri Rose, et al. Targeted learning: causal inference for observational and experimental data. Vol. 4. Springer, 2011

work page 2011
[1]

Invited commentary: demystif ying statistical inference when using machine learning in causal research

Laura B Balzer and Ted Westling. “Invited commentary: demystif ying statistical inference when using machine learning in causal research”. In: American Journal of Epidemiology 192.9 (2023), pp. 1545– 1549

work page 2023
[2]

Double/debiased machine learning fo r treatment and structural parame- ters: Double/debiased machine learning

Victor Chernozhukov et al. “Double/debiased machine learning fo r treatment and structural parame- ters: Double/debiased machine learning”. In: The Econometrics Journal 21.1 (2018)

work page 2018
[3]

Asymptotic properties of high-dimensional random forests

Chien-Ming Chi et al. “Asymptotic properties of high-dimensional random forests”. In: The Annals of Statistics 50.6 (2022), pp. 3415–3438

work page 2022
[4]

Double robustness

Rhian M Daniel. “Double robustness”. In: Wiley StatsRef: Statistics Reference Online (2014), pp. 1–14

work page 2014
[5]

Machine learning in the estimation of causal eﬀects: targeted minimum loss-based estima- tion and double/debiased machine learning

Iv´ an D ´ ıaz. “Machine learning in the estimation of causal eﬀects: targeted minimum loss-based estima- tion and double/debiased machine learning”. In: Biostatistics 21.2 (2020), pp. 353–358

work page 2020

Show all 21 references

[6]

Deep neural n etworks for estimation and infer- ence

Max H Farrell, Tengyuan Liang, and Sanjog Misra. “Deep neural n etworks for estimation and infer- ence”. In: Econometrica 89.1 (2021), pp. 181–213

2021
[7]

Causal dia grams for epidemiologic research

Sander Greenland, Judea Pearl, and James M Robins. “Causal dia grams for epidemiologic research”. In: Epidemiology 10.1 (1999), pp. 37–48

1999
[8]

The inﬂuence curve and its role in robust estimat ion

Frank R Hampel. “The inﬂuence curve and its role in robust estimat ion”. In: Journal of the american statistical association 69.346 (1974), pp. 383–393

1974
[10]

Eﬃcient estimation of longitudinal treatment ef- fects using diﬀerence-in-diﬀerences and machine learning

Nicholas Illenberger, Iv´ an D ´ ıaz, and Audrey Renson. “Eﬃcient estimation of longitudinal treatment ef- fects using diﬀerence-in-diﬀerences and machine learning”. In: arXiv preprint arXiv:2406.16234 (2024)

2024 arXiv
[12]

Causal inf erence based on undersmoothing the highly adaptive lasso

Mark J van der Laan, David Benkeser, and Weixin Cai. “Causal inf erence based on undersmoothing the highly adaptive lasso”. In: Association for the Advancement of Artiﬁcial Intelligence Spring Symposia, Palo Alto, California . 2019

2019
[13]

Challenges in obtaining valid causal eﬀect estimates with machine learning algorithms

Ashley I Naimi, Alan E Mishler, and Edward H Kennedy. “Challenges in obtaining valid causal eﬀect estimates with machine learning algorithms”. In: American Journal of Epidemiology 192.9 (2023), pp. 1536–1544

2023
[14]

Cross-ﬁtting and fast re mainder rates for semiparametric estimation

Whitney K Newey and James R Robins. “Cross-ﬁtting and fast re mainder rates for semiparametric estimation”. In: arXiv preprint arXiv:1801.09138 (2018)

2018 arXiv
[15]

Trans porting treatment eﬀects from diﬀerence-in-diﬀerences studies

Audrey Renson, Ellicott C Matthay, and Kara E Rudolph. “Trans porting treatment eﬀects from diﬀerence-in-diﬀerences studies”. In: arXiv preprint arXiv:2310.17806 (2023)

2023 arXiv
[16]

Data, design, and background knowledge in et iologic inference

James M Robins. “Data, design, and background knowledge in et iologic inference”. In: Epidemiology 12.3 (2001), pp. 313–320

2001
[17]

Non-response models for the analysis of non- monotone non-ignorable missing data

James M Robins. “Non-response models for the analysis of non- monotone non-ignorable missing data”. In: Statistics in medicine 16.1 (1997), pp. 21–37

1997
[18]

Leveraging External Validation Data: Th e Challenges of Transporting Mea- surement Error Parameters

Rachael K Ross et al. “Leveraging External Validation Data: Th e Challenges of Transporting Mea- surement Error Parameters”. In: Epidemiology 35.2 (2024), pp. 196–207

2024
[19]

Double Robust Variance Estimation

Bonnie E Shook-Sa et al. “Double Robust Variance Estimation”. I n: arXiv preprint arXiv:2404.16166 (2024)

2024 arXiv
[21]

Machine learning for causa l inference: on the use of cross-ﬁt estimators

Paul N Zivich and Alexander Breskin. “Machine learning for causa l inference: on the use of cross-ﬁt estimators”. In: Epidemiology 32.3 (2021), pp. 393–401. 8 A Deﬁning the EIF and pathwise diﬀerentiability To formally deﬁne the EIF, we must introduce parametric submodel . Sup...

2021

Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.

[1] [9]

Demystifying statistical learning based on eﬃcie nt inﬂuence functions

Oliver Hines et al. “Demystifying statistical learning based on eﬃcie nt inﬂuence functions”. In: The American Statistician 76.3 (2022), pp. 292–304

work page 2022

[2] [11]

Semiparametric doubly robust targeted d ouble machine learning: a review

Edward H Kennedy. “Semiparametric doubly robust targeted d ouble machine learning: a review”. In: arXiv preprint arXiv:2203.06469 (2022)

arXiv 2022

[3] [20]

Targeted learning: causal inference for observational and experimental data

Mark J Van der Laan, Sherri Rose, et al. Targeted learning: causal inference for observational and experimental data. Vol. 4. Springer, 2011

work page 2011

[4] [1]

Invited commentary: demystif ying statistical inference when using machine learning in causal research

Laura B Balzer and Ted Westling. “Invited commentary: demystif ying statistical inference when using machine learning in causal research”. In: American Journal of Epidemiology 192.9 (2023), pp. 1545– 1549

work page 2023

[5] [2]

Double/debiased machine learning fo r treatment and structural parame- ters: Double/debiased machine learning

Victor Chernozhukov et al. “Double/debiased machine learning fo r treatment and structural parame- ters: Double/debiased machine learning”. In: The Econometrics Journal 21.1 (2018)

work page 2018

[6] [3]

Asymptotic properties of high-dimensional random forests

Chien-Ming Chi et al. “Asymptotic properties of high-dimensional random forests”. In: The Annals of Statistics 50.6 (2022), pp. 3415–3438

work page 2022

[7] [4]

Double robustness

Rhian M Daniel. “Double robustness”. In: Wiley StatsRef: Statistics Reference Online (2014), pp. 1–14

work page 2014

[8] [5]

Machine learning in the estimation of causal eﬀects: targeted minimum loss-based estima- tion and double/debiased machine learning

Iv´ an D ´ ıaz. “Machine learning in the estimation of causal eﬀects: targeted minimum loss-based estima- tion and double/debiased machine learning”. In: Biostatistics 21.2 (2020), pp. 353–358

work page 2020