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Multi-arm bandits are gaining popularity as they enable real-world sequential decision-making across application areas, including clinical trials, recommender systems, and online decision-making. Consequently, there is an increased desire to use the available adaptively collected datasets to distinguish whether one arm was more effective than the other, e.g., which product or treatment was more effective. Unfortunately, existing tools fail to provide valid inference when data is collected adaptively or require many untestable and technical assumptions, e.g., stationarity, iid rewards, bounded random variables, etc. Our paper introduces the design-based approach to inference for multi-arm bandits, where we condition the full set of potential outcomes and perform inference on the obtained sample. Our paper constructs valid confidence intervals for both the reward mean of any arm and the mean reward difference between any arms in an assumption-light manner, allowing the rewards to be arbitrarily distributed, non-iid, and from non-stationary distributions. In addition to confidence intervals, we also provide valid design-based confidence sequences, sequences of confidence intervals that have uniform type-1 error guarantees over time. Confidence sequences allow the agent to perform a hypothesis test as the data arrives sequentially and stop the experiment as soon as the agent is satisfied with the inference, e.g., the mean reward of an arm is statistically significantly higher than a desired threshold.

Randomized experiments have become the standard method for companies to evaluate the performance of new products or services. In addition to augmenting managers' decision-making, experimentation mitigates risk by limiting the proportion of customers exposed to innovation. Since many experiments are on customers arriving sequentially, a potential solution is to allow managers to "peek" at the results when new data becomes available and stop the test if the results are statistically significant. Unfortunately, peeking invalidates the statistical guarantees for standard statistical analysis and leads to uncontrolled type-1 error. Our paper provides valid design-based confidence sequences, sequences of confidence intervals with uniform type-1 error guarantees over time for various sequential experiments in an assumption-light manner. In particular, we focus on finite-sample estimands defined on the study participants as a direct measure of the incurred risks by companies. Our proposed confidence sequences are valid for a large class of experiments, including multi-arm bandits, time series, and panel experiments. We further provide a variance reduction technique incorporating modeling assumptions and covariates. Finally, we demonstrate the effectiveness of our proposed approach through a simulation study and three real-world applications from Netflix. Our results show that by using our confidence sequence, harmful experiments could be stopped after only observing a handful of units; for instance, an experiment that Netflix ran on its sign-up page on 30,000 potential customers would have been stopped by our method on the first day before 100 observations.

Linear regression adjustment is commonly used to analyse randomised controlled experiments due to its efficiency and robustness against model misspecification. Current testing and interval estimation procedures leverage the asymptotic distribution of such estimators to provide Type-I error and coverage guarantees that hold only at a single sample size. Here, we develop the theory for the anytime-valid analogues of such procedures, enabling linear regression adjustment in the sequential analysis of randomised experiments. We first provide sequential $F$-tests and confidence sequences for the parametric linear model, which provide time-uniform Type-I error and coverage guarantees that hold for all sample sizes. We then relax all linear model parametric assumptions in randomised designs and provide nonparametric model-free sequential tests and confidence sequences for treatment effects. This formally allows experiments to be continuously monitored for significance, stopped early, and safeguards against statistical malpractices in data collection. A particular feature of our results is their simplicity. Our test statistics and confidence sequences all emit closed-form expressions, which are functions of statistics directly available from a standard linear regression table. We illustrate our methodology with the sequential analysis of software A/B experiments at Netflix, performing regression adjustment with pre-treatment outcomes.

We demonstrate the effectiveness of democratization and efficient computation as key concepts of our experimentation platform (XP) by presenting four new models supported by the platform: 1) Weighted least squares, 2) Quantile bootstrapping, 3) Bayesian shrinkage, and 4) Dynamic treatment effects. Each model is motivated by a specific business problem but is generalizable and extensible. The modular structure of our platform allows independent innovation on statistical and computational methods. In practice, a technical symbiosis is created where increasingly advanced user contributions inspire innovations to the software that in turn enable further methodological improvements. This cycle adds further value to how the XP contributes to business solutions.

We produce new reconstructions of Northern Hemisphere annually averaged temperature anomalies back to 1000 AD, and explore the effects of including external climate forcings within the reconstruction and of accounting for short-memory and long-memory features. Our reconstructions are based on two linear models, with the first linking the latent temperature series to three main external forcings (solar irradiance, greenhouse gas concentration and volcanism), and the second linking the observed temperature proxy data (tree rings, sediment record, ice cores, etc.) to the unobserved temperature series. Uncertainty is captured with additive noise, and a rigorous statistical investigation of the correlation structure in the regression errors is conducted through systematic comparisons between reconstructions that assume no memory, short-memory autoregressive models, and long-memory fractional Gaussian noise models. We use Bayesian estimation to fit the model parameters and to perform separate reconstructions of land-only and combined land-and-marine temperature anomalies. For model formulations that include forcings, both exploratory and Bayesian data analysis provide evidence against models with no memory. Model assessments indicate that models with no memory underestimate uncertainty. However, no single line of evidence is sufficient to favor short-memory models over long-memory ones, or to favor the opposite choice. When forcings are not included, the long-memory models appear to be necessary. While including external climate forcings substantially improves the reconstruction, accurate reconstructions that exclude these forcings are vital for testing the fidelity of climate models used for future projections., Comment: Published in at http://dx.doi.org/10.1214/14-AOAS785 the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org)

Discussion of "A statistical analysis of multiple temperature proxies: Are reconstructions of surface temperatures over the last 1000 years reliable?" by B.B. McShane and A.J. Wyner [arXiv:1104.4002], Comment: Published in at http://dx.doi.org/10.1214/10-AOAS398E the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org)