Hyperparameters: Optimize, or Integrate Out?

Abstract

I examine two approximate methods for computational implementation of Bayesian hierarchical models, that is, models which include unknown hyperparameters such as regularization constants. In the ‘evidence framework’ the model parameters are integrated over, and the resulting evidence is maximized over the hyperparameters. The optimized hyperparameters are used to define a Gaussian approximation to the posterior distribution. In the alternative ‘MAP’ method, the true posterior probability is found by integrating over the hyperparameters. The true posterior is then maximized over the model parameters, and a Gaussian approximation is made. The similarities of the two approaches, and their relative merits, are discussed, and comparisons are made with the ideal hierarchical Bayesian solution.

In moderately ill-posed problems, integration over hyperparameters yields a probability distribution with a skew peak which causes significant biases to arise in the MAP method. In contrast, the evidence framework is shown to introduce negligible predictive error, under straightforward conditions.

General lessons are drawn concerning the distinctive properties of inference in many dimensions.

“Integrating over a nuisance parameter is very much like estimsting the parameter from the data, and then using that estimate in our equations.” G.L. Bretthorst

“This integration would be counter-productive as far as practical manipulation is concerned.” S.F. Gull