  The entropy of a finite probability space $X$ measures the observable
cardinality of large independent products $X^{\otimes n}$ of the probability
space. If two probability spaces $X$ and $Y$ have the same entropy, there is an
almost measure-preserving bijection between large parts of $X^{\otimes n}$ and
$Y^{\otimes n}$. In this way, $X$ and $Y$ are asymptotically equivalent.
  It turns out to be challenging to generalize this notion of asymptotic
equivalence to configurations of probability spaces, which are collections of
probability spaces with measure-preserving maps between some of them.
  In this article we introduce the intrinsic Kolmogorov-Sinai distance on the
space of configurations of probability spaces. Concentrating on the large-scale
geometry we pass to the asymptotic Kolmogorov-Sinai distance. It induces an
asymptotic equivalence relation on sequences of configurations of probability
spaces. We will call the equivalence classes \emph{tropical probability
spaces}.
  In this context we prove an Asymptotic Equipartition Property for
configurations. It states that tropical configurations can always be
approximated by homogeneous configurations. In addition, we show that the
solutions to certain Information-Optimization problems are
Lipschitz-con\-tinuous with respect to the asymptotic Kolmogorov-Sinai
distance. It follows from these two statements that in order to solve an
Information-Optimization problem, it suffices to consider homogeneous
configurations.
  Finally, we show that spaces of trajectories of length $n$ of certain
stochastic processes, in particular stationary Markov chains, have a tropical
limit.
