Ultimate goal is to understand the time-varying effects on the smaller dataset.
The idea is not to use different models for the small and large datasets but instead the same one.
Since the large datasets has a lot of data we can fit complex models, models with more parameters than actual datapoints in the small dataset. We then construct our priors from our inference on the larger datasets and use them as informative priors on the small dataset.
The other idea is to use an model that is suitable for the small dataset(thus much less parameters than the one above) and do inference with it on the larger dataset as to construct priors for our rollout on the smaller dataset.
The priors are based on inference on the larger datasets.