  This paper studies various notions of approximate probabilistic bisimulation
on labeled Markov chains (LMCs). We introduce approximate versions of weak and
branching bisimulation, as well as a notion of $\varepsilon$-perturbed
bisimulation that relates LMCs that can be made (exactly) probabilistically
bisimilar by small perturbations of their transition probabilities. We explore
how the notions interrelate and establish their connections to other well-known
notions like $\varepsilon$-bisimulation.
