Bayesian inference with geodetic applications
This introduction to Bayesian inference places special emphasis on applications. All basic concepts are presented: Bayes' theorem, prior density functions, point estimation, confidence region, hypothesis testing and predictive analysis. In addition, Monte Carlo methods are discussed since the applications mostly rely on the numerical integration of the posterior distribution. Furthermore, Bayesian inference in the linear model, nonlinear model, mixed model and in the model with unknown variance and covariance components is considered. Solutions are supplied for the classification, for the posterior analysis based on distributions of robust maximum likelihood type estimates, and for the reconstruction of digital images.
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Maximum Entropy Priors
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according applied assume Bayes estimate Bayes estimate BB Bayes rule Bayesian analysis Bayesian inference classiﬁcation conﬁdence interval conﬁdence region conjugate prior covariance components covariance matrix deﬁned denotes density values derived epoch Example expected value ﬁnally ﬁnd ﬁrst ﬁxed follows fulﬁlled gamma distribution geodesy given gives gray levels Hence identical integration with respect introduce iteration Koch likelihood function line elements linear model M-estimate marginal density marginal distribution marginal posterior density marginal posterior distribution mixed model Monte Carlo integration multivariate t-distribution noninformative priors normal-gamma distribution null hypothesis numerical observation vector obtain parame parameter space parameter vector pixel positive deﬁnite posterior density function posterior distribution prior density function prior distribution prior information rameters random numbers random variable random vector robust estimation Section signal solved standard statistical techniques statistical inference subspace substituting theorem tion transformation unknown parameters variance and covariance variance components variance factor versus H1 weight