Henry P. Wynn, Noha Youssef

Maximum Entropy Sampling (MES) provides a useful framework for studying sequential design for computer experiments in a Bayesian framework. However there is some technical difficulty in making the procedure fully adaptive in the sense of making proper use of previous output as well as input data. In the simple Gaussian set-up only previous input values need be used, even in the case of unknown process variance. The approach discussed is to use a full hieratchical model for the parameter/process covariances. This allows both spatial adaptation and adaptation to smoothness, which can be thought of as frequency adaptation. An additional feature is to take advantage of the Karhumen-Loeve (K-L) expansion to approximate the process covariance function using an orthogonal function basis. It is argued that this may make it easier to use Bayes hierarchical models, rather than estimating the covariance parameters directly, the traditional approach. It is shown how to reduce the full MES method to a simple algorithm by using a special empirical Bayes approximation, rather than using time-consuming integration. Simulations and a real engineering case study show that full adaptation is beneficial. Other design optimality criteria are also considered.

**Keywords:** Computer experiments; Experimental design; Maximum entropy sampling; Adaptive sampling

**Biography:** Henry Wynn is Professor Emeritus at the London School of Economics (LSE). Following an MA in mathematics at the University of Oxford and a PhD at Imperial College London, in 1970. He joined Imperial College in 1972 as a Lecturer and later became Reader. He took the Chair in Statistics at City University in 1985 and then was appointed Professor of Industrial Statistics at the University of Warwick and finally a Chair in Statistics at LSE in 2003. He is a Fellow of the IMS, holds the Guy Medal in Silver of the Royal Statístical Society and is an Honorary Fellow of the Institute of Actuaries. He has published widely in statistics, mathematics, optimisation and engineering journals, has a number of monographs and edited volumes to his name and has directed well-funded research centres. Current main interests are: experimental design, algebraic statistics, risk theory and robust engineering design.