  Observing stochastic trajectories with rare transitions between states,
practically undetectable on time scales accessible to experiments, makes it
impossible to directly quantify the entropy production and thus infer whether
and how far systems are from equilibrium. To solve this issue for Markovian
jump dynamics, we show a lower bound that outperforms any other estimation of
entropy production (including Bayesian approaches) in regimes lacking data due
to the strong irreversibility of state transitions. Moreover, in the limit of
complete irreversibility, our novel effective version of the thermodynamic
uncertainty relation sets a lower bound to entropy production that depends only
on non-dissipative aspects of the dynamics. Such an approach is also valuable
when dealing with jump dynamics with a deterministic limit, such as
irreversible chemical reactions.
