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We investigate the challenge of modeling the belief state of a partially observable Markov system, given sample-access to its dynamics model. This problem setting is often approached using parametric sequential generative modeling methods. However, these methods do not leverage any additional computation at inference time to increase their accuracy. Moreover, applying these methods to belief state modeling in certain multi-agent settings would require passing policies into the belief model---at the time of writing, there have been no successful demonstrations of this. Toward addressing these shortcomings, we propose an inference-time improvement framework for parametric sequential generative modeling methods called belief fine-tuning (BFT). BFT leverages approximate dynamic programming in the form of fine-tuning to determine the model parameters at each time step. It can improve the accuracy of the belief model at test time because it specializes the model to the space of local observations. Furthermore, because this specialization occurs after the action or policy has already been decided, BFT does not require the belief model to process it as input. As a result of the latter point, BFT enables, for the first time, approximate public belief state search in imperfect-information games where the number of possible information states is too large to track tabularly. We exhibit these findings on large-scale variants of the benchmark game Hanabi.