The main topic of the graduation seminar is the treatment of conditional independence in the language of algebra. We find out how conditional independence statements constrain distributions and densities. We present ideals and varieties and some theorems and observations related to them. A link between ideals and algebraic varieties is established on the basis of the Nullstelensatz. The ideal of conditional independence can be used to better represent the restrictions of conditional independence. We are looking for the variety of the conditional independence ideal. The practical utility of a primary decomposition is shown, which allows ideals to be decomposed into primary ideals. We learn about binomial ideals, which are important for the primary decomposition in our case because of their properties. We also look at the case of primary decomposition for pairwise independent random variables by introducing a linear coordinate substitution.
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