Many randomized encouragement designs have imperfect compliance, where only a fraction of people comply with their assignment. Examples include phone-bank get-out-the-vote (GOTV) campaigns and draft lotteries like the Vietnam Draft Lottery. In these settings, it is common to use instrumental variable (IV) regression for analysis. Instrumental variables identify the Local

Consider a common scenario: you observe how units (users, devices, firms) interact with items (websites, products, apps), and each item has an unknown binary trait (is it harmful? is it premium content?). You want to understand how these traits relate to unit characteristics through regression, but labeling items is expensive.

An experimentation program is a portfolio problem: at each time $t$, given what we currently know and today's constraints (backlog, traffic budget, test and implementation costs), we must choose a slate of ideas to test, decide how much traffic to allocate, and determine which experiments to continue, stop,

In Basu's circus story (PDF), the owner wants to estimate the weight of fifty elephants and suggests weighing Sambo, who looked “average” three years ago, and reporting $50y$. A statistician counters with a contrived sampling plan: choose Sambo with probability $99/100$ and pick any of the other

We often need to adjust a fitted $E[Y|X]$ so that the aggregate $E[Y]$ (or class totals) matches known targets (see here). In binary classification, a single logit shift (or a global temperature) is a monotone transform of the model scores, so if person A had a higher

Textbook bias–variance intuition implies that worst‑case test error should eventually rise as model capacity overtakes the sample size because the variance term in the bound grows while the bias term has already bottomed out (see Vapnik and Chervonenkis, 1971; Bartlett and Mendelson, 2002). Those worst‑case guarantees shrink