Self-adapting mixture posteriors with Hellinger distance

In clinical trials, evidence from historical data is often employed with current data to improve inference on a parameter of interest, such as the effect of a new treatment. Several methods incorporate historical information in a prior while accounting for potential conflict with current data. This prior is then updated into a posterior used for inference. Conversely, in this article we directly consider a posterior distribution of the parameter defined as the weighted average between the posterior given all data available (historical and current) and the one based on current data only. Specifically, we propose a dynamic approach to choose the weight of the components according to a measure of compatibility based on the Hellinger distance between historical and current data. Under normality assumptions, we explore features of this posterior distribution as well as frequentist operating characteristics of the resulting testing procedure.