Evidence functions: a compositional approach to information

The discrete case of Bayes’ formula is considered the paradigm of information acquisition. Prior and posterior probability functions, as well as likelihood functions, called evidence functions, are compositions following the Aitchison geometry of the simplex, and have thus vector character. Bayes’ formula becomes a vector addition. The Aitchison norm of an evidence function is introduced as a scalar measurement of information. A fictitious fire scenario serves as illustration. Two different inspections of affected houses are considered. Two questions are addressed: (a) which is the information provided by the outcomes of inspections, and (b) which is the most informative inspection.Peer Reviewe

Evidence functions: a compositional approach to information

The discrete case of Bayes’ formula is considered the paradigm of information acquisition. Prior and posterior probability functions, as well as likelihood functions, called evidence functions, are compositions following the Aitchison geometry of the simplex, and have thus vector character. Bayes’ formula becomes a vector addition. The Aitchison norm of an evidence function is introduced as a scalar measurement of information. A fictitious fire scenario serves as illustration. Two different inspections of affected houses are considered. Two questions are addressed: (a) which is the information provided by the outcomes of inspections, and (b) which is the most informative inspection.Peer ReviewedPostprint (author's final draft

Geostatistical interpretation of paleoceanographic data over large ocean basins - Reality and fiction

A promising approach to reconstruct oceanographic scenarios of past time slices is to drive numerical ocean circulation models with sea surface temperatures, salinities, and ice distributions derived from sediment core data. Set up properly, this combination of boundary conditions provided by the data and physical constraints represented by the model can yield physically consistent sets of three-dimensional water mass distribution and circulation patterns. This idea is not only promising but dangerous, too. Numerical models cannot be fed directly with data from single core locations distributed unevenly and, as it is the common case, scarcely in space. Conversely, most models require forcing data sets on a regular grid with no missing points, and some method of interpolation between punctual source data and model grid has to be employed. An ideal gridding scheme must retain as much of the information present in the sediment core data while generating as few artifacts in the interpolated field as possible. Based on a set of oxygen isotope ratios, we discuss several standard interpolation strategies, namely nearest neighbour schemes, bicubic splines, Delaunay triangulation, and ordinary and indicator kriging. We assess the gridded fields with regard to their physical consistence and their implications for the oceanic circulation

General approach to coordinate representation of compositional tables

This is the peer reviewed version which has been published in final form at [https://doi.org/10.1111/sjos.12326]. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Use of Self-Archived Versions.Compositional tables can be considered a continuous counterpart to the well-known contingency tables. Their cells, which generally contain positive real numbers rather than just counts, carry relative information about relationships between two factors. Hence, compositional tables can be seen as a generalization of (vector) compositional data. Due to their relative character, compositions are commonly expressed in orthonormal coordinates using a sequential binary partition prior to being further processed by standard statistical tools. Unfortunately, the resulting coordinates do not respect the two-dimensional nature of compositional tables. Information about relationship between factors is thus not well captured. The aim of this paper is to present a general system of orthonormal coordinates with respect to the Aitchison geometry, which allows for an analysis of the interactions between factors in a compositional table. This is achieved using logarithms of odds ratios, which are also widely used in the context of contingency tables

Evidence information in Bayesian updating

Bayes theorem (discrete case) is taken as a paradigm of information acquisition. As men-tioned by Aitchison, Bayes formula can be identified with perturbation of a prior probability vector and a discrete likelihood function, both vectors being compositional. Considering prior, poste-rior and likelihood as elements of the simplex, a natural choice of distance between them is the Aitchison distance. Other geometrical features can also be considered using the Aitchison geom-etry. For instance, orthogonality in the simplex allows to think of orthogonal information, or the perturbation-difference to think of opposite information. The Aitchison norm provides a size of compositional vectors, and is thus a natural scalar measure of the information conveyed by the likelihood or captured by a prior or a posterior. It is called evidence information, or e-information for short. In order to support such e-information theory some principles of e-information are discussed. They essentially coincide with those of compositional data analysis. Also, a comparison of these principles of e-information with the axiomatic Shannon-information theory is performed. Shannon-information and developments thereof do not satisfy scale invariance and also violate subcomposi-tional coherence. In general, Shannon-information theory follows the philosophy of amalgamation when relating information given by an evidence-vector and some sub-vector, while the dimension reduction for the proposed e-information corresponds to orthogonal projections in the simplex. The result of this preliminary study is a set of properties of e-information that may constitute the basis of an axiomatic theory. A synthetic example is used to motivate the ideas and the subsequent discussion

La geología de los planetas: Análisis estadístico de datos composicionales al servicio de la planetología.

Anàlisi estadístic de dades composicionals dels planete

Wave-height hazard analysis in Eastern Coast of Spain : Bayesian approach using generalized Pareto distribution

Standard practice of wave-height hazard analysis often pays little attention to the uncertainty of assessed return periods and occurrence probabilities. This fact favors the opinion that, when large events happen, the hazard assessment should change accordingly. However, uncertainty of the hazard estimates is normally able to hide the effect of those large events. This is illustrated using data from the Mediterranean coast of Spain, where the last years have been extremely disastrous. Thus, it is possible to compare the hazard assessment based on data previous to those years with the analysis including them. With our approach, no significant change is detected when the statistical uncertainty is taken into account. The hazard analysis is carried out with a standard model. Time-occurrence of events is assumed Poisson distributed. The wave-height of each event is modelled as a random variable which upper tail follows a Generalized Pareto Distribution (GPD). Moreover, wave-heights are assumed independent from event to event and also independent of their occurrence in time. A threshold for excesses is assessed empirically. The other three parameters (Poisson rate, shape and scale parameters of GPD) are jointly estimated using Bayes' theorem. Prior distribution accounts for physical features of ocean waves in the Mediterranean sea and experience with these phenomena. Posterior distribution of the parameters allows to obtain posterior distributions of other derived parameters like occurrence probabilities and return periods. Predictives are also available. Computations are carried out using the program BGPE v2.

Scale effect in hazard assessment - application to daily rainfall

International audienceDaily precipitation is recorded as the total amount of water collected by a rain-gauge in 24h. Events are modelled as a Poisson process and the 24h precipitation by a Generalized Pareto Distribution (GPD) of excesses. Hazard assessment is complete when estimates of the Poisson rate and the distribution parameters, together with a measure of their uncertainty, are obtained. The shape parameter of the GPD determines the support of the variable: Weibull domain of attraction (DA) corresponds to finite support variables, as should be for natural phenomena. However, Fréchet DA has been reported for daily precipitation, which implies an infinite support and a heavy-tailed distribution. We use the fact that a log-scale is better suited to the type of variable analyzed to overcome this inconsistency, thus showing that using the appropriate natural scale can be extremely important for proper hazard assessment. The approach is illustrated with precipitation data from the Eastern coast of the Iberian Peninsula affected by severe convective precipitation. The estimation is carried out by using Bayesian techniques