Related papers: Divergence and Deformed Exponential Family
We derive a large deviation principle for families of random variables in the basin of attraction of spectrally positive stable distributions by proving a uniform version of the Tauberian theorem for Laplace-Stieltjes transforms. The main…
We derive the Kullback-Leibler divergence for the normal-gamma distribution and show that it is identical to the Bayesian complexity penalty for the univariate general linear model with conjugate priors. Based on this finding, we provide…
We propose a number of concepts and properties related to `weighted' statistical inference where the observed data are classified in accordance with a `value' of a sample string. The motivation comes from the concepts of weighted…
This document describes concisely the ubiquitous class of exponential family distributions met in statistics. The first part recalls definitions and summarizes main properties and duality with Bregman divergences (all proofs are skipped).…
In memory of Marcos Moshinsky, who promoted the algebraic study of the harmonic oscillator, some results recently obtained on an infinite family of deformations of such a system are reviewed. This set, which was introduced by Tremblay,…
We prove a Liouville type theorem for entire maximal $m$-subharmonic functions in $\mathbb C^n$ with bounded gradient. This result, coupled with a standard blow-up argument, yields a (non-explicit) a priori gradient estimate for the complex…
The Jeffreys divergence is a renown symmetrization of the oriented Kullback-Leibler divergence broadly used in information sciences. Since the Jeffreys divergence between Gaussian mixture models is not available in closed-form, various…
We describe the underlying probabilistic interpretation of alpha and beta divergences. We first show that beta divergences are inherently tied to Tweedie distributions, a particular type of exponential family, known as exponential…
The Kullback-Leibler (KL) divergence is frequently used in data science. For discrete distributions on large state spaces, approximations of probability vectors may result in a few small negative entries, rendering the KL divergence…
We approximate a given rational spectral density by one that is consistent with prescribed second-order statistics. Such an approximation is obtained by minimizing a suitable distance from the given spectrum and under the constraints…
Score-matching generative models have proven successful at sampling from complex high-dimensional data distributions. In many applications, this distribution is believed to concentrate on a much lower $d$-dimensional manifold embedded into…
The extended Gaussian family is the closure of the Gaussian family obtained by completing the Gaussian family with the counterpart elements induced by degenerate covariance or degenerate precision matrices, or a mix of both degeneracies.…
We collect evidence in support of a conjecture of Griffiths, Green and Kerr on the arithmetic of extension classes of limiting mixed Hodge structures arising from semistable degenerations over a number field. After briefly summarizing how a…
This is a short presentation of some classical results on finite dimensional complex Lie algebras (classification of nilpotent Lie algebras, deformations and perturbations, contractions and rigidity). We present some applications to…
Historically tensor calculus emerged in an attempt to formalize Rie- mann's ideas. We show that tensor calculus can be based also on Lie's idea of a transformation group and this approach leads quite naturally to the concept of deformation…
How much one has learned from an experiment is quantifiable by the information gain, also known as the Kullback-Leibler divergence. The narrowing of the posterior parameter distribution $P(\theta|D)$ compared with the prior parameter…
In previous work the authors defined the k-th order simplicial distance between probability distributions which arises naturally from a measure of dispersion based on the squared volume of random simplices of dimension k. This theory is…
We provide an analog of the Hilbert-Chow morphism for generalized discriminants.
Generative models have achieved remarkable success across a range of applications, yet their evaluation still lacks principled uncertainty quantification. In this paper, we develop a method for comparing how close different generative…
We propose extreme value analogues of natural exponential families and exponential dispersion models, and introduce the slope function as an analogue of the variance function. The set of quadratic and power slope functions characterize…