Dave Clark writes:
I was hoping for your opinion on a topic related to hierarchical models.
I am an actuary and have generally worked with the concept of hierarchical models in the context of credibility theory. The text by Bühlmann and Gisler (A Course in Credibility Theory; Springer) sets up the mixed models under the idea of Empirical Bayes formulas, using linear approximation that only require variances. These variances can be calculated directly, without the need for iteration or MCMC.
Formulas 2.8-2.10 of the paper “Combining generalized linear models and credibility models in practice” by Esbjorn Ohlsson (Scandinavian Actuarial Journal, 2008, 4, 301-314) give this result in the most basic case. In business applications, this seems a practical way to implement the theory.
Have you done work on this direct calculation method, or have any concerns on accuracy of results?
I replied: The quick answer is that with modern programs such as Stan, it is not such a big deal to just fit the full model. If you need an approximation for reasons of computing speed, this is fine, but usually I’d just prefer to fit the full Bayesian model directly so I don’t have to worry about these approximations. In addition, point estimates of variance parameters can be noisy; this is another motivation to go full Bayes which will account for all the uncertainty in the model. If you are doing point estimation, it can be useful to do some regularization to avoid some of the biggest problems with noisy estimates; see these two papers:
Clark followed up:
The main attraction of the direct solution, rather than Bayesian simulation or ML iteration, is the convenience in a business environment. We can get a “real time” answer, which is helpful when we are in the middle of a pricing negotiation. Also, the “credibility” format is easy to communicate to non-technical users, since it easily translates into simple weighted averages and users even know how to over-write model results when they need to. As the full Bayesian approaches become more accessible I do hope to move in that direction.