# Continuous-time targeted minimum loss-based estimation of intervention-specific mean outcomes

arXiv: https://arxiv.org/abs/2105.02088 · HTML: https://mlsanity.com/paper/2105.02088/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 May 2021.

## Built on by (most cited first, 9 of 9)

- [Bayesian Neural Controlled Differential Equations for Treatment Effect Estimation](https://mlsanity.com/paper/2310.17463/) (2023)
- [IGC-Net for conditional average potential outcome estimation over time](https://mlsanity.com/paper/2405.21012/) (2024)
- [Nonparametric Conditional Local Independence Testing](https://mlsanity.com/paper/2203.13559/) (2022)
- [Stabilized Neural Prediction of Potential Outcomes in Continuous Time](https://mlsanity.com/paper/2410.03514/) (2024)
- [Automated Efficient Estimation using Monte Carlo Efficient Influence Functions](https://mlsanity.com/paper/2403.00158/) (2024)
- [Efficient Targeted Maximum Likelihood Estimators for Two-Phase Design Problems](https://mlsanity.com/paper/2602.24131/) (2026)
- [Causal inference for the expected number of recurrent events in the presence of a terminal event](https://mlsanity.com/paper/2306.16571/) (2023)
- [Time After Time: Deep-Q Effect Estimation for Interventions on When and What to do](https://mlsanity.com/paper/2503.15890/) (2025)
- [Observable Neural ODEs for Identifiable Causal Forecasting in Continuous Time](https://mlsanity.com/paper/2604.26070/) (2026)

## Abstract (arXiv)

This paper studies the generalization of the targeted minimum loss-based estimation (TMLE) framework to estimation of effects of time-varying interventions in settings where both interventions, covariates, and outcome can happen at subject-specific time-points on an arbitrarily fine time-scale. TMLE is a general template for constructing asymptotically linear substitution estimators for smooth low-dimensional parameters in infinite-dimensional models. Existing longitudinal TMLE methods are developed for data where observations are made on a discrete time-grid. We consider a continuous-time counting process model where intensity measures track the monitoring of subjects, and focus on a low-dimensional target parameter defined as the intervention-specific mean outcome at the end of follow-up. To construct our TMLE algorithm for the given statistical estimation problem we derive an expression for the efficient influence curve and represent the target parameter as a functional of intensities and conditional expectations. The high-dimensional nuisance parameters of our model are estimated and updated in an iterative manner according to separate targeting steps for the involved intensities and conditional expectations. The resulting estimator solves the efficient influence curve equation. We state a general efficiency theorem and describe a highly adaptive lasso estimator for nuisance parameters that allows us to establish asymptotic linearity and efficiency of our estimator under minimal conditions on the underlying statistical model.