Related papers: Free Exponential Families as Kernel Families
This paper continues the study of a kernel family which uses the Cauchy-Stieltjes kernel in place of the celebrated exponential kernel of the exponential families theory. We extend the theory to cover generating measures with support that…
In this paper, we determine the effect of the free multiplicative convolution on the pseudo-variance function of a Cauchy-Stieltjes kernel family. We then use the machinery of variance functions to establish some limit theorems related to…
We explore properties of Cauchy-Stieltjes families that have no counterpart in exponential families. We relate the variance function of the iterated Cauchy-Stieltjes family to the pseudo-variance function of the initial Cauchy-Stieltjes…
Using the technique developed in approximation theory, we construct examples of exponential families of infinitely divisible laws which can be viewed as deformations of the normal, gamma, and Poisson exponential families. Replacing the…
A nonparametric family of conditional distributions is introduced, which generalizes conditional exponential families using functional parameters in a suitable RKHS. An algorithm is provided for learning the generalized natural parameter,…
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…
We study a family of free stochastic processes whose covariance kernels $K$ may be derived as a transform of a tempered measure $\sigma$. These processes arise, for example, in consideration non-commutative analysis involving free…
This paper studies variance functions of Cauchy-Stieltjes Kernel families generated by compactly supported centered probability measures. We describe several operations that allow us to construct additional variance functions from known…
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 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…
We show that all the possible pairs of integers occur as exponents for free or nearly free irreducible plane curves and line arrangements, by producing only two types of simple families of examples. The topology of the complements of these…
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…
Exponential families encompass the distributions central to modern machine learning -- softmax, Gaussians, and Boltzmann distributions -- and underlie the theory of variational inference, entropy-regularized reinforcement learning, and…
Exponential families are a particular class of statistical manifolds which are particularly important in statistical inference, and which appear very frequently in statistics. For example, the set of normal distributions, with mean {\mu}…
Motivated by the asymptotic collective behavior of random and deterministic matrices, we propose an approximation (called "free deterministic equivalent") to quite general random matrix models, by replacing the matrices with operators…
In this paper we study a class of exponential family on permutations, which includes some of the commonly studied Mallows models. We show that the pseudo-likelihood estimator for the natural parameter in the exponential family is…
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…
This paper summarizes some known results about Appell polynomials and investigates their various analogs. The primary of these are the free Appell polynomials. In the multivariate case, they can be considered as natural analogs of the…
We describe groups elementarily equivalent to a free metabelian group with n generators. We also explore an exponentiation that naturally occurs in metabelian groups.