This chapter presents solutions to eight problems. Problems 6.1 and 6.2 are concerned with Markov bases and can be solved based on the material presented in Chapter 1 alone. Problem 6.3 is about a Gaussian graphical model and relies on Section 2.1. Problems 6.4 and 6.5 complement Section 2.3 by providing a detailed study of the asymptotics of the likelihood ratio test in a Gaussian model and a calculation of a Fisher-information matrix. Problems 6.6 and 6.7 illustrate the use of algebraic techniques to study implications among conditional independence relations as discussed in Chapter 3. Problem 6.8 concerns a mixture model (recall Section 4.1) and involves in particular questions about exact computation of Bayesian integrals (recall Section 5.2). For each problem we list the names of the team members who worked on this problem during the week in Oberwolfach. These teams submitted their solutions to Florian Block, Dustin Cartwright, Filip Cools and Alex Engström, who helped compile them into the current chapter.
KeywordsIrreducible Component Tangent Cone Basic Move Sample Covariance Matrix Primary Decomposition
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