Algebraic statistics is a relatively new field full of exciting new developments. It combines questions and techniques from stochastics with those from (computational) algebraic geometry as well as convexity and polyhedral combinatorics.
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Sampler of problems: hypothesis testing for contingency tables, likelihood, MLE, sequence alignment? Sampler of models: independence ... conditional independence, hierarchical models, log-linear models, hidden variables?
Variety - ideal correspondence, prime ideals, primary decomposition, Groebner bases, examples from stats
|[CLO 1.4, 2.2, 2.3, 4.3-7]
[AHT Ch.3] [DSS], [ASCB]
|30.4.||Intro Convexity & Optimization|
Polytopes, occur as marginal polytopes of statistical models, convex hull pb, Legendre duality, faces
Fisher exact test, contingency table sampling, generalization to general Markov/Groebner/Graver-bases. Proof of fibers connected iff ideal generated
[AHT Ch.4], [J Ch.8], [dLHK]
Conditional statements, graphical models, parametrizations of graphical models, Markov properties, factorizations Hammersley-Clifford
|[L Ch 3.1, 3.2]
([DSS Ch.3] [J Ch.2,16])
... as log-linear models, marginal polytopes, trees, chordal graphs, junction tree, treewidth
[GMS], [L Ch.4]
|11.6.||Structure & Symmetry of Markov Bases||[AHT Ch.7 (Ch.5)]
No-3-way-interaction model, facet complexity, Markov complexity
|25.6.||Linear Concentration Models|
Gaussian distributions, concentration matrix, convex and algebraic geometry of MLE
|25.6.||Gaussian Graphical Models||[U] ([SU])||Kathlén|
|16.7.||Message Passing Algorithms|
Cumulant function & its dual, marginalisation
|[WJ, Ch. 2.5,2.6]||Yuanchen|
|[DSS]||Drton, Sturmfels and Sullivant
Lectures on Algebraic Statistics
Birkhäuser Basel. Series: Oberwolfach Seminars, Vol. 39 (2009)
|[PS]||Pachter and Sturmfels, eds.
Algebraic Statistics for Computational Biology
Cambridge University Press, New York (2005)
|[WJ]||Wainwright and Jordan (2008)
Graphical Models, Exponential Families, and Variational Inference
Foundations and Trends in Machine Learning Vol. 1, Nos. 1-2. 1-305 (2008)