Submitted to Statistical Science Bayesian Adaptive Randomization with Compound Utility Functions Alessandra Giovagnoli and Isabella Verdinelli Abstract. Bayesian adaptive designs formalize the use of previous knowledge at the planning stage of an experiment, permitting recursive updating of the prior information. They often make use of utility functions, while also allowing for randomization. We review frequentist and Bayesian adaptive design methods and show that some of the frequentist adaptive design methodology can also be employed in a Bayesian context. We use compound utility functions for the Bayesian designs, that are a trade-off between an optimal design infor- mation criterion, that represents the acquisition of scientific knowledge, and some ethical or utilitarian gain. We focus on binary response models on two groups with independent Beta prior distributions on the success probabilities. The treatment allocation is shown to converge to the allocation that produces the maximum utility. Special cases are the Bayesian Randomized (simply) Adaptive Compound (BRAC) design, an extension of the frequentist Se- quential Maximum Likelihood (SML) design and the Bayesian Randomized (doubly) Adaptive Compound Efficient (BRACE) design, a generalization of the Efficient Randomized Adaptive DEsign (ERADE). Numerical simulation studies compare BRAC with BRACE when D-optimality is the information criterion chosen. In analogy with the frequentist theory, the BRACE-D de- sign appears more efficient than the BRAC-D design. Key words and phrases: Bayesian Designs, Binary Response Model, Dou- bly Adaptive designs, Optimal Design Criteria, Optimal Target, Utility Func- tions. 1. INTRODUCTION The Bayesian approach is appropriate at the planning stage of an experiment, when previous information, such as historical data, findings from previous studies, and ex- pert opinions, is often available and can be expressed through a prior distribution on the parameters of the sta- tistical model. Bayesian design theory has an established tradition, and most of the design methods reviewed in [14] are still in use, although nowadays a more “objec- tive” data analysis is preferred, where the prior informa- tion is used at the design stage, while data are analyzed by frequentist statistical tools. Several authors, for example [55] have presented ways to combine Bayesian models, Alessandra Giovagnoli is retired Professor, Department of Statistical Sciences, Alma Mater Studiorum, Università di Bologna, Bologna, Italy, (e-mail: alessandra.giovagnoli@unibo.it). Isabella Verdinelli is Professor in Residence, Department of Statistics, Carnegie Mellon University, Pittsburgh, PA USA, (e-mail: isabella@stat.cmu.edu). utility functions and frequentist analyses. These ideas can be seen as special cases of design prior versus analysis prior proposed in [23]. Randomization is now recognized to play a useful role in the Bayesian approach too (see [31]). Response-adaptive designs have fully entered the sta- tistical literature in the past two decades. In response- adaptive experiments the current data are used at each step to update the next planning decision. There are simi- larities with the Bayesian paradigm, where previous data are used to update the prior to a posterior distribution. But adaptive design methods are conceptually unrelated to the Bayesian philosophy that treats unknown quantities as random, in fact Bayesian and adaptive designs have de- veloped independently of each other. This paper deals with model-based designs that are both Bayesian and adaptive, where, as new data become avail- able, the posterior distribution is updated and is also used to modify the execution of the trial. This is different from the typical Bayesian way of designing an experiment un- conditionally on the data. So these designs, despite a pi- 1