I am trying to find good definitions and examples for both these concepts regarding frequentist vs Bayesian statistics. Can anyone please shed light on them and explain them? Furthermore, why are Bayesian methods often considered coherent, while frequentist methods seem to focus on calibration. Finally are default Bayesian methods well calibrated and coherent?
A set of probabilities is coherent if a bookie who is willing to take all finite bets at stated prices cannot be forced to lose in all states of nature.
This is the "you cut; I choose" problem. You cut the cake in half, and I get to choose the half I want. If you state the odds, then I get to choose the bets.
Calibration, unfortunately, has multiple meanings. However, with reference to coherence, a model is well calibrated if it predicts an occurrence $k$ with probability $\alpha$ and $k$ actually happens with long-run frequency $\alpha$.
Additionally, de Finetti also stated the coherence principle with respect to losses. Given a quadratic loss function, if a set of probabilities are coherent then there doesn't exist an assignment of probabilities to outcomes that would uniformly reduce the penalty.
If a model is a well calibrated Bayesian model, then it is intrinsically coherent.
Just a footnote to the second definition, for some density functions all possible choices on the real number line minimize quadratic loss equally well, so there is a degenerate case in the mix.