This chapter is devoted to algebraic aspects of maximum likelihood estimation and likelihood ratio tests. Both of these statistical techniques rely on maximization of the likelihood function, which maps the parameters indexing the probability distributions in a statistical model to the likelihood of observing the data at hand. Algebra enters the playing field in two ways. First, computing maximum likelihood (ML) estimates often requires solving algebraic critical equations. Second, many models can be described as semi-algebraic subsets of the parameter space of a nice ambient model. In that setting, algebraic techniques are helpful for determining the behavior of statistical procedures such as the likelihood ratio test.
KeywordsTangent Cone Likelihood Ratio Statistic Multivariate Normal Distribution Likelihood Equation Junction Tree
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