Ville Anytime E-Process

In sequential hypothesis testing, we often want to track the evidence against a null hypothesis as data arrives, without being constrained to a fixed sample size. An e-process is a sequence of non-negative random variables where the expectation under the null hypothesis is always bounded by 1, making it a test martingale. Ville's inequality allows us to use this e-process to safely stop the experiment early if sufficient evidence is observed, maintaining valid error guarantees at any stopping time.

Experiment

You can interactively explore this e-process in the Ville Anytime E-Process module. Follow these steps to observe Ville's inequality in action:

1. Set the Horizon to 6.
2. Set the Alternative probability to 0.75.
3. Set the Test level to 0.5.
4. Observe the Observed path set to All heads.
5. Note that the evidence threshold is 1/0.5 = 2.0. The e-process trace crosses this threshold at t = 2.
6. Change the Observed path to Alternating. The trace will go up on the first flip, down on the second, and so on. It never crosses the threshold.
7. Regardless of the chosen path, observe the Exact null crossing probability. This value is computed by examining all 2^N possible paths and remains valid for the chosen horizon, test level, and alternative.