Numerical Algebraic Geometry for Maximum Likelihood Estimation

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Connects maximum likelihood estimation to conormal and dual varieties and gives a concrete computational recipe: form likelihood equations, clear denominators, saturate by the singular locus, and solve with homotopy continuation from numerical algebraic geometry. For the 2×2×2 coin model the count is 13 complex critical points (real ones vary 3–9). A projective bijection via coordinate-wise product maps primal and dual critical points, enabling dual computations (example: hyperdeterminant model gives ML degree 71). Matrix examples: determinant vs permanent ML degrees are 10 vs 36 for 3×3.

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" chapters (14)

1. 0:00Intro
2. 2:11Let's play a game
3. 3:45Generating Data
4. 5:13A Parameterized Statistical Madel
5. 9:09Likelihood Function
6. 10:49Algebraic MLE
7. 14:10Likelihood Equations (cont.)
8. 15:23Three experiments
9. 17:29Dual Varieties
10. 19:07MLE and Conormal Varieties
11. 20:18A bijection between critical points
12. 21:15Using the Bijection
13. 22:04Motivations and 71
14. 24:51Determinants and Permanents