A Bayesian Nonlinear Regression Model Based on t-Distributed Errors
In this contribution, a robust Bayesian approach to adjusting a nonlinear regression model with t-distributed errors is presented. In this approach the calculation of the posterior model parameters is feasible without linearisation of the functional model
- PDF / 522,656 Bytes
- 9 Pages / 595.516 x 790.987 pts Page_size
- 100 Downloads / 227 Views
Abstract
In this contribution, a robust Bayesian approach to adjusting a nonlinear regression model with t-distributed errors is presented. In this approach the calculation of the posterior model parameters is feasible without linearisation of the functional model. Furthermore, the integration of prior model parameters in the form of any family of prior distributions is demonstrated. Since the posterior density is then generally non-conjugated, Monte Carlo methods are used to solve for the posterior numerically. The desired parameters are approximated by means of Markov chain Monte Carlo using Gibbs samplers and Metropolis-Hastings algorithms. The result of the presented approach is analysed by means of a closed-loop simulation and a real world application involving GNSS observations with synthetic outliers. Keywords
Bayesian nonlinear regression model Gibbs sampler Markov Chain Monte Carlo Metropolis-Hastings algorithm Scaled t-distribution
1
Introduction
The estimation of model parameters is a fundamental task in geodetic applications. One possibility for accomplishing this task is provided by Bayesian inference, which is based on Bayes’ theorem and utilizes probability density functions of observations and parameters. Bayesian inference also enables hypothesis testing and the determination of confidence regions. In comparison to classical non-Bayesian statistics, Bayesian inference is more intuitive and “methods become apparent which in traditional statistics give the impression of arbitrary computational rules” (according to Koch 2007, p. 1). The fields of application of Bayesian statistics are diverse and include disciplines such as biological, social and economic sciences (see Gelman et al. 2014, for the fundamental basics of Bayesian inference and some appliA. Dorndorf () · B. Kargoll · J.-A. Paffenholz · H. Alkhatib Geodetic Institute, Leibniz University Hannover, Hannover, Germany e-mail: [email protected]; [email protected]; [email protected]; [email protected]
cation examples from these disciplines). Bayesian statistics has also been used for different geodetic applications for decades (Bossler 1972; Koch 1988, 2007, 2018; Riesmeier 1984; Schaffrin 1987; Yang 1991; Zhu et al. 2005). However, in all of these studies simple linear functions are used, simple conjugate prior density functions are assumed or outlier-affected observations are not considered. A particular problem that arises with nonlinear models or non-conjugate priors in connection with popular classes of algorithms, such as Markov chain Monte Carlo (MCMC) methods (cf. Gamerman and Lopes 2006; Gelman et al. 2014), one generally cannot sample directly from the posterior density function. The class of MCMC methods includes the well known Metropolis-Hastings algorithm and Gibbs sampler. The latter has been originally developed for the Bayesian restoration of digital images and later used for a variety of problems of Bayesian inference. Such problems include nonlinear inverse problems (se
Data Loading...
