General V alue Functions (GVFs) (Sutton et al., 2011) represent predictive knowledge in reinforcement learning.

Weprovide asemi-supervised estimation procedure of the optimal rule involving two datasets: a firstlabeled dataset is used to estimate both regression function and conditional variance function while a secondunlabeleddataset is exploited to calibrate the desired rejection rate.

We investigate the problem of regression where one is allowed to abstain from predicting.

We thank all reviewers for their valuable comments. Hereafter, we address comments shared by several reviewers. However, our general Theorem 3.2 asks for the continuity of the cumulative distribution function In particular Section 4.2 presents a randomized This randomization technique used to circumvent Assumption 3.1 is rather simple Besides, we mention that Assumption 3.1 is one of the limitation The obtained rates of convergence relies on Proposition C.1 (given in the supplementary material), Section 4. That is to say the rate of convergence given in Theorem 4.4 applies to estimators which can be written as Theorem 4.4 shows, from a theoretical perspective, the dependency with respect to the sample size of labeled and

While Bayesian neural networks (BNNs) hold the promise of being flexible, well-calibrated statistical models, inference often requires approximations whose consequences are poorly understood.

This thesis contains a series of independent contributions to statistics, unified by a model-free perspective. The first chapter elaborates on how a model-free perspective can be used to formulate flexible methods that leverage prediction techniques from machine learning. Mathematical insights are obtained from concrete examples, and these insights are generalized to principles that permeate the rest of the thesis. The second chapter studies the concept of local independence, which describes whether the evolution of one stochastic process is directly influenced by another. To test local independence, we define a model-free parameter called the Local Covariance Measure (LCM). We formulate an estimator for the LCM, from which a test of local independence is proposed. We discuss how the size and power of the proposed test can be controlled uniformly and investigate the test in a simulation study. The third chapter focuses on covariate adjustment, a method used to estimate the effect of a treatment by accounting for observed confounding. We formulate a general framework that facilitates adjustment for any subset of covariate information. We identify the optimal covariate information for adjustment and, based on this, introduce the Debiased Outcome-adapted Propensity Estimator (DOPE) for efficient estimation of treatment effects. An instance of DOPE is implemented using neural networks, and we demonstrate its performance on simulated and real data. The fourth and final chapter introduces a model-free measure of the conditional association between an exposure and a time-to-event, which we call the Aalen Covariance Measure (ACM). We develop a model-free estimation method and show that it is doubly robust, ensuring $\sqrt{n}$-consistency provided that the nuisance functions can be estimated with modest rates. A simulation study demonstrates the use of our estimator in several settings.