Comment: DoIt and Do It Well

227 matrices based on cross-validation took roughly 12 min, with the main computational burden being the repeated evaluation and inversions of the 550 × 550 matrices G() and G(). Due to the increased computation time, finding the " optimal " and matrices in each step of the sequential algorithm appeared no longer practically feasible. When adding 450 additional function evaluations (based on the components selected by iterLap), computations were considerably slowed down: inversion of the involved 1000 × 1000 matrix took six times longer compared with the 550 × 550 matrix, and one might expect an increase in the total computation time by a similar factor. Note that this is roughly in agreement with the fact that matrix inversion is roughly a O(n 3) process (if n denotes the size of the matrix). From these considerations, it becomes clear that the interpolation technique, as presented now, will become quickly infeasible when a large number of function evaluations is required to obtain an adequate approximation, as matrix inversions are the main factor driving computation time. The essential difference between the two methods is thus in their usage of target density evaluations. If it is cheap to perform a large number of evaluations, it appears iterLap (and also well-chosen and efficiently implemented MCMC algorithms) will outperform the interpolation technique, because the interpolation technique will itself get computationally intensive due to the required matrix inversions. If evaluations of the target are extremely expensive, so that only few evaluations are possible anyway, the interpolation technique seems to make better usage of the evaluations performed. The main computational bottleneck of the proposed procedure is the need to evaluate and invert large-dimensional matrices repeatedly (as in all kriging-type interpolation approaches). This can get quite computationally expensive, but might pay off, for example, when the posterior density is extremely time-consuming to evaluate or when it is of great interest to obtain a high-accuracy approximation of the posterior density itself. However, an improvement of the procedure in this regard seems possible and would certainly be of high interest. In summary, I would like to congratulate Professor Joseph for an interesting article that provides an innovative approach on how to apply kriging-type techniques for interpolation of positive functions, and I hope Professor Joseph's article stimulates further research in the application of these methods for Bayesian computational problems. I would like to end my discussion with the wish that an …