- Thesis:
- Master in Mathematics
- Author:
- Tom Schmid
- Title:
- Bayesian contraction and the Bernstein-von Mises theorem
- Supervisors:
- Jan JOHANNES
- Abstract:
- In this thesis we combine a Bayesian approach with a frequentist method to estimate the distribution of a random variable of interest and applying the Bernstein-von Mises theorem, after which we will investigate an application of a Bayesian approach in machine learning. We will start by studying different conjugate families and determining their contraction rates in models with one-dimensional parameters, before applying the Bernstein-von Mises theorem in the Binomial model and proving it in a normal model. Furthermore, we will continue to extend our theory to a model dependent on a multivariate parameter and investigate the conjugacy and contraction of that model before going one step further and do the same analysis in a model dependent on an infinite-dimensional parameter. We will end this thesis by examining a machine learning algorithm based on Bayes’ theory, testing it on two generated datasets, and discussing our results.
- Reference:
- S. Boucheron et E. Gassiat. A Bernstein-Von Mises Theorem for discrete probability distributions, Electronic Journal of Statistics 3:114-148, 2009
| Statistics Group | Institute for Mathematics | Faculty of Mathematics and Computer Science | University Heidelberg |