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From scientific experiments to online A/B testing, the previously observed data often affects how future experiments are performed, which in turn affects which data will be collected. Such adaptivity introduces complex correlations between the data and the collection procedure. In this paper, we prove that when the data collection procedure satisfies natural conditions, then sample means of the data have systematic \emph{negative} biases. As an example, consider an adaptive clinical trial where additional data points are more likely to be tested for treatments that show initial promise. Our surprising result implies that the average observed treatment effects would underestimate the true effects of each treatment. We quantitatively analyze the magnitude and behavior of this negative bias in a variety of settings. We also propose a novel debiasing algorithm based on selective inference techniques. In experiments, our method can effectively reduce bias and estimation error., Comment: Accepted to the 21st International Conference on Artificial Intelligence and Statistics (AISTATS) 2018, Lanzarote, Spain

Selective inference is a recent research topic that tries to perform valid inference after using the data to select a reasonable statistical model. We propose MAGIC, a new method for selective inference that is general, powerful and tractable. MAGIC is a method for selective inference after solving a convex optimization problem with smooth loss and $\ell_1$ penalty. Randomization is incorporated into the optimization problem to boost statistical power. Through reparametrization, MAGIC reduces the problem into a sampling problem with simple constraints. MAGIC applies to many $\ell_1$ penalized optimization problem including the Lasso, logistic Lasso and neighborhood selection in graphical models, all of which we consider in this paper.

Inspired by sample splitting and the reusable holdout introduced in the field of differential privacy, we consider selective inference with a randomized response. We discuss two major advantages of using a randomized response for model selection. First, the selectively valid tests are more powerful after randomized selection. Second, it allows consistent estimation and weak convergence of selective inference procedures. Under independent sampling, we prove a selective (or privatized) central limit theorem that transfers procedures valid under asymptotic normality without selection to their corresponding selective counterparts. This allows selective inference in nonparametric settings. Finally, we propose a framework of inference after combining multiple randomized selection procedures. We focus on the classical asymptotic setting, leaving the interesting high-dimensional asymptotic questions for future work.

There has been much recent work on inference after model selection when the noise level is known, however, $\sigma$ is rarely known in practice and its estimation is difficult in high-dimensional settings. In this work we propose using the square-root LASSO (also known as the scaled LASSO) to perform selective inference for the coefficients and the noise level simultaneously. The square-root LASSO has the property that choosing a reasonable tuning parameter is scale-free, namely it does not depend on the noise level in the data. We provide valid p-values and confidence intervals for the coefficients after selection, and estimates for model specific variance. Our estimates perform better than other estimates of $\sigma^2$ in simulation.

In this paper, we seek to establish asymptotic results for selective inference procedures removing the assumption of Gaussianity. The class of selection procedures we consider are determined by affine inequalities, which we refer to as affine selection procedures. Examples of affine selection procedures include post-selection inference along the solution path of the LASSO, as well as post-selection inference after fitting the LASSO at a fixed value of the regularization parameter. We also consider some tests in penalized generalized linear models. Our method of proof adapts a method of Chatterjee (2005).