  The entropy accumulation theorem, and its subsequent generalized version, is
a powerful tool in the security analysis of many device-dependent and
device-independent cryptography protocols. However, it has the drawback that
the finite-size bounds it yields are not necessarily optimal, and furthermore
it relies on the construction of an affine min-tradeoff function, which can
often be challenging to construct optimally in practice. In this work, we
address both of these challenges simultaneously by deriving a new entropy
accumulation bound. Our bound yields significantly better finite-size
performance, and can be computed as an intuitively interpretable convex
optimization, without any specification of affine min-tradeoff functions.
Furthermore, it can be applied directly at the level of R\'enyi entropies if
desired, yielding fully-R\'enyi security proofs. Our proof techniques are based
on elaborating on a connection between entropy accumulation and the frameworks
of quantum probability estimation or $f$-weighted R\'enyi entropies, and in the
process we obtain some new results with respect to those frameworks as well. In
particular, those findings imply that our bounds apply to prepare-and-measure
protocols without the virtual tomography procedures or repetition-rate
restrictions previously required for entropy accumulation.
