  The inference of thermodynamic quantities from the description of an only
partially accessible physical system is a central challenge in stochastic
thermodynamics. A common approach is coarse-graining, which maps the dynamics
of such a system to a reduced effective one. While coarse-graining states of
the system into compound ones is a well studied concept, recent evidence hints
at a complementary description by considering observable transitions and
waiting times. In this work, we consider waiting time distributions between two
consecutive transitions of a partially observable Markov network. We formulate
an entropy estimator using their ratios to quantify irreversibility. Depending
on the complexity of the underlying network, we formulate criteria to infer
whether the entropy estimator recovers the full physical entropy production or
whether it just provides a lower bound that improves on established results.
This conceptual approach, which is based on the irreversibility of underlying
cycles, additionally enables us to derive estimators for the topology of the
network, i.e., the presence of a hidden cycle, its number of states and its
driving affinity. Adopting an equivalent semi-Markov description, our results
can be condensed into a fluctuation theorem for the corresponding semi-Markov
process. This mathematical perspective provides a unifying framework for the
entropy estimators considered here and established earlier ones. The crucial
role of the correct version of time-reversal helps to clarify a recent debate
on the meaning of formal versus physical irreversibility. Extensive numerical
calculations based on a direct evaluation of waiting-time distributions
illustrate our exact results and provide an estimate on the quality of the
bounds for affinities of hidden cycles.
