Complete collineations for maximum likelihood estimation
We import the algebro-geometric notion of a complete collineation into the study of maximum likelihood estimation in directed Gaussian graphical models. A complete collineation produces a perturbation of sample data, which we call a stabilisation of the sample. While a maximum likelihood estimate (M...
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Zusammenfassung: | We import the algebro-geometric notion of a complete collineation into the
study of maximum likelihood estimation in directed Gaussian graphical models. A
complete collineation produces a perturbation of sample data, which we call a
stabilisation of the sample. While a maximum likelihood estimate (MLE) may not
exist or be unique given sample data, it is always unique given a
stabilisation. We relate the MLE given a stabilisation to the MLE given
original sample data, when one exists, providing necessary and sufficient
conditions for the MLE given a stabilisation to be one given the original
sample. For linear regression models, we show that the MLE given any
stabilisation is the minimal norm choice among the MLEs given an original
sample. We show that the MLE has a well-defined limit as the stabilisation of a
sample tends to the original sample, and that the limit is an MLE given the
original sample, when one exists. Finally, we study which MLEs given a sample
can arise as such limits. We reduce this to a question regarding the
non-emptiness of certain algebraic varieties. |
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DOI: | 10.48550/arxiv.2311.03329 |