This paper proposes an information-theoretic cost function for aggregating a Markov chain via a (possibly stochastic) mapping. The cost function is motivated by two objectives: 1) The process obtained by observing the Markov chain through the mapping should be close to a Markov chain, and 2) the aggregated Markov chain should retain as much of the temporal dependence structure of the original Markov chain as possible. We discuss properties of this parameterized cost function and show that it contains the cost functions previously proposed by Deng et al., Xu et al., and Geiger et al. as special cases. We moreover discuss these special cases providing a better understanding and highlighting potential shortcomings: For example, the cost function proposed by Geiger et al. is tightly connected to approximate probabilistic bisimulation, but leads to trivial solutions if optimized without regularization. We furthermore propose a simple heuristic to optimize our cost function for deterministic aggregations and illustrate its performance on a set of synthetic examples.