On adaption with noisy information

When observations can be made without noise, it is known that adaptive information is no more powerful than nonadaptive information for approximation of linear problems with Gaussian measure. When the noise is additive, independent of the true value, and normal, once again adaption does not help (Theorem 1 in Section 4). However, when those conditions are not satisfied, Examples 1 and 2 of Section 4 show that adaptive information can be much more powerful than nonadaptive information. Finally if orthogonal observations are used with the sample size as well as the number of repetitions fixed, and only the directions of observations are chosen adaptively, then once again adaption does not help (Theorem 2 in Section 5). The issue is analogous to whether sequential designs are more powerful than fixed sample size designs in Bayesian statistics.