Combining evidence under dependence

The quantification of uncertainty is a central element of statistical inference, as it enables the assessment of the reliability of inferential conclusions. In practice, uncertainty may arise from multiple sources, which are often dependent on each other, posing new challenges both methodologically and in applications. An example of this occurs when different p-values, obtained from distinct statistical tests, are available for the same hypothesis. This work is situated within this context, with the aim of contributing to the development of frequentist tools capable of handling complex scenarios where dependence among different measures of uncertainty cannot be neglected. In this perspective, new procedures are introduced for combining dependent p-values, with particular attention to the case in which they are exchangeable. Such situations arise, for instance, when p-values are generated through random splitting techniques, and therefore hold concrete relevance in many applications. In addition, methods are proposed for the combination of uncertainty sets --- such as confidence regions or prediction sets --- that may be arbitrarily dependent. Finally, the results are extended and placed within the framework of conformal prediction, an approach that allows for the construction of distribution-free prediction set under mild assumption.