In many real-world applications, we are interested in approximating black-box, costly functions as accurately as possible with the smallest number of function evaluations. A complex computer code is an example of such a function. In this work, a Gaussian process (GP) emulator is used to approximate the output of complex computer code. We consider the problem of extending an initial experiment (set of model runs) sequentially to improve the emulator. A sequential sampling approach based on leave-one-out (LOO) cross-validation is proposed that can be easily extended to a batch mode. This is a desirable property since it saves the user time when parallel computing is available. After fitting a GP to training data points, the expected squared LOO (ES-LOO) error is calculated at each design point. ES-LOO is used as a measure to identify important data points. More precisely, when this quantity is large at a point it means that the quality of prediction depends a great deal on that point and adding more samples nearby could improve the accuracy of the GP. As a result, it is reasonable to select the next sample where ES-LOO is maximised. However, ES-LOO is only known at the experimental design and needs to be estimated at unobserved points. To do this, a second GP is fitted to the ES-LOO errors and where the maximum of the modified expected improvement (EI) criterion occurs is chosen as the next sample. EI is a popular acquisition function in Bayesian optimisation and is used to trade-off between local/global search. However, it has a tendency towards exploitation, meaning that its maximum is close to the (current) "best" sample. To avoid clustering, a modified version of EI, called pseudo expected improvement, is employed which is more explorative than EI yet allows us to discover unexplored regions. Our results show that the proposed sampling method is promising.
Consistency of the Adaptive Multiple Importance Sampling
在蒙特卡罗技术中，重要性抽样需要对建议分布进行微调，这现在可以通过迭代方案顺利解决。Cornuet等人（2012）的自适应多重要性抽样（AMIS）由于引入了回收程序，在稳定性和有效样本量方面提供了显著改善。然而，非盟特派团信息系统估计数的一致性在很大程度上仍然没有定论。在这项工作中，我们证明了AMIS的收敛性，在学习过程中略有修改的成本。与Douc et al.（2007 a）相反，这里的结果是在渐近状态下得到的，其中迭代次数趋于无穷大，而每次迭代的绘图次数是固定的，但整数序列不断增长。因此，在最后一个政权的自适应人口蒙特卡罗算法的一些结果揭示了新的光。
