One of the goals of algebraic statistics is to extend classical statistical theory for smooth statistical models to models that have singularities. Typically, these singularities arise in statistical models with hidden variables, in which a smooth model involving both observed and hidden variables is projected, via marginalization, to a model for the observed variables only. Hidden variable models are ubiquitous in statistical applications, but standard asymptotic theory usually does not apply because of model singularities. For example, we saw in Chapter 2 that the chi-square asymptotics for the likelihood ratio test are typically not valid at a singular point.
KeywordsMixture Model Hide Variable Factor Analysis Model Zariski Closure Complete Independence
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