Recent zbMATH articles in MSC 62Ahttps://zbmath.org/atom/cc/62A2022-11-17T18:59:28.764376ZUnknown authorWerkzeugSome applications of fuzzy set theory in data analysis. 2https://zbmath.org/1496.620022022-11-17T18:59:28.764376ZThe articles of this volume will be reviewed individually within the series ``Freiberger Forschungsh., Reihe D 197 (1990)''. For Part 1 see [Zbl 1495.62004].Combining information across diverse sources: the II-CC-FF paradigmhttps://zbmath.org/1496.620262022-11-17T18:59:28.764376Z"Cunen, Céline"https://zbmath.org/authors/?q=ai:cunen.celine"Hjort, Nils Lid"https://zbmath.org/authors/?q=ai:hjort.nils-lidSummary: We introduce and develop a general paradigm for combining information across diverse data sources. In broad terms, suppose \(\phi\) is a parameter of interest, built up via components \(\psi_1, \dots, \psi_k\) from data sources \(1, \dots, k\). The proposed scheme has three steps. First, the independent inspection (II) step amounts to investigating each separate data source, translating statistical information to a confidence distribution (CD) \(C_j(\psi_j)\) for the relevant focus parameter \(\psi_j\) associated with data source \(j\). Second, confidence conversion (CC) techniques are used to translate the CDs to confidence log-likelihood functions. Finally, the focused fusion (FF) step uses relevant and context-driven techniques to construct a confidence distribution for the primary focus parameter \(\phi =\phi (\psi_1, \dots,\psi_k)\), acting on the combined confidence log-likelihood. In traditional setups, the II-CC-FF strategy amounts to versions of meta-analysis, and turns out to be competitive against state-of-the-art methods. Its potential lies in applications to harder problems, however. Illustrations are presented, related to actual applications.Improper priors and improper posteriorshttps://zbmath.org/1496.620272022-11-17T18:59:28.764376Z"Taraldsen, Gunnar"https://zbmath.org/authors/?q=ai:taraldsen.gunnar"Tufto, Jarle"https://zbmath.org/authors/?q=ai:tufto.jarle"Lindqvist, Bo H."https://zbmath.org/authors/?q=ai:lindqvist.bo-henrySummary: What is a good prior? Actual prior knowledge should be used, but for complex models this is often not easily available. The knowledge can be in the form of symmetry assumptions, and then the choice will typically be an improper prior. Also more generally, it is quite common to choose improper priors. Motivated by this we consider a theoretical framework for statistics that includes both improper priors and improper posteriors. Knowledge is then represented by a possibly unbounded measure with interpretation as explained by Rényi in [Acta Math. Acad. Sci. Hung. 6, 285--335 (1955; Zbl 0067.10401)]. The main mathematical result here is a constructive proof of existence of a transformation from prior to posterior knowledge. The posterior always exists and is uniquely defined by the prior, the observed data, and the statistical model. The transformation is, as it should be, an extension of conventional Bayesian inference as defined by the axioms of Kolmogorov. It is an extension since the novel construction is valid also when replacing the axioms of Kolmogorov by the axioms of Rényi for a conditional probability space. A concrete case based on Markov Chain Monte Carlo simulations and data for different species of tropical butterflies illustrate that an improper posterior may appear naturally and is useful. The theory is also exemplified by more elementary examples.