Aitchison Geometry for Probability and Likelihood as a new approach to mathematical statistics

The Aitchison vector space structure for the simplex is generalized to a Hilbert space structure A2(P) for distributions and likelihoods on arbitrary spaces. Central notations of statistics, such as Information or Likelihood, can be identified in the algebraical structure of A2(P) and their corresponding notions in compositional data analysis, such as Aitchison distance or centered log ratio transform. In this way very elaborated aspects of mathematical statistics can be understood easily in the light of a simple vector space structure and of compositional data analysis. E.g. combination of statistical information such as Bayesian updating, combination of likelihood and robust M-estimation functions are simple additions/ perturbations in A2(Pprior). Weighting observations corresponds to a weighted addition of the corresponding evidence. Likelihood based statistics for general exponential families turns out to have a particularly easy interpretation in terms of A2(P). Regular exponential families form finite dimensional linear subspaces of A2(P) and they correspond to finite dimensional subspaces formed by their posterior in the dual information space A2(Pprior). The Aitchison norm can identified with mean Fisher information. The closing constant itself is identified with a generalization of the cummulant function and shown to be Kullback Leiblers directed information. Fisher information is the local geometry of the manifold induced by the A2(P) derivative of the Kullback Leibler information and the space A2(P) can therefore be seen as the tangential geometry of statistical inference at the distribution P. The discussion of A2(P) valued random variables, such as estimation functions or likelihoods, give a further interpretation of Fisher information as the expected squared norm of evidence and a scale free understanding of unbiased reasoning

Graphical simulators for AUV development

A long development time is needed from the design to the implementation of an AUV. During the first steps, simulation plays an important role, since it allows for the development of preliminary versions of the control system to be integrated. Once the robot is ready, the control systems are implemented, tuned and tested. The use of a real-time simulator can help closing the gap between off-line simulation and real testing using the already implemented robot. When properly interfaced with the robot hardware, a real-time graphical simulation with a "hardware in the loop" configuration, can allow for the testing of the implemented control system running in the actual robot hardware. Hence, the development time is drastically reduced. These paper overviews the field of graphical simulators used for AUV development proposing a classification. It also presents NEPTUNE, a multi-vehicle, real-time, graphical simulator based on OpenGL that allows hardware in the loop simulations