Related papers: Approximation Operators, Exponential, q-Exponentia…
Free exponential families have been previously introduced as a special case of the q-exponential family. We show that free exponential families arise also from a procedure analogous to the definition of exponential families by using the…
The exponential family of models is defined in a general setting, not relying on probability theory. Some results of information geometry are shown to remain valid. Exponential families both of classical and of quantum mechanical…
We develop an analog of the exponential families of Wilf in which the label sets are finite dimensional vector spaces over a finite field rather than finite sets of positive integers. The essential features of exponential families are…
A generalised notion of exponential families is introduced. It is based on the variational principle, borrowed from statistical physics. It is shown that inequivalent generalised entropy functions lead to distinct generalised exponential…
We provide a general condition under which e-variables in the form of a simple-vs.-simple likelihood ratio exist when the null hypothesis is a composite, multivariate exponential family. Such `simple' e-variables are easy to compute and…
The notion of generalised exponential family is considered in the restricted context of nonextensive statistical physics. Examples are given of models belonging to this family. In particular, the q-Gaussians are discussed and it is shown…
In this paper, we give a method to construct "good" exponential families systematically by representation theory. More precisely, we consider a homogeneous space $G/H$ as a sample space and construct an exponential family invariant under…
We introduce generalized notions of a divergence function and a Fisher information matrix. We propose to generalize the notion of an exponential family of models by reformulating it in terms of the Fisher information matrix. Our methods are…
We introduce a finite version of free probability and show the link between recent results using polynomial convolutions and the traditional theory of free probability. One tool for accomplishing this is a seemingly new transformation that…
This article studies exponential families $\mathcal{E}$ on finite sets such that the information divergence $D(P\|\mathcal{E})$ of an arbitrary probability distribution from $\mathcal{E}$ is bounded by some constant $D>0$. A particular…
We introduce three one-parameter semigroups of operators and determine their spectra. Two of them are fractional integrals associated with the Askey-Wilson operator. We also study these families as families of positive linear approximation…
The {\lambda}-exponential family has recently been proposed to generalize the exponential family. While the exponential family is well-understood and widely used, this it not the case of the {\lambda}-exponential family. However, many…
Exponential family distributions are highly useful in machine learning since their calculation can be performed efficiently through natural parameters. The exponential family has recently been extended to the t-exponential family, which…
We consider (self-adjoint) families of infinite matrices of noncommutative random variables such that the joint distribution of their entries is invariant under conjugation by a free quantum group. For the free orthogonal and…
This investigation pertains to the construction of a class of generalised deformed derivative operators which furnish the familiar finite difference and the q-derivatives as special cases. The procedure involves the introduction of a linear…
This article gives a brief introduction to $q$-special functions, i.e., $q$-analogues of the classical special functions. Here $q$ is a deformation parameter, usually $0<q<1$, where $q=1$ is the classical case. The main topics to be treated…
Following the pioneering work of Duistermaat and Gr\"unbaum, we call a family $\{p_n(x)\}_{n=0}^{\infty}$ of polynomials bispectral, if the polynomials are simultaneously eigenfunctions of two commutative algebras of operators: one…
Exponential families form the backbone of modern statistics and machine learning, but textbooks seldom derive them from first principles in an accessible way. Although minimal sufficiency and the principle of maximum entropy, originating in…
This paper presents a novel systematic methodology to obtain new simple and tight approximations, lower bounds, and upper bounds for the Gaussian Q-function, and functions thereof, in the form of a weighted sum of exponential functions.…
In a first part, we give a method for solving a family of fuchsian systems of operators of pseudo-derivations associated to a family of homographies with two parameters which unify and generalize the differential, the difference and the…