Bayesian methods for borrowing historical information

Nowadays, the availability of large amounts of historical data makes their inclusion in current studies useful for improving the precision of inferential conclusions and assessing the credibility of previous findings. From a Bayesian perspective, power priors are increasingly used in clinical trials and similar studies to incorporate historical information. A key element here is represented by the discount parameter, which controls the extent of historical information included in the current analysis. In the first chapter of the thesis, we propose a novel method for eliciting its prior distribution using a simulation-based calibrated Bayes factor procedure. This method updates the prior distribution based on the data's evidence strength, encouraging borrowing from historical data when aligned with current information and limiting it when conflicts arise. We evaluate the method through simulations studies and real clinical trial data. In the second chapter we introduce an intuitive Bayesian approach based on mixture priors to assess success in replication studies. The idea is to use a mixture of the posterior distribution based on the original study and a non-informative distribution as the prior for the analysis of the replication study. We present strategies with both fixed and random mixture weights. Analyzing data from three replication studies, we find that mixture priors offer a valuable alternative to hierarchical models and power priors. In addition, we provide the free and open source R package repmix that implements the proposed methodology. In the third chapter we compare the predictive performances of statistical and machine learning models for the 2022 FIFA World Cup and for the 2023 CAF Africa Cup of Nations by evaluating alternative summaries of historical performances related to the involved teams. More specifically, we consider the Bayesian Bradley-Terry-Davidson model, which is a widely used statistical framework for ranking items based on paired comparisons that have been applied successfully in various domains, including football. The analysis was performed including both the Bradley-Terry-Davidson derived ranking and the widely recognized Coca-Cola FIFA ranking, commonly adopted by football fans and amateurs, in some canonical goal-based models and result-based algorithms.