Optical local Gaussian approximation of an exponential family

1987 ◽  
Vol 76 (1) ◽  
pp. 103-119
Author(s):  
Enno Mammen
Keyword(s):  
Exponential Family ◽  
Gaussian Approximation

scholarly journals Gaussian approximation of general non-parametric posterior distributions

2017 ◽  
Vol 7 (3) ◽  
pp. 509-529
Author(s):  
Zuofeng Shang ◽  
Guang Cheng
Keyword(s):  
Exponential Family ◽  
Approximation Theorem ◽  
Gaussian Approximation ◽  
Parametric Models ◽  
Smoothing Spline ◽  
Bayesian Procedures ◽  
Credible Intervals ◽  
Non Gaussian ◽  
Gaussian Regression ◽  
Non Parametric

AbstractIn a general class of Bayesian non-parametric models, we prove that the posterior distribution can be asymptotically approximated by a Gaussian process (GP). Our results apply to non-parametric exponential family that contains both Gaussian and non-Gaussian regression and also hold for both efficient (root-$n$) and inefficient (non-root-$n$) estimations. Our general approximation theorem does not rely on posterior conjugacy and can be verified in a class of GP priors that has a smoothing spline interpretation. In particular, the limiting posterior measure becomes prior free under a Bayesian version of ‘under-smoothing’ condition. Finally, we apply our approximation theorem to examine the asymptotic frequentist properties of Bayesian procedures such as credible regions and credible intervals.

An Introduction to Statistical Models for Networks

2020 ◽  
pp. 218-233
Author(s):  
Valentina Kuskova ◽  
Stanley Wasserman
Keyword(s):  
Social System ◽  
Exponential Family ◽  
Behavioral Science ◽  
Social Science Research ◽  
Science Research ◽  
Uniform Distributions ◽  
The Social ◽  
Special Cases ◽  
Social And Behavioral Science ◽  
Network Methods

Network theoretical and analytic approaches have reached a new level of sophistication in this decade, accompanied by a rapid growth of interest in adopting these approaches in social science research generally. Of course, much social and behavioral science focuses on individuals, but there are often situations where the social environment—the social system—affects individual responses. In these circumstances, to treat individuals as isolated social atoms, a necessary assumption for the application of standard statistical analysis is simply incorrect. Network methods should be part of the theoretical and analytic arsenal available to sociologists. Our focus here will be on the exponential family of random graph distributions, p*, because of its inclusiveness. It includes conditional uniform distributions as special cases.

scholarly journals On the Notion of Reproducibility and Its Full Implementation to Natural Exponential Families

Mathematics ◽  
2021 ◽  
Vol 9 (13) ◽  
pp. 1568
Author(s):  
Shaul K. Bar-Lev
Keyword(s):  
Power Function ◽  
Exponential Family ◽  
Probability Distributions ◽  
Infinite Divisibility ◽  
Variance Function ◽  
Exponential Families ◽  
Natural Exponential Family ◽  
Natural Exponential Families ◽  
Convolution Properties ◽  
The Relationship

Let F=Fθ:θ∈Θ⊂R be a family of probability distributions indexed by a parameter θ and let X1,⋯,Xn be i.i.d. r.v.’s with L(X1)=Fθ∈F. Then, F is said to be reproducible if for all θ∈Θ and n∈N, there exists a sequence (αn)n≥1 and a mapping gn:Θ→Θ,θ⟼gn(θ) such that L(αn∑i=1nXi)=Fgn(θ)∈F. In this paper, we prove that a natural exponential family F is reproducible iff it possesses a variance function which is a power function of its mean. Such a result generalizes that of Bar-Lev and Enis (1986, The Annals of Statistics) who proved a similar but partial statement under the assumption that F is steep as and under rather restricted constraints on the forms of αn and gn(θ). We show that such restrictions are not required. In addition, we examine various aspects of reproducibility, both theoretically and practically, and discuss the relationship between reproducibility, convolution and infinite divisibility. We suggest new avenues for characterizing other classes of families of distributions with respect to their reproducibility and convolution properties .

