In safe hypothesis testing with test supermartingals, Ville's inequality provides anytime-valid type-I error guarantees for every significance level $α\in(0,1]$, if one rejects the null hypothesis whenever the wealth process first exceeds $\frac{1}α$.
Due to an inherent asymmetry, the type-II error does not have such guarantees: a heavy concentration of the probability on the lower tail of the log-increments can lead to one catastrophic bet that undoes any amount of accumulated evidence.
This paper studies how different hypotheses on those lower-tail probabilities lead to different bounds on the type-II error of the sequential test.
They all reduce to one master inequality, which bounds the type-II error at level $α$, at a fixed horizon and sequentially, in terms of a one-sided Legendre transform of the (inverse-)moment generating function of the e-variables, evaluated at one number: the amount by which the lower bound of the accumulated e-powers exceeds $\log\frac{1}α$.
And, the step is lossless, in the sense, that it extracts exactly a constrained information projection.
Every bound presented here is a corollary, obtained by a certain majorant of the above function.
The hypotheses are:
- a finite negative moment;
- an exponentially small crash probability with a moment on the winning side;
- a wealth floor with a conditional variance, and its Bernstein variant, which interpolates between a Gaussian regime set by the variance and an exponential one set by the scale;
- a sub-Gaussian or bounded-tilt lower tail;
- bounded log-increments;
- and i.i.d. increments, where the majorant is the truth.
We also provide an empirical-Bernstein variant.
Each hypothesis may either be read as a condition on the e-variables one has, or as the price of betting with an approximation to the likelihood ratio rather than the ratio itself, which satisfies the weakest condition for free.