I examine two approximate methods for computational implementation of Bayesian hierarchical models, that is, models that include unknown hyperparameters such as regularization constants and noise levels. 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 signifi-cant 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 inference in many dimensions.