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In the Bayesian framework power prior distributions are increasingly adopted in clinical trials and similar studies to incorporate external and past information, typically to inform the parameter associated to a treatment effect. Their use is particularly effective in scenarios with small sample sizes and where robust prior information is actually available. A crucial component of this methodology is represented by its weight parameter, which controls the volume of historical information incorporated into the current analysis. This parameter can be considered as either fixed or random. Although various strategies exist for its determination, eliciting the prior distribution of the weight parameter according to a full Bayesian approach remains a challenge. In general, this parameter should be carefully selected to accurately reflect the available prior information without dominating the posterior inferential conclusions. To this aim, we propose a novel method for eliciting the prior distribution of the weight parameter through a simulation-based calibrated Bayes factor procedure. This approach allows for the prior distribution to be updated based on the strength of evidence provided by the data: The goal is to facilitate the integration of historical data when it aligns with current information and to limit it when discrepancies arise in terms, for instance, of prior-data conflicts. The performance of the proposed method is tested through simulation studies and applied to real data from clinical trials.

Replication of scientific studies is important for assessing the credibility of their results. However, there is no consensus on how to quantify the extent to which a replication study replicates an original result. We propose a novel Bayesian approach based on mixture priors. 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. The mixture weight then determines the extent to which the original and replication data are pooled. Two distinct strategies are presented: one with fixed mixture weights, and one that introduces uncertainty by assigning a prior distribution to the mixture weight itself. Furthermore, it is shown how within this framework Bayes factors can be used for formal testing of scientific hypotheses, such as tests regarding the presence or absence of an effect. To showcase the practical application of the methodology, we analyze data from three replication studies. Our findings suggest that mixture priors are a valuable and intuitive alternative to other Bayesian methods for analyzing replication studies, such as hierarchical models and power priors. We provide the free and open source R package repmix that implements the proposed methodology.

Over the last few years, there has been a growing interest in the prediction and modelling of competitive sports outcomes, with particular emphasis placed on this area by the Bayesian statistics and machine learning communities. In this paper, we have carried out a comparative evaluation of statistical and machine learning models to assess their predictive performance for the 2022 FIFA World Cup and for the 2023 CAF Africa Cup of Nations by evaluating alternative summaries of past 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 in some canonical goal-based models both the Bradley-Terry-Davidson derived ranking and the widely recognized Coca-Cola FIFA ranking commonly adopted by football fans and amateurs.