The 4-Sample Theorem on Planar Graphs

Key Points

• To explore the relationship between the 4-Sample Theorem and the 4-Color Theorem in planar graphs.
• Utilized theoretical frameworks from graph theory and algebraic statistics.
• Examined conditions under which the maximum likelihood estimator exists for Gaussian graphical models in planar graphs.
• Analyzed implications of the 4-Sample Theorem on matrix completion and rigidity theory.
• Confirmed that at least four samples guarantee the maximum likelihood estimator with probability 1.
• Established connections between maximum likelihood thresholds and graph invariants.
• Highlighted the relevance of this threshold in addressing parameter estimation and matrix completion.

Abstract