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  "body": "\n## Summary\n\nAlgebraic statistics uses algebraic geometry and commutative algebra to study statistical models and procedures.\nMany models for discrete data are sets of probability vectors cut out by polynomial equations.\nExamples are independence models for [[contingency tables|PANSMHE0SW]], [[graphical models|Q3FQAD1BXG]] and log-linear models.\nTools such as [[polynomial ideals|096Z9MP2ME]], [[Gröbner bases|C83KPT47R0]] and [[algebraic varieties|HV6YBS0YDH]] then give information on identifiability, maximum likelihood estimation and exact tests.\nA well-known example is the use of Markov bases, introduced by Persi Diaconis and Bernd Sturmfels, to sample tables with fixed margins.\n\n## Explanations\n\n### L1 human\n\nToss two coins that may be biased and record the four outcomes HH, HT, TH, TT.\nIf the coins are independent, the probabilities satisfy $p_{HH} p_{TT} = p_{HT} p_{TH}$.\nSo the independent models form a curved surface defined by one polynomial equation inside the set of all probability vectors.\nAlgebraic statistics studies such surfaces with the tools of algebra and uses them to answer statistical questions.\n",
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