Related papers: Chentsov's theorem for exponential families
We give a simple proof of the exponential de Finetti theorem due to Renner. Like Renner's proof, ours combines the post-selection de Finetti theorem, the Gentle Measurement lemma, and the Chernoff bound, but avoids virtually all…
I define a natural measure of the complexity of a parametric distribution relative to a given true distribution called the {\it razor} of a model family. The Minimum Description Length principle (MDL) and Bayesian inference are shown to…
The Boltzmann-Gibbs probability distribution, seen as a statistical model, belongs to the exponential family. Recently, the latter concept has been generalized. The q-exponential family has been shown to be relevant for the statistical…
We introduce a new definition of exponential family of Markov chains, and show that many characteristic properties of the usual exponential family of probability distributions are properly extended to Markov chains. The method of…
We define the equivariant family index of a family of elliptic operators invariant with respect to the free action of a bundle $\GR$ of Lie groups. If the fibers of $\GR \to B$ are simply-connected solvable, we then compute the Chern…
The Chernoff bound is an important inequality relation in probability theory. The original version of the Chernoff bound is to give an exponential decreasing bound on the tail distribution of sums of independent random variables. Recent…
We consider the parameter estimation of Markov chain when the unknown transition matrix belongs to an exponential family of transition matrices. Then, we show that the sample mean of the generator of the exponential family is an…
The quantum Fisher information represents the continuous family of metrics on the space of quantum states and places the fundamental limit on the accuracy of quantum state estimation. We show that the entire family of the quantum Fisher…
Braunstein and Caves (1994) proposed to use Helstrom's {\em quantum information} number to define, meaningfully, a metric on the set of all possible states of a given quantum system. They showed that the quantum information is nothing else…
In this paper I discuss the relation between the concept of the Fisher metric and the concept of differentiability of a family of probability measures. I compare the concepts of smooth statistical manifolds, differentiable families of…
Random Fisher matrices arise naturally in multivariate statistical analysis and understanding the properties of its eigenvalues is of primary importance for many hypothesis testing problems like testing the equality between two multivariate…
This work studies the large sample properties of the posterior-based inference in the curved exponential family under increasing dimension. The curved structure arises from the imposition of various restrictions on the model, such as moment…
Point estimators may not exist, need not be unique, and their distributions are not parameter invariant. Generalized estimators provide distributions that are parameter invariant, unique, and exist when point estimates do not. Comparing…
A classic theorem in combinatorial design theory is Fisher's inequality, which states that a family $\mathcal F$ of subsets of $[n]$ with all pairwise intersections of size $\lambda$ can have at most $n$ non-empty sets. One may weaken the…
We develope a new and general notion of parametric measure models and statistical models on an arbitrary sample space $\Omega$ which does not assume that all measures of the model have the same null sets. This is given by a diffferentiable…
It is well known that the independence of the sample mean and the sample variance characterizes the normal distribution. By using Anosov's theorem, we further investigate the analogous characteristic properties in terms of the sample mean…
In this note we introduce the notion of a $C^k$-diffeological statistical model, which allows us to apply the theory of diffeological spaces to (possibly singular) statistical models. In particular, we introduce a class of almost…
Famously, the quantum Fisher information -- the maximum Fisher information over all physical measurements -- is additive for independent copies of a system and the optimal measurement acts locally. We are left to wonder: does the same hold…
Among Lie submodels of the (real symmetric potential) dispersionless Nyzhnyk equation, we single out a remarkable submodel as such that, despite being the only one, is associated with a family of in general inequivalent one-dimensional…
For perturbations of integrable Hamiltonians systems, the Nekhoroshev theorem shows that all solutions are stable for an exponentially long interval of time, provided the integrable part satisfies a steepness condition and the system is…