In recent years, there has been growing interest in selective inference—valid inferences about parameters chosen based on the data. I will begin this talk with an overview of existing approaches, focusing on a class of methods I refer to as "infer-and-widen". These methods involve simple modifications to classical inference procedures but tend to be overly conservative due to biased point estimates. As an alternative, I will discuss conditional selective inference methods, and illustrate how one such method can be used to construct confidence intervals for the proportion of variance explained by principal components. However, these conditional approaches often rely on strong parametric assumptions or require bespoke inferential approaches. To address these limitations, I will introduce a randomization-based approach that decomposes an asymptotically normal statistic into asymptotically independent components. In the context of inference after the lasso, this method yields asymptotically valid conditional inference using the classical inference procedure.
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