Given sequentially observed data, anytime-valid methods guarantee valid inference at arbitrary stopping times, as opposed to pre-specified sample sizes, thereby allowing the experimenter to stop experiments early. In this talk, I will present two recent advances in the emerging field of sequential anytime-valid inference.

First, consider two forecasters, each making a prediction for a sequence of events over time. How can we rigorously compare these forecasters as the events unfold, while avoiding restrictive assumptions such as stationarity? I will address this question by designing a novel statistical inference procedure for estimating the time-varying difference in mean forecast scores. The procedure utilizes confidence sequences, which are sequences of confidence intervals that are anytime-valid and can be continuously monitored over time. I will demonstrate applications of this approach to real-world sports and weather forecasters.

Next, given a composite null hypothesis over sequentially observed data, consider two or more testing procedures for the null based on e-processes, which are the anytime-valid notion of statistical evidence. How can we combine these e-processes so that we can leverage their collective statistical power, particularly when they are constructed in different information sets (i.e., filtrations)? This question is motivated by various sequential testing problems, including randomness testing and multi-step forecast evaluation. I will show how a simple approach based on adjusters allows us to combine arbitrary e-processes across filtrations at a logarithmic cost, and demonstrate its use case in financial time series modeling.