AGNES ALGEBRAIC GEOMETRY FREE

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Two mathematical solutions and three applications to statistics based on algebraic geometry reported in this article are now being used in many practical fields in data science and artificial intelligence.Īkaike, H.: A new look at the statistical model identification.

AGNES ALGEBRAIC GEOMETRY FREE

The posterior distribution is represented by the renormalized form, the asymptotic free energy is derived, and the universal formula among the generalization loss, the cross validation, and the information criterion is established. Third, three applications to statistics are introduced. An appropriate parameter space can be found by a resolution map, which makes the posterior distribution be normal crossing and the log likelihood ratio function be well-defined. Second, two mathematical solutions are derived based on algebraic geometry. First, we explain the framework of Bayesian statistics and introduce a new perspective from the birational geometry.

In this article, the following results in recent advances are reported. However, from the end of the 20th century, new theory and methodology based on algebraic geometry have been established which enable us to investigate such models and machines in the real world. In nonidentifiable models, both the likelihood function and the posterior distribution have singularities in general, hence it was difficult to analyze their statistical properties. Many statistical models and learning machines which contain hierarchical structures or latent variables are called nonidentifiable, because the map from a parameter to a statistical model is not one-to-one. This article is a review of theoretical advances in the research field of algebraic geometry and Bayesian statistics in the last two decades.

Agnes algebraic geometry