• 제목/요약/키워드: Laplace approximation

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Choosing the Tuning Constant by Laplace Approximation

  • Ahn, Sung-Mahn;Kwon, Suhn-Beom
    • Communications for Statistical Applications and Methods
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    • 제19권4호
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    • pp.597-605
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    • 2012
  • Evidence framework enables us to determine the tuning constant in a penalized likelihood formula. We apply the framework to the estimating parameters of normal mixtures. Evidence, which is a solely data-dependent measure, can be evaluated by Laplace approximation. According to a synthetic data simulation, we found that the proper values of the tuning constant can be systematically obtained.

Maximum Likelihood Estimation Using Laplace Approximation in Poisson GLMMs

  • Ha, Il-Do
    • Communications for Statistical Applications and Methods
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    • 제16권6호
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    • pp.971-978
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    • 2009
  • Poisson generalized linear mixed models(GLMMs) have been widely used for the analysis of clustered or correlated count data. For the inference marginal likelihood, which is obtained by integrating out random effects is often used. It gives maximum likelihood(ML) estimator, but the integration is usually intractable. In this paper, we propose how to obtain the ML estimator via Laplace approximation based on hierarchical-likelihood (h-likelihood) approach under the Poisson GLMMs. In particular, the h-likelihood avoids the integration itself and gives a statistically efficient procedure for various random-effect models including GLMMs. The proposed method is illustrated using two practical examples and simulation studies.

Review on statistical methods for large spatial Gaussian data

  • Park, Jincheol
    • Journal of the Korean Data and Information Science Society
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    • 제26권2호
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    • pp.495-504
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    • 2015
  • The Gaussian geostatistical model has been widely used for modeling spatial data. However, this model suffers from a severe difficulty in computation because inference requires to invert a large covariance matrix in evaluating log-likelihood. In addressing this computational challenge, three strategies have been employed: likelihood approximation, lower dimensional space approximation, and Markov random field approximation. In this paper, we reviewed statistical approaches attacking the computational challenge. As an illustration, we also applied integrated nested Laplace approximation (INLA) technology, one of Markov approximation approach, to real data to provide an example of its use in practice dealing with large spatial data.

Fourier Series Approximation for the Generalized Baumgartner Statistic

  • Ha, Hyung-Tae
    • Communications for Statistical Applications and Methods
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    • 제19권3호
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    • pp.451-457
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    • 2012
  • Baumgartner et al. (1998) proposed a novel statistical test for the null hypothesis that two independently drawn samples of data originate from the same population, and Murakami (2006) generalized the test statistic for more than two samples. Whereas the expressions of the exact density and distribution functions of the generalized Baumgartner statistic are not yet found, the characteristic function of its limiting distribution has been obtained. Due to the development of computational power, the Fourier series approximation can be readily utilized to accurately and efficiently approximate its density function based on its Laplace transform. Numerical examples show that the Fourier series method provides an accurate approximation for statistical quantities of the generalized Baumgartner statistic.

New Inference for a Multiclass Gaussian Process Classification Model using a Variational Bayesian EM Algorithm and Laplace Approximation

  • Cho, Wanhyun;Kim, Sangkyoon;Park, Soonyoung
    • IEIE Transactions on Smart Processing and Computing
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    • 제4권4호
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    • pp.202-208
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    • 2015
  • In this study, we propose a new inference algorithm for a multiclass Gaussian process classification model using a variational EM framework and the Laplace approximation (LA) technique. This is performed in two steps, called expectation and maximization. First, in the expectation step (E-step), using Bayes' theorem and the LA technique, we derive the approximate posterior distribution of the latent function, indicating the possibility that each observation belongs to a certain class in the Gaussian process classification model. In the maximization step, we compute the maximum likelihood estimators for hyper-parameters of a covariance matrix necessary to define the prior distribution of the latent function by using the posterior distribution derived in the E-step. These steps iteratively repeat until a convergence condition is satisfied. Moreover, we conducted the experiments by using synthetic data and Iris data in order to verify the performance of the proposed algorithm. Experimental results reveal that the proposed algorithm shows good performance on these datasets.

Hierarchical Bayes Estimators of Exchangeable Poisson Mean using Laplace Approximation

  • Chung, Youn-Shik
    • Communications for Statistical Applications and Methods
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    • 제2권1호
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    • pp.137-144
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    • 1995
  • Hierarchical Bayes estimations of exchangeable mean vector of a multivariate Poisson distribution are obtained. Since sophiscated analytic integration procedures are needed, the Laplace method is employed in order tocompute these estimations approximately. An example is presented.

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Bayesian Estimation of the Reliability Function of the Burr Type XII Model under Asymmetric Loss Function

  • Kim, Chan-Soo
    • Communications for Statistical Applications and Methods
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    • 제14권2호
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    • pp.389-399
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    • 2007
  • In this paper, Bayes estimates for the parameters k, c and reliability function of the Burr type XII model based on a type II censored samples under asymmetric loss functions viz., LINEX and SQUAREX loss functions are obtained. An approximation based on the Laplace approximation method (Tierney and Kadane, 1986) is used for obtaining the Bayes estimators of the parameters and reliability function. In order to compare the Bayes estimators under squared error loss, LINEX and SQUAREX loss functions respectively and the maximum likelihood estimator of the parameters and reliability function, Monte Carlo simulations are used.

베이지안 비선형회귀모형의 선택과 진단 (Bayesian Mode1 Selection and Diagnostics for Nonlinear Regression Model)

  • 나종화;김정숙
    • 응용통계연구
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    • 제15권1호
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    • pp.139-151
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    • 2002
  • 본 논문에서는 베이지안 기법을 이용한 비선형회귀모형의 선택법을 제안하였다. 베이즈요인에 기초한 이 방법은 주로 대표본의 경우에 이용되는 고전적 모형선택법에 비해 사전정보를 이용하는 측면과 비내포모형 및 소표본의 경우에 대해서도 효과적으로 사용될 수 있다는 장점을 가진다. 본 논문에서는 정보적 사전분포를 고려하였으며, 베이즈요인의 추정 방법으로 Laplace - Metropolis 추정 법을 제안하였다. 또한 MCMC 과정을 통해 추정된 모수의 수렴진단에 대해서도 고려하였다. 실제자료에 대한 최적의 모형선택 및 진단과정을 구체적으로 제시하였다.

ON THE WEAK LIMIT THEOREMS FOR GEOMETRIC SUMMATIONS OF INDEPENDENT RANDOM VARIABLES TOGETHER WITH CONVERGENCE RATES TO ASYMMETRIC LAPLACE DISTRIBUTIONS

  • Hung, Tran Loc
    • 대한수학회보
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    • 제58권6호
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    • pp.1419-1443
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    • 2021
  • The asymmetric Laplace distribution arises as a limiting distribution of geometric summations of independent and identically distributed random variables with finite second moments. The main purpose of this paper is to study the weak limit theorems for geometric summations of independent (not necessarily identically distributed) random variables together with convergence rates to asymmetric Laplace distributions. Using Trotter-operator method, the orders of approximations of the distributions of geometric summations by the asymmetric Laplace distributions are established in term of the "large-𝒪" and "small-o" approximation estimates. The obtained results are extensions of some known ones.