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Every day, weather forecasting centres around the world make use of noisy, incomplete observations of the atmosphere to update their weather forecasts. This process is known as data assimilation, data fusion or state estimation and is best expressed as Bayesian inference: given a set of observations, some prior beliefs and a model of the target system, what is the probability distribution of some set of unobserved quantities or latent variables at some time, possibly in the future? While data assimilation has developed rapidly in some areas, relatively little progress has been made in performing data assimilation with agent-based models. This has hampered the use of agent-based models to make quantitative claims about real-world systems. Here we present an algorithm that uses Markov-Chain-Monte-Carlo methods to generate samples of the parameters and trajectories of an agent-based model over a window of time given a set of possibly noisy, aggregated and incomplete observations of the system. This can be used as-is, or as part of a data assimilation cycle or sequential-MCMC algorithm. Our algorithm is applicable to time-stepping, agent-based models whose agents have a finite set of states and a finite number of ways of acting on the world. As presented the algorithm is only practical for agents with a few bytes of internal state although we discuss ways of removing this restriction. We demonstrate the algorithm by performing data assimilation with an agent-based, spatial predator-prey model.