  In anytime-valid sequential inference, it is known that any admissible
procedure must be based on e-processes, which are composite generalizations of
test martingales that quantify the accumulated evidence against a composite
null hypothesis at any arbitrary stopping time. This paper studies methods for
combining e-processes constructed using different information sets
(filtrations) for the same null. Although e-processes constructed in the same
filtration can be combined effortlessly (e.g., by averaging), e-processes
constructed in different filtrations cannot, because their validity in a
coarser filtration does not translate to validity in a finer filtration. This
issue arises in exchangeability tests, independence tests, and tests for
comparing forecasts with lags. We first establish that a class of functions
called adjusters allows us to lift e-processes from a coarser filtration into
any finer filtration. We then introduce a characterization theorem for
adjusters, formalizing a sense in which using adjusters is necessary. There are
two major implications. First, if we have a powerful e-process in a coarsened
filtration, then we readily have a powerful e-process in the original
filtration. Second, when we coarsen the filtration to construct an e-process,
there is an asymptotically logarithmic cost of recovering anytime-validity in
the original filtration.
