Updating a hierarchical model in the face of new data


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:

y_{ij} \sim N(\alpha_j, \sigma_y) \\ \alpha_j \sim N(\mu_\alpha, \sigma_\alpha) \\ \mu_{\alpha} \sim N(0, 1000)

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?