This model has two hyperparameters: the inverse of the common variance to the components and the inverse of the variance of the prior probability . In the empirical Bayesian method, hyperparameters are estimated by maximizing the evidence for the hyperparameters:
For the BSOM model, however, it is difficult to calculate this integral exactly, thus we need to use an approximated integral. Here, we use a Gaussian approximation (Laplace method) at the MAP estimates . Now, the approximated log evidence is given by
where is the negative Hessian of the log posterior probability at the MAP estimates. The formula to calculate this Hessian is given in .