Multi-linear Regression Model Of Bayesian Optimal Design

Experimental design is a technique for arranging experiments economically and scientificallybased on the theories of probability and statistics, which has extensive application in industry pro-ducing and the engineering design. Bayesian design is an important branch of the experimentaldesign. The classical approach for experimental design doesn't take account of prior knowledgeabout the regression parameter, the error distribution, requirements concerning the accuracy ofprediction, etc. However, the use of such additional information allows for a more precise estima-tion and a possibly considerable reduction of experimental efforts, which is of special importancein small sample situations. Bayesian decision theory which forms the subject of this paper is thefoundation of Bayesian experimental design. It defines a decision ruleδand use a loss function Lto measure the goodness ofδ. Design criterion is to get the design that minimize the average valueof loss function with respective to the regression parameter and response variables. Generally,δcan be an estimator of regression parameter or a predictor of response variables.We considered the construction of Bayesian optimal design for multiresponse linear regres-sion models. First, we set the distance between two probability distributions as the loss functionand use a predictive approach to derive the design criterion. We discuss convexity of the criterionfor continuous region and derive the equivalent condition of optimal design. An algorithm is con-structed to generate the Bayesian optimal design for a two-response linear regression model. Theoptimal design for discrete region which is generated by another algorithm is very similar withthat of continuous region.Second, we set the average bias of response predictor as the loss function and derive the designcriterion which is based on the mean squared error. In classical optimal design theory, many resultsare obtained under the assumption of exactly correct response and homoscedasticity. In mostsituation, however, the assumption is not always true and an unknown bias may exist between theassumed response and the true response. So it's more dangerous to put the design obtained fromthe ideal model to practice. To reduce the risk, we consider the construction of Bayesian robustoptimal design and derive the criterion by minimax method. The sensitivity testing is done for thecorrelation between responses and the prior information of unknown parameters to observe theimpact on the optimal design. Finally, we certify the necessity of robust optimal design by meansof the efficiency of all-variance design and all-bias design.