scholarly journals Multiple particle filtering for tracking wireless agents via Monte Carlo likelihood approximation

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Stephan Schlupkothen ◽  
Gerd Ascheid
Keyword(s):  
Monte Carlo ◽  
Computational Complexity ◽  
Particle Filtering ◽  
Gaussian Approximation ◽  
State Of The Art ◽  
Approximation Techniques ◽  
Accurate Localization ◽  
Likelihood Approximation ◽  
Current State ◽  
Multiple Particle

Abstract The localization of multiple wireless agents via, for example, distance and/or bearing measurements is challenging, particularly if relying on beacon-to-agent measurements alone is insufficient to guarantee accurate localization. In these cases, agent-to-agent measurements also need to be considered to improve the localization quality. In the context of particle filtering, the computational complexity of tracking many wireless agents is high when relying on conventional schemes. This is because in such schemes, all agents’ states are estimated simultaneously using a single filter. To overcome this problem, the concept of multiple particle filtering (MPF), in which an individual filter is used for each agent, has been proposed in the literature. However, due to the necessity of considering agent-to-agent measurements, additional effort is required to derive information on each individual filter from the available likelihoods. This is necessary because the distance and bearing measurements naturally depend on the states of two agents, which, in MPF, are estimated by two separate filters. Because the required likelihood cannot be analytically derived in general, an approximation is needed. To this end, this work extends current state-of-the-art likelihood approximation techniques based on Gaussian approximation under the assumption that the number of agents to be tracked is fixed and known. Moreover, a novel likelihood approximation method is proposed that enables efficient and accurate tracking. The simulations show that the proposed method achieves up to 22% higher accuracy with the same computational complexity as that of existing methods. Thus, efficient and accurate tracking of wireless agents is achieved.

scholarly journals KR20 and KR21 for Some Nondichotomous Data (It’s Not Just Cronbach’s Alpha)

2021 ◽  
pp. 001316442199253
Author(s):  
Robert C. Foster
Keyword(s):  
Exponential Family ◽  
Exponential Families ◽  
Natural Exponential Family ◽  
Cronbach’S Alpha ◽  
Natural Exponential Families ◽  
Cronbach's Alpha ◽  
Generalized Framework ◽  
The Common ◽  
Poisson Data ◽  
Poisson Count

This article presents some equivalent forms of the common Kuder–Richardson Formula 21 and 20 estimators for nondichotomous data belonging to certain other exponential families, such as Poisson count data, exponential data, or geometric counts of trials until failure. Using the generalized framework of Foster (2020), an equation for the reliability for a subset of the natural exponential family have quadratic variance function is derived for known population parameters, and both formulas are shown to be different plug-in estimators of this quantity. The equivalent Kuder–Richardson Formulas 20 and 21 are given for six different natural exponential families, and these match earlier derivations in the case of binomial and Poisson data. Simulations show performance exceeding that of Cronbach’s alpha in terms of root mean square error when the formula matching the correct exponential family is used, and a discussion of Jensen’s inequality suggests explanations for peculiarities of the bias and standard error of the simulations across the different exponential families.

scholarly journals Medium induced QCD cascades: broadening and rescattering during branching

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
E. Blanco ◽  
K. Kutak ◽  
W. Płaczek ◽  
M. Rohrmoser ◽  
R. Straka
Keyword(s):  
Momentum Transfer ◽  
Quark Gluon Plasma ◽  
Evolution Equations ◽  
Gaussian Approximation ◽  
Gluon Plasma ◽  
Jet Quenching ◽  
Diffusive Approximation

Abstract We study evolution equations describing jet propagation through quark-gluon plasma (QGP). In particular we investigate the contribution of momentum transfer during branching and find that such a contribution is sizeable. Furthermore, we study various approximations, such as the Gaussian approximation and the diffusive approximation to the jet-broadening term. We notice that in order to reproduce the BDIM equation (without the momentum transfer in the branching) the diffusive approximation requires a very large value of the jet-quenching parameter $$ \hat{q} $$ q ̂ .

Sign in / Sign up

Export Citation Format

Share Document