Hello!
I have the following problem. I have data with an hierarchical structure and I know the parameter values change over time. I get new data every month and I would like to update the model with the new data, where the new data has a higher weight than the previous data points.
Suppose we start with the following hierarchical model:
where j \in \{1, ..., N\} are N groups.
I train this model on an initial dataset, and obtain posterior distributions for the parameters \{\alpha_j \mid j \in N\}, \mu_\alpha, \sigma_\alpha.
I would then like to update the model with data from the next month. Ideally, the hierarchical structure would be preserved. However, assuming I want to update the hierarchical distribution by using the posteriors as priors, I don’t see how to incorporate the information on the alphas that I learned in the earlier stage.
Currently, I lose the hierarchical structure after the initial model training, and use the posteriors on the alphas as priors for the new model, estimating a separate (non-hierarchical) model for each of the groups. To assign more weight to the new month, the standard deviation of the posterior is multiplied by a small factor.
I would rather not abandon the initial hierarchical structure; but I cannot come up with a model that retains it while preserving the information on the individual alphas at the same time. Could you please give suggestions on how to do this?