  Information projections have found important applications in probability
theory, statistics, and related areas. In the field of hypothesis testing in
particular, the reverse information projection (RIPr) has recently been shown
to lead to growth-rate optimal (GRO) e-statistics for testing simple
alternatives against composite null hypotheses. However, the RIPr as well as
the GRO criterion are undefined whenever the infimum information divergence
between the null and alternative is infinite. We show that in such scenarios,
under some assumptions, there still exists a measure in the null that is
closest to the alternative in a specific sense. Whenever the information
divergence is finite, this measure coincides with the usual RIPr. It therefore
gives a natural extension of the RIPr to certain cases where the latter was
previously not defined. This extended notion of the RIPr is shown to lead to
optimal e-statistics in a sense that is a novel, but natural, extension of the
GRO criterion. We also give conditions under which the (extension of the) RIPr
is a strict sub-probability measure, as well as conditions under which an
approximation of the RIPr leads to approximate e-statistics. For this case we
provide tight relations between the corresponding approximation rates.
