Principles of the Bayesian automatic adaptive quadrature

The addition of Bayesian inference to the automatic adaptive quadrature scheme developed in QUADPACK (R. Piessens, E. deDoncker-Kapenga, C.W. Uberhuber, and D.K. Kahaner, QUADPACK, a quadrature package for automatic integration, Springer, Berlin, 1983) significantly increases the output reliability under the lack of \it a priori knowledge on the behavior of the function over the integration domain. In the present paper, we review the progress obtained until now along these lines. Keywords: numerical integration, automatic adaptive quadrature, integrand profile analysis, reliability, Bayesian inference