Combining Evidence Across Filtrations
Yo Joong Choe Aaditya Ramdas
In anytime-valid sequential inference, it is known that any admissible
inference procedure must be based on test martingales and their composite
generalization, called e-processes, which are nonnegative processes whose
expectation at any arbitrary stopping time is upper-bounded by one. An
e-process quantifies the accumulated evidence against a composite null
hypothesis over a sequence of outcomes. This paper studies methods for
combining e-processes that are computed using different information sets, i.e.,
filtrations, for a null hypothesis. Even though e-processes constructed on the
same filtration can be combined effortlessly (e.g., by averaging), e-processes
constructed on different filtrations cannot be combined as easily because their
validity in a coarser filtration does not translate to validity in a finer
filtration. We discuss three concrete examples of such e-processes in the
literature: exchangeability tests, independence tests, and tests for evaluating
and comparing forecasts with lags. Our main result establishes that these
e-processes can be lifted into any finer filtration using adjusters, which are
functions that allow betting on the running maximum of the accumulated wealth
(thereby insuring against the loss of evidence). We also develop randomized
adjusters that can improve the power of the resulting sequential inference
procedure.