Hierarchische Bayes-Regression bei Einbettung großer Datensätze

Abstract

Subspace embedding is a well-known method of data reduction preserving the essential information. Its applicability in a Bayesian linear regression framework has already been proven. The posterior distribution of the coefficients estimated from the original data is approximated by that from the so-called sketch up to a small, controlled error. As a generalization, some hierarchical regression models are analyzed in this thesis. Apart from the regression coefficients, hyperparameters are estimated, each with comparison between original and sketched data. Simulation studies suggest a good approximation of both the regression coefficients and hyperparameters for all linear regression models with normal likelihood and various priors. For a generalized model with a logit-based link function, the approximation seems worse. For a normal likelihood, a bounded distance between the two posterior distributions from the original and the sketched data can be derived from the non-hierarchical results, although without exact quantification. For the marginal distribution of the regression coefficients with normal priors, the previous result is reproduced exactly. If a location hyperparameter is the expected value of a regression coefficient, the same bound holds for it. A non-linear link function raises certain problems if a linear embedding is applied.