Final Report Summary - INFINITEBAYESIAN (Bayesian Statistics in Infinite Dimensions: TargetingPriors by Mathematical Analysis)
Quantifying the uncertainty in scientific or societal conclusions is important in many applications and is the bread and butter of mathematical statistics. The Bayesian statistical paradigm has gained in popularity in many areas, including climate science, genomics, astronomy or life sciences, which all have to deal with high-dimensional phenomena and modelling. In this project we have shown that uncertainty quantification for such high-dimensional problems, by any method and hence also the Bayesian method, is always conditional on prior assumptions on the unknown response function or signal. When using the Bayesian method such assumptions can be minimised by making a prior probability distribution flexible towards complexity, by making it dependent on a hyperparameter, but some prior and non-verifiable hypotheses remain necessary. In this project we have described the latter hypotheses in a precise mathematical way, and shown them to be dependent on the priors. Our results apply to curve and surface estimation, the estimation of sparse signals, and to both direct and so-called inverse problems, including diffusion equations. We have also shown the benefit of incorporating external information in high dimensional estimation problems and investigated mixed Bayesian-nonBayesian strategies. The key outcomes of the project are formulated as precise mathematical theorems and proofs.