Computational issues

The application of Bayesian ideas leads to computational problems, mostly related to the calculation of integrals for normalizing the posterior pdf and for obtaining credibility regions, or simply the moments of the distribution (and, hence, expectations, variances and covariances). The difficulties become challenging for problems involving many parameters. This is one of the reasons why Bayesian inference was abandoned at the beginning of the 20$^{\rm th}$ century in favor of simplified - and simplistic - methods. Indeed, the Bayesian renaissance over the past few decades is largely due to the emergence of new numerical methods and the dramatic increases in computational power, along with clarifying work on the foundations of the theory.