  The vast majority of the literature on learning dynamical systems or
stochastic processes from time series has focused on stable or ergodic systems,
for both Bayesian and frequentist inference procedures. However, most
real-world systems are only metastable, that is, the dynamics appear to be
stable on some time scale, but are in fact unstable over longer time scales.
Consistency of inference for metastable systems may not be possible, but one
can ask about metaconsistency: Do inference procedures converge when
observations are taken over a large but finite time interval, but diverge on
longer time scales? In this paper we introduce, discuss, and quantify
metaconsistency in a Bayesian framework. We discuss how metaconsistency can be
exploited to efficiently infer a model for a sub-system of a larger system,
where inference on the global behavior may require much more data. We also
discuss the relation between meta-consistency and the spectral properties of
the model dynamical system in the case of uniformly ergodic diffusions.
