Book ChapterDOI
Differential Geometry of Statistical Models
Shun-ichi Amari
- pp 11-65
TLDR
In this article, the tangent space, the Riemannian metric and the α-connections are introduced in a statistical manifold, and the present chapter is devoted to the introduction of fundamental differential geometrical structures of statistical models.Abstract:
The present chapter is devoted to the introduction of fundamental differential-geometrical structures of statistical models. The tangent space, the Riemannian metric and the α-connections are introduced in a statistical manifold. No differential-geometrical background is required for reading this monograph, because the present chapter provides a readable introduction to differential geometry.read more
Citations
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Journal ArticleDOI
Riemann manifold Langevin and Hamiltonian Monte Carlo methods
Mark Girolami,Ben Calderhead +1 more
TL;DR: In this article, the authors proposed Metropolis adjusted Langevin and Hamiltonian Monte Carlo sampling methods on the Riemann manifold to resolve the shortcomings of existing Monte Carlo algorithms when sampling from target densities that may be high dimensional and exhibit strong correlations.
Journal Article
Riemann manifold Langevin and Hamiltonian Monte Carlo methods
Mark Girolami,Ben Calderhead +1 more
TL;DR: The methodology proposed automatically adapts to the local structure when simulating paths across this manifold, providing highly efficient convergence and exploration of the target density, and substantial improvements in the time‐normalized effective sample size are reported when compared with alternative sampling approaches.
Journal ArticleDOI
Backpropagation and stochastic gradient descent method
TL;DR: The wide applicability of the stochastic gradient descent method to various types of models and loss functions is reviewed, and in particular, it is applied to the pattern recognition problem, obtaining a new learning algorithm based on the information criterion.
Journal ArticleDOI
Exponential statistical manifold
Alberto Cena,Giovanni Pistone +1 more
TL;DR: In this article, the authors consider the non-parametric statistical model e(p) of all positive densities q that are connected to a given positive density p by an open exponential arc, i.e., a one-parameter exponential model p(t), t ∈ I, where I is an open interval.
Journal ArticleDOI
Dualistic geometry of the manifold of higher-order neurons
TL;DR: An information geometrical method, which can be applied to more general neural network manifolds, is proposed and the accuracy of statistical estimation is shown in terms of the dimensionality of a model and the number of examples.