We introduce a novel paradigm for Bayesian learning based on optimal transport theory. Namely, we propose to use the Wasserstein barycenter of the posterior law on models as a predictive posterior, thus introducing an alternative to classical choices like the maximum a posteriori estimator and the Bayesian model average. We exhibit conditions granting the existence and statistical consistency of this estimator, discuss some of its basic and specific properties, and provide insight into its theoretical advantages. Finally, we introduce a novel numerical method which is ideally suited for the computation of our estimator, and we explicitly discuss its implementations for specific families of models. This method can be seen as a stochastic gradient descent algorithm in the Wasserstein space, and is of independent interest and applicability for the computation of Wasserstein barycenters. We also provide an illustrative numerical example for experimental validation of the proposed method.

We present and study a novel algorithm for the computation of 2-Wasserstein population barycenters. The proposed method admits, at least, two relevant interpretations: it can be seen as a stochastic gradient descent procedure in the 2-Wasserstein space, and also as a manifestation of a Law of Large Numbers in the same space. Building on stochastic gradient descent, the proposed algorithm is online and computationally inexpensive. Furthermore, we show that the noise in the method can be attenuated via the use of mini-batches, and also analyse those cases where the iterations of the algorithm have a semi-explicit form. The accompanying paper [7] develops a statistical application (and numerical implementation) of the proposed method aiming at Bayesian learning.

BackgroundAlthough Tellinidae is one of the largest and most diverse families of bivalves, its taxonomy is utterly chaotic. This is mainly due to the morphological diversity and homoplasy displayed by their shells and to the scarcity of the molecular phylogenetic studies performed on them. A molecular cytogenetic analysis of four tellin shell species, Bosemprella incarnata, Macomangulus tenuis, Moerella donacina and Serratina serrata, was performed. To molecularly characterize the analyzed specimens, the sequence of a fragment of the mitochondrial cytochrome c oxidase subunit I (COI) was also studied.ResultsThe karyotypes of the four species were composed of different amounts of bi-armed and telocentric chromosomes. The chromosomal mapping of 45S and 5S rDNA and H3 histone gene clusters by fluorescent in situ hybridization also revealed conspicuous differences on the distribution of these DNA sequences on their karyotypes. Vertebrate type telomeric sequences were located solely on both ends of each chromosome in all four tellin shells.ConclusionWe present clear evidence of the valuable information provided by FISH signals in both analyzing chromosome evolution in Tellinidae and as a further tool in identifying tellin shell specimens for molecular phylogenies.

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This research try to determine what factors influence the well-being of various communities and city council in Chile, based on urban national life quality indicators, the above carried out for communes that are close to the threshold of 50,000 inhabitants, which represents close to 100 communes of the country, reaching approximately 80% of the population. To measure efficiency, an econometric model is developed with a series of variables that explain the levels of well-being, as well as to also find measures of significance and importance in the relationship structure, in order to understand the sensitivity of some factors in the national well-being and quality of life.